As we have seen, Book 1 of Euclid’s Elements lists five Common Notions: axioms that are shared by geometry and several other branches of logic and mathematics. Over the past two millennia the authenticity of some or all of these axioms has been called into question. But for every mathematician who would strike out one or more of these Common Notions there is another who believes that Euclid’s text would be improved if their number were increased. In this article we will be taking a brief look at some of these additional axioms.
We have already learned that some of the manuscript sources for the Elements contain as many as nine common notions, so this practice of adding to the original five is not a recent phenomenon. In his 1814 edition of the Elements, the French mathematician François Peyrard lists nine Common Notions. The first three and the last two correspond to Euclid’s Five as found in modern editions. The remaining four are:
And if unequal things are added to equal things then the wholes are unequal.
And if from unequal things equal things are subtracted then the remainders are unequal.
And those things which are equal to twice the same thing are equal to each other.
And those things which are equal to half the same thing are equal to each other.
In some manuscripts of the Elements the third and fourth of these are quoted in the proofs of Propositions 1:37 and 1:47 (the Pythagorean Theorem)―which is probably the reason they were added to Euclid’s list of Common Notions (Fitzpatrick 37, 48).
In his critical notes at the end of Volume 1, Peyrard also records three more Common Notions that can be found in some manuscripts:
And all right angles are equal to each other.
And 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.
And two straight lines do not enclose a region.
Of these, the first two are Euclid’s Fourth and Fifth Postulates. The third is known as Euclid’s Sixth Postulate, as it was sometimes included among the Postulates rather than the Common Notions.
Christopher Clavius, a German Jesuit scholar of the late 16th century, is better known today as a geocentric astronomer, who resisted the Copernican Revolution and played a pivotal rôle in the introduction of the Gregorian Calendar. But he was also a mathematician and the author of a commentary on Euclid. Clavius took the liberty of adding several common notions to Euclid’s Five, bringing the total number to nineteen. The first twelve of these correspond to the twelve in Peyrard. The remaining seven are:
Two straight lines cannot have one and the same line segment in common.
If unequals are added to equals, the excess of the wholes will be equal to the excess of the things added.
If equals are added to unequals, the excess of the wholes will be equal to the excess of the things we started with.
If unequals are taken from equals, the excess of the remainders will be equal to the excess of the things removed.
If equals are taken from unequals, the excess of the remainders will be equal to the excess of the original things.
Every whole is equal to all its parts taken together.
If a whole is twice another whole, and we take away from the first twice what is taken away from the second, then the remainder of the first shall be twice the remainder of the second.
In his translation and commentary on the Elements the British mathematician Thomas Little Heath reviews the axioms that were introduced after Euclid’s time. Among those that we haven’t met already are six that he attributes to Pappus of Alexandria, who flourished in the 4th century of the Common Era:
All the parts of a plane, or of a straight line, coincide with one another.
A point divides a line, a line a surface, and a surface a solid.
A surface cuts a surface in a line.
If two surfaces which cut one another are plane, they cut one another in a straight line.
A line cuts a line in a point.
Magnitudes are susceptible of the infinite (or unlimited) both by way of addition and by way of successive diminution, but in both cases potentially only. This refers to the way in which a line (for example) can be extended indefinitely without limit, or can be made as small as one wishes, again without limit. Heath suggests that this axiom was prompted by Aristotle’s discussion of the infinite in the Physics 3:5-8.
Heath also includes a lengthy discussion of the Principle of Continuity, which he believes is necessary to make good some deficiencies in Euclid’s proofs. For example, in the very first Proposition in the Elements, Euclid assumes that the two circle in the diagram intersect:
It is a commonplace that Euclid has no right to assume, without premising some postulate, that the two circles will meet in a point C. To supply what is wanted we must invoke the Principle of Continuity (see note thereon above, p. 235). It is sufficient for the purpose of this proposition and of 1.21, where there is a similar tacit assumption, to use the form of postulate suggested by Killing. “If a line [in this case e.g. the circumference ACE] belongs entirely to a figure [in this case a plane] which is divided into two parts [namely the part enclosed within the circumference of the circle BCD and the part outside that circle], and if the line has at least one point common with each part, it must also meet the boundary between the parts [i.e. the circumference ACE must meet the circumference BCD].” (Heath 242)
Related to this is the Postulate of Dedekind, named for the German mathematician Richard Dedekind, who first enunciated this axiom in 1872:
If a segment of a straight line AB is divided into two parts so that (i) every point of the segment AB belongs to one of the parts, (2) the extremity A belongs to the first part and B to the second, and (3) any point whatever of the first part precedes any point whatever of the second part, in the order AB of the segment, there exists a point C of the segment AB (which may belong either to one part or to the other) such that every point of AB that precedes C belongs to the first part, and every point of AB that follows C belongs to the second part in the division originally assumed.
(If one of the two parts consists of the single point A or B, the point C is the said extremity A or B of the segment.)
This is the Postulate of Dedekind, which was enunciated by Dedekind himself in the following slightly different form (_Stetigkeit und irrationale Zahlen, 1872, new edition 1905, p. 11).
“If all points of a straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, there exists one and only one point which produces this division of all the points into two classes, this division of the straight line into two parts.”
The above enunciation may be said to correspond to the intuitive notion which we have that, if in a segment of a straight line two points start from the ends and describe the segment in opposite senses, they meet in a point. The point of meeting might be regarded as belonging to both parts, but for the present purpose we must regard it as belonging to one only and subtracted from the other part. (Heath 236)
In his Grundlagen der Geometrie [Foundations of Geometry] of 1899 the German mathematician David Hilbert set out to formally axiomatize geometry by substituting for Euclid’s Postulates and Common Notions a formal set of axioms. There are twenty or so of Hilbert’s Axioms―the precise number varies from edition to edition. The following list of twenty-one is taken from E J Townsend’s 1910 authorized translation of Grundlagen der Geometrie:
I1 Two distinct points A and B always completely determine a straight line a. [Euclid’s First Postulate]
I2 Any two distinct points of a straight line completely determine that line.
I3 Three points A, B, C not situated in the same straight line always completely determine a plane.
I4 Any three points A, B, C of a plane α which do not lie in the same straight line, completely determine that plane.
I5 If two points A, B of a straight line a lie in a plane α, then every point of a lies in α.
I6 If two planes α, β have a point A in common, then they have at least a second point B in common.
I7 Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane.
II1 If A, B, C are points of a straight line and B lies between A and C, then B lies also between C and A.
II2 If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D.
II3 Of any three points situated on a straight line, there is always one and only one which lies between the other two.
II4 Any four points A, B, C, D of a straight line can always be so arranged that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D. Hilbert discarded this axiom when the American mathematicians E H Moore and R L Moore independently proved in 1902 that it is redundant.
II5 Pasch’s Axiom: Let A, B, C be three points not lying in the same straight line and let a be a straight line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the straight line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.
III Playfair’s Axiom: In a plane a there can be drawn through any point A, lying outside of a straight line a, one and only one straight line which does not intersect the line a. This straight line is called the parallel to a through the given point A. [Euclid’s Fifth Postulate]
IV1 If A, B are two points on a straight line a, and if A´ is a point upon the same or another straight line a´, then, upon a given side of A´ on the straight line a´ , we can always find one and only one point B´ so that the segment AB (or BA) is congruent to the segment A´B´. We indicate this relation by writing AB≡A´B´. Every segment is congruent to itself.
IV2 If a segment AB is congruent to the segment A´B´ and also to the segment AʺBʺ, then the segment A´B´ is congruent to the segment AʺBʺ.
IV3 Let AB and BC be two segments of a straight line a which have no points in common aside from the point B, and, furthermore, let A´B´ and B´C´ be two segments of the same or of another straight line a´ having, likewise, no point other than B´ in common. Then, if AB≡A´B´ and BC≡B´C´, we have AC≡A´C´.
IV4 Let an angle (h, k) be given in the plane α and let a straight line a´ be given in a plane α´. Suppose also that, in the plane α´, a definite side of the straight line a´ be assigned. Denote by h´ a half -ray of the straight line a´ emanating from a point O´ of this line. Then in the plane α´ there is one and only one half ray k´ such that the angle (h, k), or (k, h), is congruent to the angle (h´, k´) and at the same time all interior points of the angle (h´, k´) lie upon the given side of a´. We express this relation by means of the notation ∠(h, k)≡∠(h´, k´). Every angle is congruent to itself; that is, ∠(h, k)≡∠(h, k) or ∠(h, k)≡∠(k, h).
IV5 If the angle (h, k) is congruent to the angle (h´, k´) and to the angle (hʺ, kʺ), then the angle (h´, k´) is congruent to the angle (hʺ, kʺ).
IV6 If, in the two triangles ABC and A´B´C´ , the following congruences hold: AB≡A´B´, AC≡A´C´, ∠BAC≡∠B´A´C´, then the congruences ∠ABC≡∠A´B´C´ and ∠ACB≡∠A´C´B´ also hold.
V1 Archimedes’ Axiom: Let A1 be any point upon a straight line between the arbitrarily chosen points A and B. Take the points A2, A3, A4 ... so that A1 lies between A and A2, A2 lies between A1 and A3, A3 lies between A2 and A4, etc. Moreover, let the segments AA1, A1A2, A2A3, A3A4, etc be equal to one another. Then, among this series of points there always exists a certain point An such that B lies between A and An
V2 Axiom of Completeness: To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid. This axiom was added by Hilbert to the French translation of Grundlagen der Geometrie.
Only Euclid’s First and Fifth Postulates are included among Hilbert’s Axioms. None of his Common Notions are explicitly included, though IV2 and IV5 are closely related to CN1. Hilbert was attempting to create a rigorous axiomatic foundation for Euclidean geometry, which required a much more sophisticated set of axioms than those found in the Elements.
Nevertheless, there are some curious omissions among Hilbert’s Axioms. Take, for example, Euclid’s “proof” of Proposition 1:1 on the construction of an equilateral triangle. Euclid assumes that the two circle intersect, but this is an unwarranted assumption:
The assumption that the circles do indeed cut one another should be counted as an additional postulate. (Fitzpatrick 8)
The Italian historian of philosophy and science Vincenzo De Risi has investigated this matter:
It has been well known, at least since the time of Pasch onward, that in the Elements there are no explicit principles governing the existence of the points of intersections between lines, so that in several propositions of Euclid the simple crossing of two lines (two circles, for instance) is regarded as the actual meeting of such lines, it being simply assumed that the point of their intersection exists. Such assumptions are labelled, today, as implicit claims about the continuity of the lines, or about the continuity of the underlying space. Euclid’s proofs, therefore, would seem to have some demonstrative gaps that need to be filled by a set of continuity axioms (as we do, in fact, find in modern axiomatizations). (De Risi 233)
De Risi, however, contests the claim that Euclid’s theory of intersections was based on some notion of continuity:
Had Euclid been asked to explain why the points of intersections of lines and circles should exist, it would have never occurred to him to mention continuity in this connection. Ancient geometry was very different from ours, and it is only our modern views on continuity that tend to give rise to the expectation that this latter must be included in the foundations of elementary mathematics. (De Risi 233)
This is a question we will discuss at greater length when we come to study Proposition 1:1. In the meantime De Risi suggests the following “fix”:
In modern terms, the simplest and most economical way of assuming that all the intersection points required in elementary geometry exist is to adopt one of the following (equivalent) geometrical axioms: the so-called Line-Circle intersection axiom, stating that
If a straight line has one point inside a circle, the line will meet the circle (ie the point of their intersection exists).
Or the Circle-Circle intersection axiom,stating that:
If one of two circles has one point inside the other and one point outside it, the two circles will meet.
These latter axioms may be immediately adduced in order to prove Elements I, 1 in a rigorous way and it can be easily proven that all the remaining propositions of Euclid’s book may also be demonstrated using these axioms―at least as far as the problem of intersections is concerned. (De Risi 236)
Why then does Hilbert not include either of these axioms? It might be argued that the reason is that he includes more general Axioms of Continuity (ie Archimedes’ Axiom and the Axiom of Completeness), which do the job:
This axiom makes possible the introduction into geometry of the idea of continuity. (Hilbert 24)
But Hilbert explicitly contradicts this neat solution in the Preface to Grundlagen der Geometrie:
Among the important results obtained, the following are worthy of special mention ...
4. The significance of several of the most important axioms and theorems in the development of the euclidean geometry is clearly shown; for example, it is shown that the whole of the euclidean geometry may be developed without the use of the axiom of continuity ... (Hilbert iii-iv)
De Risi does not explicitly solve this mystery. He does, however, mention that Dedekind’s Axiom is a sufficient axiom of continuity to satisfy all of euclidean geometry:
Dedekind’s axiom is sufficient to rigorously prove all the Euclidean propositions in the Elements (and much more), but it has been regarded as a threat to the purity of methods in geometry, as it requires a difficult roundabout path through number theory and analysis. (De Risi 236-237)
As De Risi points out in a footnote, Thomas Heath provides a proof of this statement in his translation of the Elements (Heath 237-240). Dedekind’s Axiom of Continuity is equivalent to Hilbert’s two Axioms of Continuity (V1 and V2).
Unfortunately, Hilbert never proves―or even mentions―Euclid’s Proposition 1:1 in Grundlagen der Geometrie, so I do not know how he would have proved it without using an Axiom of Continuity, such as Archimedes’ Axiom or Dedekind’s Axiom. After extensively researching the matter online, I came across only a few helpful remarks:
... the proof of Euclid’s first proposition ... requires finding a point where the two circles intersect. Euclid seems to assume that this is always possible on the basis of the diagram, but none of his postulates appear to require the circles to intersect. Hilbert’s axiomatization was meant to make it possible to eliminate all such unstated assumptions. In fact, Hilbert showed that there is a unique geometry that satisfies his axioms, so that any fact that is true in that geometry is a logical consequence of his axioms. However, a proof from Hilbert’s axioms may not look anything like Euclid’s proof of the same fact. For example, Hilbert’s axioms do not mention circles, so any proof of Euclid’s first proposition will have to be very different from Euclid’s proof. (Miller 9-10)
A key point of Hilbert’s foundations involved a distinction that was fully formulated only 20 years after his geometry. The distinction is between first and second order logic. First order sentences only quantify over individuals. Thus, Hilbert would rephrase Euclid’s first proposition, I.1, as: for any two points A, B there is a third point C, not on AB, so that AB, BC and CA are congruent (ABC is an equilateral triangle.). (Baldwin & Mueller 12)
As I understand things, then, Euclid’s proof of Proposition 1:1 requires something like an Axiom of Continuity or one of De Risi’s Axioms of Intersection to complete it, but without these it is still possible to prove the existence of equilateral triangles using only Hilbert’s first nineteen Axioms (categories I-IV)―but the proof will be quite different from Euclid’s.
And that’s a good place to stop.
