Commentary on Book I Common Notions¶
Back to Book I Common Notions.
AXIOMS OR COMMON NOTIONS.¶
In a paper Sur l’authenticité des axiomes d’Euclide in the Bulletin des sciences math. et astron. 1884, p. 162 sq. (Mémoires scientifiques, II., pp. 48—63), Paul Tannery maintained that the Common Notions (including the first three) were not in Euclid’s work but were interpolated later. The following are his main arguments. (1) If Euclid had set about distinguishing between indemonstrable principles (a) common to all demonstrative sciences and (b) peculiar to geometry, he would, says Tannery, certainly not have placed the common principles second and the special principles (the Postulates) first. (2) If the Common Notions are Euclid’s, this designation of them must be his too; for he must have used some name to distinguish them from the Postulates and, if he had used another name, such as Axioms, it is impossible to imagine why that name was changed afterwards for a less suitable one. The word ἔννοια (notion), says Tannery, never signified a notion in the sense of a proposition, but a notion of some object; nor is it found in any technical sense in Plato and Aristotle. (3) Tannery’s own view was that the formulation of the Common Notions dates from the time of Apollonius, and that it was inspired by his work relating to the Elements (we know from Proclus that Apollonius tried to prove the Common Notions). This idea, Tannery thought, was confirmed by a “fortunate coincidence” furnished by the occurrence of the word ἔννοια (notion) in a quotation by Proclus (p. 100, 6): “we shall agree with Apollonius when he says that we have a notion (ἔννοιαν) of a line when we order the lengths, only, of roads or walls to be measured.”
In reply to argument (1) that it is an unnatural order to place the purely geometrical Postulates first, and the Common Notions, which are not peculiar to geometry, last, it may be pointed out that it would surely have been a still more awkward arrangement to give the Definitions first and then to separate from them, by the interposition of the Common Notions, the Postulates, which are so closely connected with the Definitions in that they proceed to postulate the existence of certain of the things defined, namely straight lines and circles.
Though it is true that ἔννοια in Plato and Aristotle is generally a notion of an object, not of a fact or proposition, there are instances in Aristotle where it does mean a notion of a fact: thus in the Eth. Nic. IX. 11, 1171 a 32 he speaks of “the notion (or consciousness) that friends sympathise” (ἡ ἔννοια τοῦ συναλγεῖν τοὺς φίλους) and again, b 14, of “the notion (or consciousness) that they are pleased at his good fortune.” It is true that Plato and Aristotle do not use the word in a technical sense; but neither was there apparently in Aristotle’s time any fixed technical term for what we call “axioms,” since he speaks of them variously as “the so-called axioms in mathematics,” “the so-called common axioms,” “the common (things)” (τὰ κοινά), and even “the common opinions” (κοιναὶ δόξαι). I see therefore no reason why Euclid should not himself have given a technical sense to “Common Notions,” which is at least a distinct improvement upon “common opinions.”
The use of ἔννοια in Proclus’ quotation from Apollonius seems to me to be an unfortunate, rather than a fortunate, coincidence from Tannery’s point of view, for it is there used precisely in the old sense of the notion of an object (in that case a line).
No doubt it is difficult to feel certain that Euclid did himself use the term Common Notions, seeing that Proclus’ commentary generally speaks of Axioms. But even Proclus (p. 194, 8), after explaining the meaning of the word “axiom,” first as used by the Stoics, and secondly as used by “Aristotle and the geometers,” goes on to say: “For in their view (that of Aristotle and the geometers) axiom and common notion are the same thing.” This, as it seems to me, may be a sort of apology for using the word “axiom” exclusively in what has gone before, as if Proclus had suddenly bethought himself that he had described both Aristotle and the geometers as using the one term “axiom,” whereas he should have said that Aristotle spoke of “axioms,” while “the geometers” (in fact Euclid), though meaning the same thing, called them Common Notions. It may be for a like reason that in another passage (p. 76, 16), after quoting Aristotle’s view of an “axiom,” as distinct from a postulate and a hypothesis, he proceeds: “For it is not by virtue of a common notion that, without being taught, we preconceive the circle to be such and such a figure.” If this view of the two passages just quoted is correct, it would strengthen rather than weaken the case for the genuineness of Common Notions as the Euclidean term.
Again, it is clear from Aristotle’s allusions to the “common axioms in mathematics” that more than one axiom of this kind had a place in the text-books of his day; and as he constantly quotes the particular axiom that, if equals be taken from equals, the remainders are equal, which is Euclid’s Common Notion 3, it would seem that at least the first three Common Notions were adopted by Euclid from earlier text-books. It is, besides, scarcely credible that, if the Common Notions which Apollonius tried to prove had not been introduced earlier (e.g. by Euclid), they would then have been interpolated as axioms and not as propositions to be proved. The line taken by Apollonius is much better explained on the assumption that he was directly attacking axioms which he found already admitted into the Elements.
Proclus, who recognised the five Common Notions given in the text, warns us, not only against the error of unnecessarily multiplying the axioms, but against the contrary error of reducing their number unduly (p. 196, 15), “as Heron does in enunciating three only; for it is also an axiom that the whole is greater than the part, and indeed the geometer employs this in many places for his demonstrations, and again that things which coincide are equal.”
Thus Heron recognised the first three of the Common Notions; and this fact, together with Aristotle’s allusions to “common axioms” (in the plural), and in particular to our Common Notion 3, may satisfy us that at least the first three Common Notions were contained in the Elements as they left Euclid’s hands.
COMMON NOTION 1.¶
Τὰ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα.
Things which are equal to the same thing are also equal to one another.
Aristotle throughout emphasises the fact that axioms are self-evident truths, which it is impossible to demonstrate. If, he says, any one should attempt to prove them, it could only be through ignorance. Aristotle therefore would undoubtedly have agreed in Proclus’ strictures on Apollonius for attempting to prove the axioms. Proclus gives (p. 194, 25), as a specimen of these attempted proofs by Apollonius, that of the first of the Common Notions. “Let A be equal to B, and the latter to C; I say that A is also equal to C. For, since A is equal to B, it occupies the same space with it; and since B is equal to C, it occupies the same space with it.
Therefore A also occupies the same space with C.”
Proclus rightly remarks (p. 194, 22) that “the middle term is no more intelligible (better known, γνωριμώτερον) than the conclusion, if it is not actually more disputable.” Again (p. 195, 6), the proof assumes two things, (1) that things which “occupy the same space” (τόπος) are equal to one another, and (2) that things which occupy the same space with one and the same thing occupy the same space with one another; which is to explain the obvious by something much more obscure, for space is an entity more unknown to us than the things which exist in space.
Aristotle would also have objected to the proof that it is partial and not general (καθόλου), since it refers only to things which can be supposed to occupy a space (or take up room), whereas the axiom is, as Proclus says (p. 196, 1), true of numbers, speeds, and periods of time as well, though of course each science uses axioms in relation to its own subject-matter only.
COMMON NOTIONS 2, 3.¶
Καὶ ἐὰν ἴσοις ἴσα προστεθῇ, τὰ ὅλα ἐστὶν ἴσα.
Καὶ ἐὰν ἀπὸ ἴσων ἴσα ἀφαιρεθῇ, τὰ καταλειπόμενά ἐστιν ἴσα.
If equals be added to equals, the wholes are equal.
If equals be subtracted from equals, the remainders are equal.
These two Common Notions are recognised by Heron and Proclus as genuine. The latter is the axiom which is so favourite an illustration with Aristotle.
Following them in the MSS. and editions there came four others of the same type as 1—3. Three of these are given by Heiberg in brackets; the fourth he omits altogether.
The three are:
(a) If equals be added to unequals, the wholes are unequal. (b) Things which are double of the same thing are equal to one another. (c) Things which are halves of the same thing are equal to one another.
The fourth, which was placed between (a) and (b), was:
(d) If equals be subtracted from unequals, the remainders are unequal.
Proclus, in observing that axioms ought not to be multiplied, indicates that all should be rejected which follow from the five admitted by him and appearing in the text above (p. 155). He mentions the second of those just quoted (b) as one of those to be excluded, since it follows from Common Notion 1. Proclus does not mention (a), (c) or (d); an-Nairīzī gives (a), (d), (b) and (c), in that order, as Euclid’s, adding a note of Simplicius that “three axioms (sententiae acceptae) only are extant in the ancient manuscripts, but the number was increased in the more recent.”
(a) stands self-condemned because “unequal” tells us nothing. It is easy to see what is wanted if we refer to I. 17, where the same angle is added to a greater and a less, and it is inferred that the first sum is greater than the second. So far however as the wording of (a) is concerned, the addition of equal to greater and less might be supposed to produce less and greater respectively. If therefore such an axiom were given at all, it should be divided into two. Heiberg conjectures that this axiom may have been taken from the commentary of Pappus, who had the axiom about equals added to unequals quoted below (e); if so, it can only be an unskilful adaptation of some remark of Pappus, for his axiom (e) has some point, whereas (a) is useless.
As regards (b), I agree with Tannery in seeing no sufficient reason why, if we reject it (as we certainly must), the words in I. 47 “But things which are double of equals are equal to one another” should be condemned as an interpolation. If they were interpolated, we should have expected to find the same interpolation in I. 42, where the axiom is tacitly assumed. I think it quite possible that Euclid may have inserted such words in one case and left them out in another, without necessarily implying either that he was quoting a formal Common Notion of his own or that he had not included among his Common Notions the particular fact stated as obvious.
The corresponding axiom (c) about the halves of equals can hardly be genuine if (b) is not, and Proclus does not mention it. Tannery acutely observes however that, when Heiberg, in I. 37, 38, brackets words stating that “the halves of equal things are equal to one another” on the ground that axiom (c) was interpolated (although before Theon’s time), and explains that Euclid used Common Notion 3 in making his inference, he is clearly mistaken. For, while axiom (b) is an obvious inference from Common Notion 2, axiom (c) is not an inference from Common Notion 3. Tannery says, in a note, that (c) would have to be established by reductio ad absurdum with the help of axiom (b), that is to say, of Common Notion 2. But, as the hypothesis in the reductio ad absurdum would be that one of the halves is greater than the other, and it would therefore be necessary to prove that the one whole is greater than the other, while axiom (b) or Common Notion 2 only refers to equals, a little argument would be necessary in addition to the reference to Common Notion 2. I think Euclid would not have gone through this process in order to prove (c), but would have assumed it as equally obvious with (b).
Proclus (pp. 197, 6—198, 5) definitely rejects two other axioms of the above kind given by Pappus, observing that, as they follow from the genuine axioms, they are rightly omitted in most copies, although Pappus said that they were “on record” with the others (συναναγράφεσθαι):
(e) If unequals be added to equals, the difference between the wholes is equal to the difference between the added parts; and (f) If equals be added to unequals, the difference between the wholes is equal to the difference between the original unequals.
Proclus and Simplicius (in an-Nairīzī) give proofs of both. The proof of the former, as given by Simplicius, is as follows:
Let AB, CD be equal magnitudes; and let EB, FD be added to them respectively, EB being greater than FD.
I say that AE exceeds CF by the same difference as that by which BE exceeds DF.
Cut off from BE the magnitude BG equal to DF.
Then, since AE exceeds AG by GE, and AG is equal to CF and BG to DF,
AE exceeds CF by the same difference as that by which BE exceeds DF.
COMMON NOTION 4.¶
Καὶ τὰ ἐφαρμόζοντα ἐπ᾽ ἄλληλα ἴσα ἀλλήλοις ἐστίν.
Things which coincide with one another are equal to one another.
The word ἐφαρμόζειν, as a geometrical term, has a different meaning according as it is used in the active or in the passive. In the passive, ἐφαρμόζεσθαι, it means “to be applied to” without any implication that the applied figure will exactly fit, or coincide with, the figure to which it is applied; on the other hand the active ἐφαρμόζειν is used intransitively and means “to fit exactly,” “to coincide with.” In Euclid and Archimedes ἐφαρμόζειν is constructed with ἐπί and the accusative, in Pappus with the dative.
On Common Notion 4 Tannery observes that it is incontestably geometrical in character, and should therefore have been excluded from the Common Notions; again, it is difficult to see why it is not accompanied by its converse, at all events for straight lines (and, it might be added, angles also), which Euclid makes use of in I. 4. As it is, says Tannery, we have here a definition of geometrical equality more or less sufficient, but not a real axiom.
It is true that Proclus seems to recognise this Common Notion and the next as proper axioms in the passage (p. 196, 15—21) where he says that we should not cut down the axioms to the minimum, as Heron does in giving only three axioms; but the statement seems to rest, not upon authority, but upon an assumption that Euclid would state explicitly at the beginning all axioms subsequently used and not reducible to others unquestionably included. Now in I. 4 this Common Notion is not quoted; it is simply inferred that “the base $BC$ will coincide with $EF$, and will be equal to it.” The position is therefore the same as it is in regard to the statement in the same proposition that, “if… the base $BC$ does not coincide with $EF$, two straight lines will enclose a space: which is impossible”; and, if we do not admit that Euclid had the axiom that “two straight lines cannot enclose a space,” neither need we infer that he had Common Notion 4. I am therefore inclined to think that the latter is more likely than not to be an interpolation.
It seems clear that the Common Notion, as here formulated, is intended to assert that superposition is a legitimate way of proving the equality of two figures which have the necessary parts respectively equal, or, in other words, to serve as an axiom of congruence.
The phraseology of the propositions, e.g. I. 4 and I. 8, in which Euclid employs the method indicated, leaves no room for doubt that he regarded one figure as actually moved and placed upon the other. Thus in I. 4 he says, “The triangle $ABC$ being applied (ἐφαρμοζομένου) to the triangle $DEF$, and the point $A$ being placed (τιθεμένου) upon the point $D$, and the straight line $AB$ on $DE$, the point $B$ will also coincide with $E$ because $AB$ is equal to $DE$”; and in I. 8, “If the sides $BA$, $AC$ do not coincide with $ED$, $DF$, but fall beside them (take a different position, παραλλάξουσιν), then” etc. At the same time, it is clear that Euclid disliked the method and avoided it wherever he could, e.g. in I. 26, where he proves the equality of two triangles which have two angles respectively equal to two angles and one side of the one equal to the corresponding side of the other. It looks as though he found the method handed down by tradition (we can hardly suppose that, if Thales proved that the diameter of a circle divides it into two equal parts, he would do so by any other method than that of superposition), and followed it, in the few cases where he does so, only because he had not been able to see his way to a satisfactory substitute. But seeing how much of the Elements depends on I. 4, directly or indirectly, the method can hardly be regarded as being, in Euclid, of only subordinate importance; on the contrary, it is fundamental. Nor, as a matter of fact, do we find in the ancient geometers any expression of doubt as to the legitimacy of the method. Archimedes uses it to prove that any spheroidal figure cut by a plane through the centre is divided into two equal parts in respect of both its surface and its volume; he also postulates in Equilibrium of Planes I. that “when equal and similar plane figures coincide if applied to one another, their centres of gravity coincide also.”
Killing (Einführung in die Grundlagen der Geometrie, II. pp. 4, 5)
contrasts the attitude of the Greek geometers with that of the philosophers, who, he says, appear to have agreed in banishing motion from geometry altogether. In support of this he refers to the view frequently expressed by Aristotle that mathematics has to do with immovable objects (ἀκίνητα), and that only where astronomy is admitted as part of mathematical science is motion mentioned as a subject for mathematics. Cf. Metaph. 989 b 32 “For mathematical objects are among things which exist apart from motion, except such as relate to astronomy”; Metaph. 1064 a 30 “Physics deals with things which have in themselves the principle of motion; mathematics is a theoretical science and one concerned with things which are stationary (μένοντα) but not separable” (sc. from matter); in Physics II. 2, 193 b 34 he speaks of the subjects of mathematics as “in thought separable from motion.”
But I doubt whether in Aristotle’s use of the words “immovable,” “without motion” etc. as applied to the subjects of mathematics there is any implication such as Killing supposes. We arrive at mathematical concepts by abstraction from material objects; and just as we, in thought, eliminate the matter, so according to Aristotle we eliminate the attributes of matter as such, e.g. qualitative change and motion. It does not appear to me that the use of “immovable” in the passages referred to means more than this. I do not think that Aristotle would have regarded it as illegitimate to move a geometrical figure from one position to another; and I infer this from a passage in De caelo III. 1 where he is criticising “those who make up every body that has an origin by putting together planes, and resolve it again into planes.” The reference must be to the Timaeus (54 B sqq.) where Plato evolves the four elements in this way. He begins with a right-angled triangle in which the hypotenuse is double of the smaller side; six of these put together in the proper way produce one equilateral triangle. Making solid angles with (a) three, (b) four, and (c) five of these equilateral triangles respectively, and taking the requisite number of these solid angles, namely four of (a), six of (b) and twelve of (c) respectively, and putting them together so as to form regular solids, he obtains ($\alpha$) a tetrahedron, ($\beta$) an octahedron, ($\gamma$) an icosahedron respectively. For the fourth element (earth), four isosceles right-angled triangles are first put together so as to form a square, and then six of these squares are put together to form a cube. Now, says Aristotle (299 b 23), “it is absurd that planes should only admit of being put together so as to touch in a line; for just as a line and a line are put together in both ways, lengthwise and breadthwise, so must a plane and a plane. A line can be combined with a line in the sense of being a line superposed, and not added”; the inference being that a plane can be superposed on a plane. Now this is precisely the sort of motion in question here; and Aristotle, so far from denying its permissibility, seems to blame Plato for not using it. Cf. also Physics V. 4, 228 b 25, where Aristotle speaks of “the spiral or other magnitude in which any part will not coincide with any other part,” and where superposition is obviously contemplated.
Motion without deformation.¶
It is well known that Helmholtz maintained that geometry requires us to assume the actual existence of rigid bodies and their free mobility in space, whence he inferred that geometry is dependent on mechanics.
Veronese exposed the fallacy in this (Fondamenti di geometria, pp. xxxv—xxxvi, 239—240 note, 615—7), his argument being as follows. Since geometry is concerned with empty space, which is immovable, it would be at least strange if it was necessary to have recourse to the real motion of bodies for a definition, and for the proof of the properties, of immovable space. We must distinguish the intuitive principle of motion in itself from that of motion without deformation. Every point of a figure which moves is transferred to another point in space. “Without deformation” means that the mutual relations between the points of the figure do not change, but the relations between them and other figures do change (for if they did not, the figure could not move). Now consider what we mean by saying that, when the figure $A$ has moved from the position $A_1$ to the position $A_2$, the relations between the points of $A$ in the position $A_2$ are unaltered from what they were in the position $A_1$, are the same in fact as if $A$ had not moved but remained at $A_1$. We can only say that, judging of the figure (or the body with its physical qualities eliminated) by the impressions it produces in us during its movement, the impressions produced in us in the two different positions (which are in time distinct) are equal. In fact, we are making use of the notion of equality between two distinct figures. Thus, if we say that two bodies are equal when they can be superposed by means of movement without deformation, we are committing a petitio principii. The notion of the equality of spaces is really prior to that of rigid bodies or of motion without deformation. Helmholtz supported his view by reference to the process of measurement in which the measure must be, at least approximately, a rigid body, but the existence of a rigid body as a standard to measure by, and the question how we discover two equal spaces to be equal, are matters of no concern to the geometer. The method of superposition, depending on motion without deformation, is only of use as a practical test; it has nothing to do with the theory of geometry.
Compare an acute observation of Schopenhauer (Die Welt als Wille, 2 ed. 1844, II. p. 130) which was a criticism in advance of Helmholtz’ theory: “I am surprised that, instead of the eleventh axiom [the Parallel-Postulate], the eighth is not rather attacked: ‘Figures which coincide (sich decken) are equal to one another.’ For coincidence (das Sichdecken) is either mere tautology, or something entirely empirical, which belongs, not to pure intuition (Anschauung), but to external sensuous experience. It presupposes in fact the mobility of figures; but that which is movable in space is matter and nothing else. Thus this appeal to coincidence means leaving pure space, the sole element of geometry, in order to pass over to the material and empirical.”
Mr Bertrand Russell observes (Encyclopaedia Britannica, Suppl. Vol. 4, 1902, Art. “Geometry, non-Euclidean”) that the apparent use of motion here is deceptive; what in geometry is called a motion is merely the transference of our attention from one figure to another. Actual superposition, which is nominally employed by Euclid, is not required; all that is required is the transference of our attention from the original figure to a new one defined by the position of some of its elements and by certain properties which it shares with the original figure.
If the method of superposition is given up as a means of defining theoretically the equality of two figures, some other definition of equality is necessary. But such a definition can be evolved out of empirical or practical observation of the result of superposing two material representations of figures. This is done by Veronese (Elementi di geometria, 1904) and Ingrami (Elementi di geometria, 1904). Ingrami says, namely (p. 66):
“If a sheet of paper be folded double, and a triangle be drawn upon it and then cut out, we obtain two triangles superposed which we in practice call equal. If points $A$, $B$, $C$, $D \dots$ be marked on one of the triangles, then, when we place this triangle upon the other (so as to coincide with it), we see that each of the particular points taken on the first is superposed on one particular point of the second in such a way that the segments $AB$, $AC$, $AD$, $BC$, $BD$, $CD$, $\dots$ are respectively superposed on as many segments in the second triangle and are therefore equal to them respectively. In this way we justify the following
“Definition of equality.
“Any two figures whatever will be called equal when to the points of one the points of the other can be made to correspond univocally [i.e. every one point in one to one distinct point in the other and vice versa] in such a way that the segments which join the points, two and two, in one figure are respectively equal to the segments which join, two and two, the corresponding points in the other.”
Ingrami has of course previously postulated as known the signification of the phrase equal (rectilineal) segments, of which we get a practical notion when we can place one upon the other or can place a third movable segment successively on both.
New systems of Congruence-Postulates.¶
In the fourth Article of Questioni riguardanti le matematiche elementari, I., pp. 93—122, a review is given of three different systems: (1) that of Pasch in Vorlesungen über neuere Geometrie, 1882, p. 101 sqq., (2) that of Veronese according to the Fondamenti di geometria, 1891, and the Elementi taken together, (3) that of Hilbert (see Grundlagen der Geometrie, 1903, pp. 7—15).
These systems differ in the particular conceptions taken by the three authors as primary. (1) Pasch considers as primary the notion of congruence or equality between any figures which are made up of a finite number of points only. The definitions of congruent segments and of congruent angles have to be deduced in the way shown on pp. 102—103 of the Article referred to, after which Eucl. I. 4 follows immediately, and Eucl. I. 26 (1) and I. 8 by a method recalling that in Eucl. I. 7, 8.
Veronese takes as primary the conception of congruence between segments (rectilineal). The transition to congruent angles, and thence to triangles is made by means of the following postulate:
“Let $AB$, $AC$ and $A'B'$, $A'C'$ be two pairs of straight lines intersecting at $A$, $A'$, and let there be determined upon them the congruent segments $AB$, $A'B'$ and the congruent segments $AC$, $A'C'$; then, if $BC$, $B'C'$ are congruent, the two pairs of straight lines are congruent.”
Hilbert takes as primary the notions of congruence between both segments and angles.
It is observed in the Article referred to that, from the theoretical standpoint, Veronese’s system is an advance upon that of Pasch, since the idea of congruence between segments is more simple than that of congruence between any figures; but, didactically, the development of the theory is more complicated when we start from Veronese’s system than when we start from that of Pasch.
The system of Hilbert offers advantages over both the others from the point of view of the teaching of geometry, and I shall therefore give a short account of his system only, following the Article above quoted.
Hilbert’s system.
The following are substantially the Postulates laid down.
If one segment is congruent with another, the second is also congruent with the first.
If an angle is congruent with another angle, the second angle is also congruent with the first.
Two segments congruent with a third are congruent with one another.
Two angles congruent with a third are congruent with one another.
Any segment $\text{AB}$ is congruent with itself, independently of its sense.
This we may express symbolically thus:
$$AB \equiv AB \equiv BA.$$
Any angle $(\text{ab})$ is congruent with itself, independently of its sense.
This we may express symbolically thus:
$$(ab) \equiv (ab) \equiv (ba).$$
On any straight line $\text{r}'$, starting from any one of its points $\text{A}'$, and on each side of it respectively, there exists one and only one segment congruent with a segment $\text{AB}$ belonging to the straight line $\text{r}$.
Given a ray $\text{a}$, issuing from a point $\text{O}$, in any plane which contains it and on each of the two sides of it, there exists one and only one ray $\text{b}$ issuing from $\text{O}$ such that the angle $(\text{ab})$ is congruent with a given angle $(\text{a}'\text{b}')$.
If $\text{AB}$, $\text{BC}$ are two consecutive segments of the same straight line $\text{r}$ (segments, that is, having an extremity and no other point common), and $\text{A}'\text{B}'$, $\text{B}'\text{C}'$ two consecutive segments on another straight line $\text{r}'$, and if $\text{AB} \equiv \text{A}'\text{B}'$, $\text{BC} \equiv \text{B}'\text{C}'$, then
$$AC \equiv A'C'.$$
If $(\text{ab})$, $(\text{bc})$ are two consecutive angles in the same plane $\pi$ (angles, that is, having the vertex and one side common), and $(\text{a}'\text{b}')$, $(\text{b}'\text{c}')$ two consecutive angles in another plane $\pi'$, and if $(\text{ab}) \equiv (\text{a}'\text{b}')$, $(\text{bc}) = (\text{b}'\text{c}')$, then
$$(ac) = (a'c').$$
If two triangles have two sides and the included angles respectively congruent, they have also their third sides congruent as well as the angles opposite to the congruent sides respectively.
As a matter of fact, Hilbert’s postulate corresponding to (11) does not assert the equality of the third sides in each, but only the equality of the two remaining angles in one triangle to the two remaining angles in the other respectively. He proves the equality of the third sides (thereby completing the theorem of Eucl. I. 4) by reductio ad absurdum thus. Let ABC, A'B'C' be the two triangles which have the sides AB, AC respectively congruent with the sides A'B', A'C' and the included angle at A congruent with the included angle at A'.
Then, by Hilbert’s own postulate, the angles ABC, A'B'C' are congruent, as also the angles ACB, A'C'B'.
If BC is not congruent with B'C', let D be taken on B'C' such that BC, B'D are congruent, and join A'D.
Then the two triangles ABC, A'B'D have two sides and the included angles congruent respectively; therefore, by the same postulate, the angles BAC, B'A'D are congruent.
But the angles BAC, B'A'C' are congruent; therefore, by (4) above, the angles B'A'C', B'A'D are congruent: which is impossible, since it contradicts (8) above.
Hence BC, B'C' cannot but be congruent.
Eucl. I. 4 is thus proved; but it seems to be as well to include all of that theorem in the postulate, as is done in (11) above, since the two parts of it are equally suggested by empirical observation of the result of one superposition.
A proof similar to that just given immediately establishes Eucl. I. 26 (1), and Hilbert next proves that
If two angles ABC, A'B'C' are congruent with one another, their supplementary angles CBD, C'B'D' are also congruent with one another.
We choose A, D on one of the straight lines forming the first angle, and A', D' on one of those forming the second angle, and again C, C' on the other
straight lines forming the angles, so that A'B' is congruent with AB, C'B' with CB, and D'B' with DB.
The triangles ABC, A'B'C' are congruent, by (11) above; and AC is congruent with A'C', and the angle CAB with the angle C'A'B'.
Thus, AD, A'D' being congruent, by (9), the triangles CAD, C'A'D' are also congruent, by (11);
whence CD is congruent with C'D', and the angle ADC with the angle A'D'C'.
Lastly, by (11), the triangles CDB, C'D'B' are congruent, and the angles CBD, C'B'D' are thus congruent.
Hilbert’s next proposition is that
Given that the angle (h, k) in the plane $\alpha$ is congruent with the angle (h', k') in the plane $\alpha'$, and that l is a half-ray in the plane $\alpha$ starting from the vertex of the angle (h, k) and lying within that angle, there always exists a half-ray l' in the second plane $\alpha'$, starting from the vertex of the angle (h', k') and lying within that angle, such that
$$(\text{h}, \text{l}) \equiv (\text{h}', \text{l}'), \text{ and } (\text{k}, \text{l}) \equiv (\text{k}', \text{l}').$$
If O, O' are the vertices, we choose points A, B on h, k, and points A', B' on h', k' respectively, such that OA, O'A' are congruent and also OB, O'B'.
The triangles OAB, O'A'B' are then congruent; and, if l meets AB in C, we can determine C' on A'B' such that A'C' is congruent with AC.
Then l' drawn from O' through C' is the half-ray required.
The congruence of the angles (h, l), (h', l') follows from (11) directly, and that of (k, l) and (k', l') follows in the same way after we have inferred by means of (9) that, AB, AC being respectively congruent with A'B', A'C', the difference BC is congruent with the difference B'C'.
It is by means of the two propositions just given that Hilbert proves that
All right angles are congruent with one another.
Let the angle BAD be congruent with its adjacent angle CAD, and likewise the angle B'A'D' congruent with its adjacent angle C'A'D'. All four angles are then right angles.
If the angle B'A'D' is not congruent with the angle BAD, let the angle with AB for one side and congruent with the angle B'A'D' be the angle BAD'', so that AD'' falls either within the angle BAD or within the angle DAC. Suppose the former.
By the last proposition but one (about adjacent angles), the angles B'A'D', BAD'' being congruent, the angles C'A'D', CAD'' are congruent.
Hence, by the hypothesis and postulate (4) above, the angles BAD'', CAD'' are also congruent.
And, since the angles BAD, CAD are congruent, we can find within the angle CAD a half-ray CAD''' such that the angles BAD'', CAD''' are congruent, and likewise the angles DAD'', DAD''' (by the last proposition).
But the angles BAD'', CAD'' were congruent (see above); and it follows, by (4), that the angles CAD'', CAD''' are congruent: which is impossible, since it contradicts postulate (8).
Therefore etc.
Euclid I. 5 follows directly by applying the postulate (11) above to $ABC$, $ACB$ as distinct triangles.
Postulates (9), (10) above give in substance the proposition that “the sums or differences of segments, or of angles, respectively equal, are equal.”
Lastly, Hilbert proves Eucl. I. 8 by means of the theorem of Eucl. I. 5 and the proposition just stated as applied to angles.
ABC, A'B'C' being the given triangles with three sides respectively congruent, we suppose an angle CBA'' to be determined, on the side of BC opposite to A, congruent with the angle A'B'C', and we make BA'' equal to A'B'.
The proof is obvious, being equivalent to the alternative proof often given in our text-books for Eucl. I. 8.
COMMON NOTION 5.¶
καὶ τὸ ὅλον τοῦ μέρους μεῖζόν [ἐστιν].
The whole is greater than the part.
Proclus includes this “axiom” on the same ground as the preceding one. I think however there is force in the objection which Tannery takes to it, namely that it replaces a different expression in Eucl. I. 6, where it is stated that “the triangle $DBC$ will be equal to the triangle $ACB$, the less to the greater: which is absurd.” The axiom appears to be an abstraction or generalisation substituted for an immediate inference from a geometrical figure, but it takes the form of a sort of definition of whole and part. The probabilities seem to be against its being genuine, notwithstanding Proclus’ approval of it.
Clavius added the axiom that the whole is the equal to the sum of its parts.
OTHER AXIOMS INTRODUCED AFTER EUCLID’S TIME.¶
[9] Two straight lines do not enclose (or contain) a space.
Proclus (p. 196, 21) mentions this in illustration of the undue multiplication of axioms, and he points out, as an objection to it, that it belongs to the subject matter of geometry, whereas axioms are of a general character, and not peculiar to any one science. The real objection to the axiom is that it is unnecessary, since the fact which it states is included in the meaning of Postulate 1. It was no doubt taken from the passage in I. 4, “if…the base $BC$ does not coincide with the base $EF$, two straight lines will enclose a space: which is impossible”; and we must certainly regard it as an interpolation, notwithstanding that two of the best MSS. have it after Postulate 5, and one gives it as Common Notion 9.
Pappus added some others which Proclus objects to (p. 198, 5) because they are either anticipated in the definitions or follow from them.
($g$) All the parts of a plane, or of a straight line, coincide with one another.
($h$) A point divides a line, a line a surface, and a surface a solid; on which Proclus remarks that everything is divided by the same things as those by which it is bounded.
An-Nairīzī (ed. Besthorn-Heiberg, p. 31, ed. Curtze, p. 38) in his version of this axiom, which he also attributes to Pappus, omits the reference to solids, but mentions planes as a particular case of surfaces.
“($\alpha$) A surface cuts a surface in a line;
($\beta$) If two surfaces which cut one another are plane, they cut one another in a straight line;
($\gamma$) A line cuts a line in a point (this last we need in the first proposition).”
($k$) 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 (τὸ ἄπειρον ἐν τοῖς μεγέθεσίν ἐστιν καὶ τῇ προσθέσει καὶ τῇ ἐπικαθαιρέσει, δυνάμει δὲ ἑκάτερον).
An-Nairīzī’s version of this refers to straight lines and plane surfaces only: “as regards the straight line and the plane surface, in consequence of their evenness, it is possible to produce them indefinitely.”
This “axiom” of Pappus, as quoted by Proclus, seems to be taken directly from the discussion of τὸ ἄπειρον in Aristotle, Physics III. 5—8, even to the wording, for, while Aristotle uses the term division (διαίρεσις) most frequently as the antithesis of addition (σύνθεσις), he occasionally speaks of subtraction (ἀφαίρεσις) and diminution (καθαίρεσις). Hankel (Zur Geschichte der Mathematik im Alterthum und Mittelalter, 1874, pp. 119—120) gave an admirable summary of Aristotle’s views on this subject; and they are stated in greater detail in Görland, Aristoteles und die Mathematik, Marburg, 1899, pp. 157—183. The infinite or unlimited (ἄπειρον) only exists potentially (δυνάμει), not in actuality (ἐνεργείᾳ). The infinite is so in virtue of its endlessly changing into something else, like day or the Olympic Games (Phys. III. 6, 206 a 15—25). The infinite is manifested in different forms in time, in Man, and in the division of magnitudes. For, in general, the infinite consists in something new being continually taken, that something being itself always finite but always different. Therefore the infinite must not be regarded as a particular thing (τόδε τι), as man, house, but as being always in course of becoming or decay, and, though finite at any moment, always different from moment to moment. But there is the distinction between the forms above referred to that, whereas in the case of magnitudes what is once taken remains, in the case of time and Man it passes or is destroyed but the succession is unbroken. The case of addition is in a sense the same as that of division; in the finite magnitude the former takes place in the converse way to the latter; for, as we see the finite magnitude divided ad infinitum, so we shall find that addition gives a sum tending to a definite limit. I mean that, in the case of a finite magnitude, you may take a definite fraction of it and add to it (continually) in the same ratio; if now the successive added terms do not include one and the same magnitude whatever it is [i.e. if the successive terms diminish in geometrical progression], you will not come to the end of the finite magnitude, but, if the ratio is increased so that each term does include one and the same magnitude whatever it is, you will come to the end of the finite magnitude, for every finite magnitude is exhausted by continually taking from it any definite fraction whatever. Thus in no other sense does the infinite exist, but only in the sense just mentioned, that is, potentially and by way of diminution (206 a 25—b 13). And in this sense you may have potentially infinite addition, the process being, as we say, in a manner, the same as with division ad infinitum: for in the case of addition you will always be able to find something outside the total for the time being, but the total will never exceed every definite (or assigned) magnitude in the way that, in the direction of division, the result will pass every definite magnitude, that is, by becoming smaller than it. The infinite therefore cannot exist even potentially in the sense of exceeding every finite magnitude as the result of successive addition (206 b 16—22). It follows that the correct view of the infinite is the opposite of that commonly held: it is not that which has nothing outside it, but that which always has something outside it (206 b 33—207 a 1).
Contrasting the case of number and magnitude, Aristotle points out that (1) in number there is a limit in the direction of smallness, namely unity, but none in the other direction: a number may exceed any assigned number however great; but (2) with magnitude the contrary is the case: you can find a magnitude smaller than any assigned magnitude, but in the other direction there is no such thing as an infinite magnitude (207 b 1—5). The latter assertion he justified by the following argument. However large a thing can be potentially, it can be as large actually. But there is no magnitude perceptible to sense that is infinite. Therefore excess over every assigned magnitude is an impossibility; otherwise there would be something larger than the universe (οὐρανός) (207 b 17—21).
Aristotle is aware that it is essentially of physical magnitudes that he is speaking. He had observed in an earlier passage (Phys. III. 5, 204 a 34) that it is perhaps a more general inquiry that would be necessary to determine whether the infinite is possible in mathematics, and in the domain of thought and of things which have no magnitude; but he excuses himself from entering upon this inquiry on the ground that his subject is physics and sensible objects. He returns however to the bearing of his conclusions on mathematics in III. 7, 207 b 27: “my argument does not even rob mathematicians of their study, although it denies the existence of the infinite in the sense of actual existence as something increased to such an extent that it cannot be gone through (ἀδιεξίτητον); for, as it is, they do not even need the infinite or use it, but only require that the finite (straight line) shall be as long as they please; and another magnitude of any size whatever can be cut in the same ratio as the greatest magnitude. Hence it will make no difference to them for the purpose of demonstration.”
Lastly, if it should be urged that the infinite exists in thought, Aristotle replies that this does not involve its existence in fact. A thing is not greater than a certain size because it is conceived to be so, but because it is; and magnitude is not infinite in virtue of increase in thought (208 a 16—22).
Hankel and Görland do not quote the passage about an infinite series of magnitudes (206 b 3—13) included in the above paraphrase; but I have thought that mathematicians would be interested in the distinct expression of Aristotle’s view that the existence of an infinite series the terms of which are magnitudes is impossible unless it is convergent, and (with reference to Riemann’s developments) in the statement that it does not matter to geometry if the straight line is not infinite in length, provided that it is as long as we please.
Aristotle’s denial of even the potential existence of a sum of magnitudes which shall exceed every definite magnitude was, as he himself implies, in conflict with the lemma or assumption used by Eudoxus (as we infer from Archimedes) to prove the theorem about the volume of a pyramid. The lemma is thus stated by Archimedes (Quadrature of a parabola, preface): “The excess by which the greater of two unequal areas exceeds the less can, if it be continually added to itself, be made to exceed any assigned finite area.” We can therefore well understand why, a century later, Archimedes felt it necessary to justify his own use of the lemma as he does in the same preface: “The earlier geometers too have used this lemma: for it is by its help that they have proved that circles have to one another the duplicate ratio of their diameters, that spheres have to one another the triplicate ratio of their diameters, and so on. And, in the result, each of the said theorems has been accepted no less than those proved without the aid of this lemma.”
Principle of continuity.
The use of actual construction as a method of proving the existence of figures having certain properties is one of the characteristics of the Elements. Now constructions are effected by means of straight lines and circles drawn in accordance with Postulates 1—3; the essence of them is that such straight lines and circles determine by their intersections other points in addition to those given, and these points again are used to determine new lines, and so on. This being so, the existence of such points of intersection must be postulated or proved in the same way as that of the lines which determine them. Yet there is no postulate of this character expressed in Euclid except Post. 5. This postulate asserts that two straight lines meet if they satisfy a certain condition. The condition is of the nature of a διορισμός (discrimination, or condition of possibility) in a problem; and, if the existence of the point of intersection were not granted, the solutions of problems in which the points of intersection of straight lines are used would not in general furnish the required proofs of the existence of the figures to be constructed.
But, equally with the intersections of straight lines, the intersections of circle with straight line, and of circle with circle, are used in constructions. Hence, in addition to Postulate 5, we require postulates asserting the actual existence of points of intersection of circle with straight line and of circle with circle. In the very first proposition the vertex of the required equilateral triangle is determined as one of the intersections of two circles, and we need therefore to be assured that the circles will intersect. Euclid seems to assume it as obvious, although it is not so; and he makes a similar assumption in I. 22. It is true that in the latter case Euclid adds to the enunciation that two of the given straight lines must be together greater than the third; but there is nothing to show that, if this condition is satisfied, the construction is always possible. In I. 12, in order to be sure that the circle with a given centre will intersect a given straight line, Euclid makes the circle pass through a point on the side of the line opposite to that where the centre is. It appears therefore as if, in this case, he based his inference in some way upon the definition of a circle combined with the fact that the point within it called the centre is on one side of the straight line and one point of the circumference on the other, and, in the case of two intersecting circles, upon similar considerations. But not even in Book III., where there are several propositions about the relative positions of two circles, do we find any discussion of the conditions under which two circles have two, one, or no point common.
The deficiency can only be made good by the Principle of Continuity.
Killing (Einführung in die Grundlagen der Geometrie, II. p. 43) gives the following forms as sufficient for most purposes.
(a) Suppose a line belongs entirely to a figure which is divided into two parts; then, if the line has at least one point common with each part, it must also meet the boundary between the parts; or
(b) If a point moves in a figure which is divided into two parts, and if it belongs at the beginning of the motion to one part and at the end of the motion to the other part, it must during the motion arrive at the boundary between the two parts.
In the Questioni riguardanti le matematiche elementari, I., Art. 5, pp. 123—143, the principle of continuity is discussed with special reference to the Postulate of Dedekind, and it is shown, first, how the Postulate may be led up to and, secondly, how it may be applied for the purposes of elementary geometry.
Suppose that in a segment $AB$ of a straight line a point $C$ determines two segments $AC$, $CB$. If we consider the point $C$ as belonging to only one of the two segments $AC$, $CB$, we have a division of the segment $AB$ into two parts with the following properties.
Every point of the segment $AB$ belongs to one of the two parts.
The point $A$ belongs to one of the two parts (which we will call the first) and the point $B$ to the other; the point $C$ may belong indifferently to one or the other of the two parts according as we choose to premise.
Every point of the first part precedes every point of the second in the order $AB$ of the segment.
(For generality we may also suppose the case in which the point $C$ falls at $A$ or at $B$. Considering $C$, in these cases respectively, as belonging to the first or second part, we still have a division into parts which have the properties above enunciated, one part being then a single point $A$ or $B$.)
Now, considering carefully the inverse of the above proposition, we see that it agrees with the idea which we have of the continuity of the straight line. Consequently we are induced to admit as a postulate the following.
If a segment of a straight line $AB$ is divided into two parts so that
every point of the segment $AB$ belongs to one of the parts,
the extremity $A$ belongs to the first part and $B$ to the second, and
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.
Application of Dedekind’s postulate to angles.¶
If we consider an angle less than two right angles bounded by two rays $a$, $b$, and draw the straight line connecting $A$, a point on $a$, with $B$, a point on $b$, we see that all points on the finite segment $AB$ correspond univocally to all the rays of the angle, the point corresponding to any ray being the point in which the ray cuts the segment $AB$; and if a ray be supposed to move about the vertex of the angle from the position $a$ to the position $b$, the corresponding points of the segment $AB$ are seen to follow in the same order as the corresponding rays of the angle $(ab)$.
Consequently, if the angle $(ab)$ is divided into two parts so that
each ray of the angle $(ab)$ belongs to one of the two parts,
the outside ray $a$ belongs to the first part and the ray $b$ to the second,
any ray whatever of the first part precedes any ray whatever of the second part,
the corresponding points of the segment $AB$ determine two parts of the segments such that
every point of the segment $AB$ belongs to one of the two parts,
the extremity $A$ belongs to the first part and $B$ to the second,
any point whatever of the first part precedes any point whatever of the second.
But in that case there exists a point $C$ of $AB$ (which may belong to one or the other of the two parts) 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.
Thus exactly the same thing holds of $c$, the ray corresponding to $C$, with reference to the division of the angle $(ab)$ into two parts.
It is not difficult to extend this to an angle $(ab)$ which is either flat or greater than two right angles; this is done (Vitali, op. cit. pp. 126—127) by supposing the angle to be divided into two, $(ad)$, $(db)$, each less than two right angles, and considering the three cases in which
the ray $d$ is such that all the rays that precede it belong to the first part and those which follow it to the second part,
the ray $d$ is followed by some rays of the first part,
the ray $d$ is preceded by some rays of the second part.
Application to circular arcs.¶
If we consider an arc $AB$ of a circle with centre $O$, the points of the arc correspond univocally, and in the same order, to the rays from the point $O$ passing through those points respectively, and the same argument by which we passed from a segment of a straight line to an angle can be used to make the transition from an angle to an arc.
Intersections of a straight line with a circle.¶
It is possible to use the Postulate of Dedekind to prove that
If a straight line has one point inside and one point outside a circle, it has two points common with the circle.
For this purpose it is necessary to assume (1) the proposition with reference to the perpendicular and obliques drawn from a given point to a given straight line, namely that of all straight lines drawn from a given point to a given straight line the perpendicular is the shortest, and of the rest (the obliques) that is the longer which has the longer projection upon the straight line, while those are equal the projections of which are equal, so that for any given length of projection there are two equal obliques and two only, one on each side of the perpendicular, and (2) the proposition that any side of a triangle is less than the sum of the other two.
Consider the circle (C) with centre O, and a straight line (r) with one point A inside and one point B outside the circle.
By the definition of the circle, if R is the radius,
$$OA < R, \quad OB > R.$$
Draw OP perpendicular to the straight line r.
Then OP < OA, so that OP is always less than R, and P is therefore within the circle C.
Now let us fix our attention on the finite segment AB of the straight line r. It can be divided into two parts, (1) that containing all the points H for which $OH < R$ (i.e. points inside C), and (2) that containing all the points K for which $OK \geqq R$ (points outside C or on the circumference of C).
Thus, remembering that, of two obliques from a given point to a given straight line, that is greater the projection of which is greater, we can assert that all the points of the segment PB which precede a point inside C are inside C, and those which follow a point on the circumference of C or outside C are outside C.
Hence, by the Postulate of Dedekind, there exists on the segment PB a point M such that all the points which precede it belong to the first part and those which follow it to the second part.
I say that M is common to the straight line r and the circle C, or
$$OM = R.$$
For suppose, e.g., that OM < R.
There will then exist a segment (or length) σ less than the difference between R and OM.
Consider the point M', one of those which follow M, such that MM' is equal to σ.
Then, because any side of a triangle is less than the sum of the other two,
$$OM' < OM + MM'.$$
But
$$OM + MM' = OM + \sigma < R,$$
whence
$$OM' < R,$$
which is absurd.
A similar absurdity would follow if we suppose that OM > R.
Therefore OM must be equal to R.
It is immediately obvious that, corresponding to the point M on the segment PB which is common to r and C, there is another point on r which has the same property, namely that which is symmetrical to M with respect to P.
And the proposition is proved.
Intersections of two circles.¶
We can likewise use the Postulate of Dedekind to prove that
If in a given plane a circle C has one point X inside and one point Y outside another circle C', the two circles intersect in two points.
We must first prove the following
Lemma.¶
If O, O' are the centres of two circles C, C', and R, R' their radii respectively, the straight line OO' meets the circle C in two points A, B, one of which is inside C' and the other outside it.
Now one of these points must fall (1) on the prolongation of O'O beyond O or (2) on OO' itself or (3) on the prolongation of OO' beyond O'.
First, suppose A to lie on O'O produced.
Then
$$AO' = AO + OO' = R + OO' \ldots\ldots (\alpha).$$
But, in the triangle OO'Y,
$$O'Y < OY + OO',$$
and, since O'Y > R', OY = R,
$$R' < R + OO'.$$
It follows from $(\alpha)$ that AO' > R'; and A therefore lies outside C'.
Secondly, suppose A to lie on OO'.
Then
$$OO' = OA + AO' = R + AO' \ldots (\beta).$$
From the triangle OO'X we have
$$OO' < OX + O'X,$$
and, since OX = R, O'X < R', it follows that
$$OO' < R + R',$$
whence, by $(\beta)$, AO' < R', so that A lies inside C'.
Thirdly, suppose A to lie on OO' produced.
Then
$$R = OA = OO' + O'A \ldots\ldots\ldots (\gamma).$$
And, in the triangle OO'X,
$$OX < OO' + O'X,$$
that is
$$R < OO' + O'X,$$
whence, by $(\gamma)$,
$$OO' + O'A < OO' + O'X,$$
or
$$O'A < O'X,$$
and A lies inside C'.
It is to be observed that one of the two points A, B is in the position of case (1) and the other in the position of either case (2) or case (3): whence we must conclude that one of the two points A, B is inside and the other outside the circle C'.
Proof of theorem.¶
The circle C is divided by the points A, B into two semicircles. Consider one of them, and suppose it to be described by a point moving from A to B.
Take two separate points P, Q on it and, to fix our ideas, suppose that P precedes Q.
Comparing the triangles OO'P, OO'Q, we observe that one side OO' is common, OP is equal to OQ, and the angle POO' is less than the angle QOO'.
Therefore O'P < O'Q.
Now, considering the semicircle APQB as divided into two parts, so that the points of the first part are inside the circle C', and those of the second part on the circumference of C' or outside it, we have the conditions necessary for the applicability of the Postulate of Dedekind (which is true for arcs of circles as for straight lines); whence there exists a point M separating the two parts.
I say that O'M = R'.
For, if not, suppose O'M < R'.
If then σ signifies the difference between R' and O'M, suppose a point M', which follows M, taken on the semicircle such that the chord MM' is not greater than σ (for a way of doing this see below).
Then, in the triangle O'MM',
$$O'M' < O'M + MM' < O'M + \sigma,$$
and therefore
$$O'M' < R'.$$
It follows that M', a point on the arc MB, is inside the circle C': which is absurd.
Similarly it may be proved that O'M is not greater than R.
Hence O'M = R.
[To find a point M' such that the chord MM' is not greater than σ, we may proceed thus.
Draw from M a straight line MP distinct from OM, and cut off MP on it equal to σ/2.
Join OP, and draw another radius OQ such that the angle POQ is equal to the angle MOP.
The intersection, M', of OQ with the circle satisfies the required condition.
For MM' meets OP at right angles in S.
Therefore, in the right-angled triangle MSP, MS is not greater than MP (it is less, unless MP coincides with MS, when it is equal).
Therefore MS is not greater than σ/2, so that MM' is not greater than σ.]