The Definitions and Axioms in Book I of Elements

Introduction

Elements by Euclid was an important work in the history of mathematics. Before the book was written, people’s mathematical knowledge was disorganised. The importance of Elements lies in the fact that it organised and arranged the various known mathematical knowledge in a logical matter, making it into a rigorous system. Of course, from modern view, Elements has its imperfections. However, its way of doing mathematics has been in use till now. The purpose of this paper is to analyse and discuss in detail the method adopted by Euclid and to investigate in depth the definitions and axioms in Book I of Elements. More importantly, we learn to understand and appreciate the work of ancient mathematicians.

1. Foreword

Geometry originated and developed from practical problems like the making of vessels, construction and surveying. During the Stone Age, men already knew to make stone tools of certain regular shapes. For instance, some stone tools of around five hundred thousand years of age found in Zhoukoudian Village in the southwest of Beijing; as well as some of tens of thousands of years of age in Dingcun Ruins, Xiangfenxian, Shanxi Province, China, were spherical in shape. As human civilization began to develop, we concern not only the shape of objects, but also their volumes. Thus we need to develop methods for calculating length, area and volume, from which some geometric formulae emerged. Those who know the mathematical culture in Babylon, ancient Egypt and China will realise that, unlike the ‘definition-theorem-proof’ style in modern textbooks, these formulae are always implicitly given by examples. The formulae were often obtained by experience or experiments and the people of the time did not seem to care to have a proof. Because of this, some formulae might give only an approximate answer. For instance, the Babylonians used A = c²/12 (where c is a circumference) as the formula for the area of a circle. From modern viewpoint, this is equivalent to taking 3 as an approximation to p.

It was the Greeks who made a comprehensive and deep study of geometry and developed it into an independent subject. Elements I written by Euclid (around 300 BC, Euclides or Eucleides in Latin, Eύĸλείδŋs in Greek) was the first piece of work which involved systematic discussions in geometry. Since the book was written, it was circulated in many different countries of the world, and had greater influence than any other work except possibly the Bible. During the next two millennia, thousands of people came to engage in scientific research via the logical training by Elements. In the foreword to his work Philosophiae Naturalis Principia Mathematica, Newton wrote, “It is the glory of geometry that from so few principles, fetched from without, it is able to accomplish so much.”

Even till today, the Elements is a great reference for all those who are engaged in the study of the history of mathematics and mathematics education.

2. The Axiomatic Approach by Euclid

Before the Elements was written, many Greek mathematicians had produced a lot of preliminary results. Most of these results, however, were based on experimental evidence without rigorous proofs. (Even in modern day, 2000 years after Euclid, many important mathematical results first appeared without a complete proof.) The axiomatic approach raised by Euclid was a way of proving propositions. It can not only prove the correctness of a proposition but can often yield more general results. For instance, the ancient Egyptians already knew that a triangle with side lengths 3, 4 and 5 is right-angled. The Greeks, on the other hand, actually proved that if the side lengths a, b, c of a triangle a²+b²=c² , then the triangle is right-angled. If we attempt to prove this conclusion by experiment, we will need infinitely many experiments, which is clearly impossible.

What is the axiomatic method? Try to imagine yourself that you want to convince someone that a proposition P₁ is true. A natural way is to give another proposition P₂ which is widely agreed, and try to argue that P₁ follows from P₂. If the truth of P₂ is in doubt, you can try to find another Proposition P₃, from which P₂ follows, and so on. When you have reached a proposition P_n that everyone agrees, you need not explain further.

If you cannot reach a proposition P_n which everyone agrees, you would reach an endless situation. The spirit of the axiomatic method is to avoid such situation, and to do so we set up the following two rules:

Rule 1

Some propositions, to be called “common notions” or “postulates”, are to be accepted without proof.

Rule 2

The meaning of “” was agreed, so that there is a convention for laws of deduction.

The greatest achievement of Euclid was that he carefully chose 10 postulates and common notions from which 465 propositions followed. Among these propositions, some (e.g. the Pythagoras’ Theorem) are highly non-trivial. It is so amazing that with only the 10 axioms, Euclid was able to explain many different geometrical phenomena and thereby reflecting his understanding of the world!

3. The 23 Definitions in Book I of Elements

When determining the validity of a proof, we need to consider, in addition to the two rules introduced above, the following.

Rule 3

The words, terms and symbols used in a proof are widely understood.

If a new term is used in a proof, or if a new meaning is attached to an existing term, it would be necessary to include, before giving the proof, a new definition. Basically, definitions can be given in any way one wishes, but it should not, in most cases, contradicts itself, other definitions or axioms. Of course, whether a definition is reasonable and useful is another matter, even if it will not give rise to any contradiction.

There are 23 definitions in the Book I of Elements. We extract some important ones from [3]:

1. A point is that which has no part.

2. A line is breadthless length.

3. The ends of a line² are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only.

6. The edges of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.

16. And the point is called the centre of the circle.

20. Of trilateral figures, an equilateral triangle is that which has its three sides equal; an isosceles triangle that which has two of its sides alone equal; and a scalene triangle that which has its three sides unequal.

23. Parallel straight lines are straight lines which, being on the same plane and being produced indefinitely³ in both directions, do not meet one another in either direction.

It is not possible to define every term we have used. To define a term, we must necessarily use other terms, which are to be defined using yet some other terms. If we do not allow the existence of some undefined terms, we will engage in the same endless situation as before. But Euclid did attempt to define every term in geometry: “a point is that which has no part”, “a line is breadthless length”, etc. These definitions cannot be used in proofs. Euclid should have realised this; perhaps all he wanted was to give an intuitive description of the geometrical objects under discussion.

Many later mathematicians axiomatised Euclidean geometry. Amongst them, the most prominent, straightforward and Euclid’s spirit retained is that from the 19-century later German mathematician, David Hilbert (1862 – 1943). In his great work Grundlagen der Geometrie (1899)⁴, he specified several undefined terms:

• point
• line
• plane
• point lies on a line
• point on a plane
• in between
• congruent⁵

With these undefined terms, we can define a line segment as the set of points in between two distinct points together with the two points; we can define what is meant by to say that two points lie on the same side of a straight line, etc. In Elements, the meaning of “two points are on the same side of a straight line” was not clearly defined, and it used, without explanation, the notion of “a point lying in between two points” in proofs (Euclid felt that this could be seen from the diagram). These were due to the fact that Euclid could not distinguish which notions need to be defined and which should be put as undefined terms⁶. Of course, we should not demand too much in Elements. As we mentioned before, the axiomatic approach was undoubtedly a great breakthrough. Without set theory at that time, it is unreasonable to expect that Euclid could give definitions like “a line segment is the set of points in between two distinct points together with the two points”.

In Definition 15, Euclid defined what a circle is. The phrase “two segments are equal” used by Euclid has, in modern day, two common (and equivalent) understandings. One means “congruent”; and some authors put “congruence of segments” as an undefined term (like “congruence of angles”). This understanding does not involve the concept of length. The other understanding, which should be Euclid’s original meaning, means “the lengths are the same”. The notion of length of line segments is intuitive and natural, and is the most important part of Euclidean geometry (as we mentioned in the Foreword, geometry originated and developed from practical problems like the making of vessels, construction and surveying, which all involved lengths and areas. At that time, Definition 15 was very clear. It clearly defined circles – an important member in Euclidean geometry (basically Euclidean geometry is to deal with straight lines, circles and figures formed from them). However, later mathematicians were not satisfied with such a definition. What is meant by length? The answer is given by a proposition in the Hilbert’s system:

Proposition: (length of line segment)

Given a line segment OI (called a unit segment), there exists a unique function , which assigns each segment AB a positive real number such that

(i) ;

(ii) if and only if AB and CD are congruent;

(iii) If a point B lies between A and C, then ;

(iv) if and only if there exists a point E lying between C and D such that AB and CE are congruent;

(v) There exists a segment AB corresponding to each positive real number l such that .

This theorem once again let us understand that all the results in Elements can be established upon a rigorous axiomatic system. It is only that Euclid was not able to do so in his time. Despite that many of the proofs in Elements depended on diagrams, one could, using the axiomatic system of Hilbert, rewrite the proofs of all propositions from the postulates and common notions without having to refer to diagrams nor without having to introduce additional assumptions which look obvious. Incidentally, it should be noted that, (i) to (v) in the proposition tell us to some extent that line segments (or straight lines) should be continuous. Axioms of continuity are also something that are in lack of in Elements, and we will come to discuss this in the next section.

According to Definition 20, an equilateral triangle is not regarded as an isosceles triangle. (Similarly, a square is not regarded as a rectangle in Elements.) This is in contrary to most modern textbooks. It is usually more convenient to regard an equilateral triangle as being isosceles, for in proving that a triangle is isosceles we need only prove that two sides are equal without having to consider the third side.

The definition of parallel lines as given in Definition 23 involves the notions of “indefinite extension” and “direction”. It is doubtful about the extent to which Euclid actually mastered these concepts. It was believed that the term “indefinite extension” appeared only in the process of translation. The lines (straight or curved) mentioned by Euclid are all finite in length, and the original definition for parallel lines should be “straight lines which do not meet no matter how we extend them”. (In the next section we will discuss the postulates in Elements, one of which says that a finite straight line can be produced indefinitely.) As regards the saying that a line segment can be produced in both directions, the notion of “direction” that Euclid made use of should depend mainly on intuition.

4. The 5 Postulates and 5 Common Notions in Book I of Elements

At the beginning of Book I of Elements, Euclid gave 23 definitions, followed by 5 postulates and 5 common notions. Euclid thought that the axioms are truths that are applicable in all sciences, while the postulates apply only in geometry. Nowadays we seldom make such distinction, and all those statements which are to be accepted without proof are known collectively as axioms. As we mentioned in Section 2, Euclid’s greatest achievements lies in his ingenious choice of the 10 axioms.

Postulates

1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and radius⁷.
4. That all right angles equal one another.
5. That, if a straight line falling on two straight lines makes 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.

Common Notions

1. Things which equal the same thing also equal one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things which coincide with one another equal one another.
5. The whole is greater than the part.

There have been widespread discussions and criticisms over the 10 axioms since they were written. In particular, the study of Postulate 5 led men to understand that Euclidean geometry is not the only geometry in the universe, and even led to the emergence of non-Euclidean geometries. Before we come to discuss the axioms, we should first point out that Euclid never assumed that there is no contradiction among the 10 axioms. Aristotle (384 – 322 BC) felt that we could define things that are contradicting each other, and the truthfulness of axioms could be verified by comparing the results that they derive with the real world.

Postulate 1 does not point out the uniqueness of the straight line passing through the two points, but Euclid made use of this in Proposition 4. The proposition is as follows.

Proposition 4 in the book I:

If two triangles have two equal corresponding sides, and the angles between them are also equal, then their bases are equal and the two triangles are congruent. Furthermore, their remained two angles are equal indeed.

The proof in Elements is roughly as follows. Let ABC and DEF be two triangles with AB = DE , AC = DF and ÐBAC = ÐDEF. “Move” triangle ABC to triangle DEF so that A, B, C coincide with D, E, F respectively. Let’s not ask what it means by to ‘move’ (of course Euclid was not able to explain this), but then Euclid said that from this we know BC and EF coincide. He said that for otherwise, the two segments would enclose an area, which is not possible. In the proof, Euclid made use of something which is not mentioned in Postulate 1, namely, the uniqueness of the straight line (or line segment) passing through two given points.

There was a similar defect in Postulate 2 as in Postulate 1. The uniqueness of the extension of the line segment was not mentioned. It would be better if this has been explicitly stated. Since all lines considered by the Greeks at that time were all finite in length, sometimes two line segments meet only after extension. This postulate ensures the possibility of extending line segments.

Postulate 3 guarantees the existence of a circles. (In fact, as in Postulates 1 and 2, Euclid emphasized ‘construction’ more than “existence”.) Proposition 1 of Book I already makes use of this postulate. From a modern mathematical viewpoint, a circle is a set of points equidistant from a fixed point (the centre). So the existence of a circle can be guaranteed by the axioms of set theory. Therefore, Hilbert did not put this postulate into his axiom system.

Postulate 4 says “that all right angles equal one another”. It is not easy, at first sight, to grasp what Euclid’s attempt was. Intuitively, a right angle can be transformed onto another right angle by translation and rotation. Postulate 4 guarantees that the perpendicularity is preserved under such translations and rotations. More generally, translations and rotations do not alter the angle between two straight lines. This fact was used implicitly in the proof of Proposition 4 of Book I that we have seen earlier. Actually it should be Proposition 4 (the SAS condition for congruent triangles) that ensures the invariance in the structure of geometrical objects under translations and rotations. Hilbert perceived this point, and so he put the SAS condition for congruence⁸ is placed into his table of axioms.

Among the 10 axioms in Book I of Elements, Postulate 5 aroused the most controversies. The other 9 axioms are all intuitive and brief. Postulate 5 looks totally different. We have reason to believe that Euclid himself did not like this postulate, as can be shown in the following:

	First proposition that invokes the postulate/axiom
Postulate 1	Statement 1
Postulate 2	Statement 2
Postulate 3	Statement 1
Postulate 4	Statement 14
Postulate 5	Statement 29
Common Notion 1	Statement 1
Common Notion 2	Statement 13
Common Notion 3	Statement 2
Common Notion 4	Statement 4
Common Notion 5	Statement 16

From the above table, we can see that Postulate 5 was first used much later than the other postulates and axioms. It was not used until Proposition 29. Propositions 1 to 28 consist of important theorems and constructions whose proofs are far more complicated than Proposition 29. That the first 28 propositions need not make use of Postulate 5 is not because they are simpler, but rather because Euclid deliberately put all those propositions which do not need Postulate 5 at the beginning, until he finally needed to invoke Postulate 5 in Proposition 29. Let’s look at this important proposition.

Proposition 29 in the Book I of Elements

If a straight line goes through a pair of parallel lines, then there are equal alternate angles and equal corresponding angles. Also the sum of the interior angles on the same side equals two right angles.

In the figure above, AB // CD and the straight line EF intersects AB, CD at G, H respectively. The proof of Proposition 29 in Elements as follows. If the alternate angles are unequal, then the sum of the interior angles on one of two sides must be less than two right angles. By Postulate 5, AB and CD meet by extending them indefinitely. This contradicts the fact that AB // CD.

Proposition 29 gives the properties of parallel lines that we are familiar with. Together with Propositions 27 and 28, we have the necessary and sufficient conditions for two lines to be parallel, from which many important results and construction problems can be proved. One famous example is that the sum of the interior angles of a triangle is equal to two right angles. Although Postulate 5 is rarely used in Elements, but it is worth to include it in the list of axioms only to prove this important Proposition 29. According to historians, most of the results in Elements were known before Euclid, but Postulate 5 was due to Euclid himself. This clearly shows the genius of Euclid, who realised the importance of this postulate.

Since Postulate 5 looks so strange, the attempt to prove this postulate using the other 4 postulates and the 5 axioms had begun as early as Elements appeared. Many people believed that Postulate 5 can be proved, yet no one ever succeeded after 2000 years. As for many other famous problems, many people had claimed that they had solved it, but after all it was found that all such proofs had implicitly used facts that cannot be proved without referring to Postulate 5. With more and more study on the postulate, people began to doubt whether it can really be proved. Throughout these 2000 years, mathematics had developed so much that analytic geometry, calculus, differential equations and other branches of mathematics had gradually emerged; many first-class mathematicians had made significant contribution in the mathematical world; and yet no one could prove Postulate 5. In 1759, the French mathematician Jean Le Rond d’Alembert (1717 – 1783) said that Postulate 5 was “a disgrace to the family of geometry”.

The famous mathematician Adrien Marie Legendre (1752 – 1833) is among the many who were fascinated by Postulate 5. He spent 29 years of study before he attempted to publish a proof of Postulate 5. Before we discuss his proof, let us first point out that in the process of studying Postulate 5, mathematicians had known many statements that are equivalent to Postulate 5, one of which is the parallel postulate.

Parallel Postulate

Given any line L and a point P not on L, there is at most one straight line lying on the same plane which is parallel to L and passes through P.

　

The parallel postulate only says that there is at most one straight line parallel to L and passing through P, but does not mention the existence of such a straight line. This is because that the existence can be proved, and so it is not necessary to put it into the axiom. In modern textbooks, Postulate 5 of Euclid is usually replaced by the parallel postulate. One of the reasons is that the parallel postulate avoided the notion of “one side of a straight line” in Postulate 5. Although this notion can be rigorously defined, but as an axiom, it would better be as concise as possible.

Legendre’s attempt to prove the parallel postulate

Let a point P lie not on a straight line L₁. Draw a perpendicular from P to L₁ meeting L₁ at Q, and draw a straight line L₂ passing through P perpendicular to PQ. Then we have L₁ // L₂. Suppose an arbitrary straight line L₃ passing through P which is different from PQ and L₂, we want to prove that L₁ and L₃ intersect.

Choose a point R (different from P) on L₃ such that the ray PR is between the ray PQ and the ray lying on L₂ with endpoint P. On the other side of PQ, take a point R‘ such that ÐQPR‘ = ÐQPR. Notice that the point Q lies in the interior of the ÐRPR‘ and L₁ passes through Q, thus L₁ must intersect one of the arms of ÐRPR‘. If L₁ and PR intersect, then L₁ and L₃ also intersect. Now suppose L₁ and PR‘ intersect at A, and take a point B lying on PR such that PB = PA. Then (SAS). It follows that ÐPQB is a right angle. Furthermore, B lies on L₁. This proves that L₁ and L₃ intersect.

Is the proof correct? To answer this question, we must carefully examine every step in the proof. First of all, we must clearly define every term used in the proof, such as “perpendicular”, “interior of an angle”, “one side of a straight line”, “congruence of triangles”, etc. Secondly, we need to explain why L₁ and L₂ are parallel to each given that as they are both perpendicular to PQ (note that we cannot use Postulate 5 here); why point R‘ exists; why Q lies in the interior of the angle; why if L₁ crosses the interior of ÐRPR‘ then it has to intersect with PR or PR‘. One can imagine how difficult it is to prove the parallel postulate. We do not intend to discuss here the gap in Legendre’s proof, but rather to point out some of the propositions that have been used in the proof which cannot be proved using axioms other than Postulate 5.

A turning point finally came after a long time. Mathematicians were amazed by the development of non-Euclidean geometries. In order to prove the parallel postulate, people first assumed the negation of the parallel postulate, and then tried to get a contradiction using the other axioms. Such attempts resulted in repeated failure, and a new geometry that is entirely different from Euclidean geometry developed at the same time. If these new geometries do not have contradictions in themselves, we can prove that the parallel postulate (or Postulate 5) is independent of the other axioms. Through many years of experience, people believe that the axioms of Euclidean geometry will not result in contradictory propositions. But how can we be convinced that these new geometries do not contain contradictions either? In 1898, Poincaré suggested a new viewpoint. He thought that the consistency of an axiomatic system could be tested by assigning to its objects some arithmetical meanings. This is because if the system contains contradictions, the same contradictions would appear in the arithmetical meanings attached. Hilbert completed this job of ‘transferring contradictions’ by assigning arithmetical meanings to non-Euclidean geometries. Since human experience led men to believe that arithmetic is consistent, we should have the same belief in non-Euclidean geometries. This answer is already very satisfactory.

As for the five axioms in Book I of Elements, some have been included in Hilbert’s axioms after modification, while some have been removed. We shall not go further into this matter, and interested readers could refer to the Appendix or the suggested references.

Finally, it must be mentioned that insufficiencies in the axioms of Elements forced Euclid to use implicitly facts that look intuitively obvious, especially those concerning the order of points on straight lines and the continuity of straight lines and circles. For instance, in the proof of Proposition 1 of Book I of Elements, Euclid assumed the fact that two circles centred at two distinct points A, B with radius AB must intersect. It is equivalent to assuming that a circle is continuous.

5. Conclusion

Euclid’s Elements is one of the most influential works in the history of mathematics. Although most of the results were known before Euclid, but the axiomatic approach it adopted had been in use till now. The 10 axioms which had been ingeniously chosen fully illustrated the genius and excellent judgment of Euclid. In particular, the introduction of the Fifth Postulate attracted a large number of first-class mathematicians to attempt to prove it. This led to the emergence of non-Euclidean geometries and also a deeper understanding of Euclidean geometry.

Although there are defects in the axiomatic system and the proofs of Elements, it made the first step towards the axiomatisation of geometry. The road from Elements to Grundlagen der Geometrie is a supplement of insufficiencies from a logical viewpoint, but is indeed a great advancement in the abstract thinking of man.

6. References

[1] Marvin Jay Greenberg, Euclidean and non-Euclidean Geometries: development and history, 3rd Edition, W. H. Freeman and Company. New York, 1994.

[2] Benno Artmann, Euclid – the creation of mathematics, Springer-Verlag New York, Inc., 1999.

[3] Euclid, translated by Lan Chicheng, Chu Enkuan, The Thirteen Books of Euclid’s Elements, Chiu Chang Publishing Company, 1996.

[4] Morris Kline, translated by Mathematical History Translation Group of Peking University, Mathematical thought from Ancient to Modern Times, Shanghai Science and Technology Publishing Company, 2002.

[5] Wang Huaichuan, The Hometown of Mathematics, Wang Huaichuan, 1997.

[6] Chiang Sheng, Euclid’s Fifth Postulate, Chiu Chang Publishing Company, 1993.

7. Appendix: The Table of Axioms from Grindlagen der Geometrie, 7th Edition, David Hilbert

The following table of axioms is highlighted from [6].

Hilbert specified the following as undefined concepts:

• point
• line
• plane
• point lies on a line
• point on a plane
• in between
• congruence of angles⁵

Hilbert’s axioms are divided into 5 groups with a total of 20 axioms.

I. Axiom of Connection

1. For two arbitrary points A, B, there exists a straight line a such that A, B lie on a.

2. For two arbitrary points A, B, there is at most one straight line a such that A, B lie on a.

3. Each straight line contains at least two points. At least three points are not collinear.

4. For three non-collinear points A, B, C, there is at least one plane a such that A, B, C lies on a. Each plane at least contains one point.

5. For three non-collinear points A, B, C, there is at most one plane a such that A, B, C lies on a.

6. If two points of a straight line lie on a plane a, then every points of the straight line lie on a.

7. If A is a coplanar point of two planes a, b, then there is at least one point B lying on a, b.

8. There are at least four points non-coplanar.

II. Axiom of Betweenness

1. If a point B lies between two points A, C, then A, B, C are distinct points lying on the same straight line and B also lies between C, A.

2. For two arbitrary points A, B, there is at least one point C lying on the straight line AB such that B lies between A, C.

3. For three arbitrary points, there is at most one of them lying between the remained two points.

   Definition

   Let two points A and B lie on a straight line a. Non-ordinal A, B or B, A are called segment AB. Point in AB is called an interior point of AB, A and B are called endpoints, other point on a (not an interior point, an endpoint) is called an exterior point.

4. (Pasch’s Axiom) Let three collinear points A, B, C lie on a plane a, there is a straight line a lying on a on which A, B, C do not lie. If a passes through one point of segment AB, then a also passes through one point of segment AC or BC.

III. Axiom of Congruence

1. Two points A, B lie on a straight line a. If a point A‘ lies on a (or another straight line a‘), then there is an another point B‘ on a (or a‘) such that the two segments AB and A‘B‘ are identical, written as AB = A‘B‘.

2. If AB = A’B’ and AB = A”B”, then A’B’ = A”B”.

3. Suppose that segments AB, BC lying on a straight line have no common interior point, and segments A‘B‘, B‘C‘ lying on a straight line a‘ have no common interior point either. If AB = A‘B‘ and BC = B‘C‘, then AC = A‘C‘.

4. Given an angle Ð(h, k) on the plane a, a straight line a’ on a plane a‘, specifying one side of a‘ and h‘ is a ray starting from a point O‘ on a‘. Thus there is a ray k‘ specified side of a’ lying on a‘ such that Ð(h, k) = Ð(h‘, k‘). Each angle is identical to itself.

5. If two triangles ABC and A‘B‘C‘ are satisfied with AB = A‘B‘, AC = A‘C‘ and ÐCAB = ÐC‘A‘B‘, then there must be ÐABC = ÐA‘B‘C‘.

IV. Axiom of Parallels

Given a line a and a point A not on a, there exists at most one straight line lying on the same plane which passes through A and is parallel to a.

V. Axiom of Continuity

1. (Archimedes’s Axiom) Given two arbitrary segments AB and CD, there are finite points A₁, A₂, …, A_n on a ray starting from A through B such that segments AA₁, A₁A₂, …, A_n-1A_n are identical to CD. Also B lies between A and A_n.

2. Suppose points on a straight line is satisfied with axioms I(1), I(2), II, III(1) and V(1), then it is impossible to expand to be a larger set similarly satisfied with these axioms.