SOURCE: "Did Euclid's Elements, Book I, Develop Geometry Axiomatically?," in Archive for History of Exact Sciences, Vol. 14, 1974/1975, pp. 263-95.

[In the following essay, Seidenberg challenges the assumption that Euclid, in Elements, developed geometry on an axiomatic basis. Seidenberg argues that, by insisting on this assumption, the work is viewed "from a false perspective" and its accomplishments are thus displayed "in a bad light."]

Historians are fond of repeating that Euclid developed geometry on an axiomatic basis, but the wonder is that any mathematician who has looked at The Elements would agree with this. Anyone who looks at The Elements with modern hindsight sees that something is wrong; but we have all been told in our childhood that Euclid had the axiomatic method, so the usual reaction is to speak of "gaps". This word is hardly right, though, if there was nothing there in the first place.

Could it be that, by insisting on the axiomatic basis, we are viewing The Elements from a false perspective and seeing its accomplishments in a bad light? This is precisely what I intend to prove.

The Greeks of Euclid's time¹ had the axiomatic method; Aristotle's description of it can be considered a close approximation to our own. Or better yet, one may consider Eudoxus' theory of magnitude as presented in Book V of The Elements: the procedure there disclosed is pretty much in accordance with our view of what an axiomatic development should be. It is known, however, that The Elements is a compilation of uneven quality, so that even with the definitions, postulates, and common notions of Book I, it is unwarranted to assumed that Book I is written from the same point of view as Book V.

At the beginning of Book I as it has come down to us there stand, besides some definitions, some fifteen statements grouped into "Postulates" and "Common Notions". J. L. Heiberg, basing himself on the best of the Greek manuscripts, the oldest of which, however (not counting some fragments), only dates back to the ninth century, brought out an edition of The Elements in which there are five Postulates and nine Common Notions.² Of the usual Common Notions Heiberg left out one without comment; he bracketed three others, thereby ascribing to them a doubtful status; and he later declared another one to be definitely an interpo lation (thus leaving five).³ T. L. Heath in his edition of The Elements gives the five Postulates and the remaining five Common Notions;⁴ these are also precisely the Postulates and Common Notions that Proclus accepts in his Commentary on Book I.⁵ Nevertheless Heath considered it probable that Common Notions 4 and 5 were interpolations; and P. Tannery maintained that none were authentic.⁶

I. Postulate 3 and Propositions 1, 2, and 3

Of the postulates, the first three are the so-called postulates of construction, the fourth says that all right angles are equal, and the fifth is the parallel postulate. Postulate 3 says that one may "describe a circle with any center and any distance". This seems clear enough, but it has been given some curious interpretations. One author, regarding especially the words "any distance", considered that Euclid was saying that space is infinite (since a distance can be arbitrarily large) and that it is continuous, rather than discrete (since a distance can be artitrarily small). Health thinks this view is exaggerated, but he himself considers that the postulate does not refer to a construction at all: it merely postulates the existence of a circle with any center A and any radius.⁷ Heath may be right or may be wrong, but certainly he is ascribing to Euclid a sophistication that is not matched by most twentieth century mathematicians. Almost any present day mathematician would grant the mere existence of the circle without more ado; and would no more think of making this assumption explicit than of making his acceptance of the tertium non datur explicit. If he were even a little careless, he would say it's obvious. But a modern day mathematician knows that he must never say "obvious". So instead he would refer to an axiom of set-theory. This axiom says, roughly, that for any property P, the set of points (within a given set, say in a plane) satisfying the property exists. Of course, he accepts the axiom because he regards it as obvious.

Although he would thus readily grant the existence of the circle, he would not without more ado—and quite rightly—grant that there are any points on it. There might well be no points on it. If the circle is the circle of center A and radius AB, then B is on the circle, but there may be no other points on it.

Let us see how Euclid uses Postulate 3. The very first proposition of Book I is To construct an equilateral triangle on a given straight line, say AB. With centers A and B, circles are drawn with radius AB. The circles intersect in a point C and ABC is a required triangle. But the question as to the existence of C is not raised. And here the polemic begins. We are invited to believe that Euclid had some subtle insight into the nature of geometry (or of reasoning) when he postulated that circles can be drawn, yet overlooked the obvious in Book I, Proposition I.

Let us look at Proposition 1 and what Euclid says in a straightforward way. Postulate 3 says nothing about existence: it says one can draw a circle. And Proposition 1 does not ask us to prove the existence of an equilateral triangle, but to draw one. Anyone who has thought about a construction problem knows very well that there is a difference between existence and construction. Even granted that an equilateral triangle on AB exists, there is still a trick to finding one: and Euclid shows us this trick.

A proposition of the form: There exists an equilateral triangle on any given straight line is a theorem, whereas a proposition of the form: To construct an equilateral triangle on a given straight line is a problem. We recognize the difference between a theorem and a problem, and so did Euclid. He does not label his propositions as theorems or problems, but each proposition ends either with the words "which was to be proved" or the words "which was to be done", so we have but to look at the last words of Proposition 1 to have incontestable evidence that Euclid considered it a problem.

Proposition 1 does indeed (if we allow the intersection) give the existence, but it also gives something more. As to the existence itself, there already were in ancient times opposing views. According to the school of Menaechmus, geometric objects, say equilateral triangles, exist because we produce them: just as a chair exists because an artisan has constructed it, so an equilateral triangle exists because the geometer constructs it. The followers of Speusippus, the successor of Plato at the Academy, on the other hand, held that geometric objects are eternal things, and hence not brought into being: it is better to say these objects exist.⁸ If we follow Speusippus, then the circles will meet, and Euclid is vindicated of every fault, at least through Proposition 1. The followers of Speusippus, however, in accordance with the view just expressed, insisted on calling all propositions "theorems" (rejecting the designation "problem" for the constructions). Hence it looks as though Euclid was inclining toward Menaechmus. In any event the issue as to whether the circles meet was not contemplated in ancient times.⁹

For axiomatic purposes with modern intentions there would have been no use for Postulate 3 : wherever Euclid says to draw a circle with center A and through B one has but to say instead: "Consider the circle with center A that passes through B". Hubert, who in Chapter I of his Foundations of Geometry follows Euclid pretty closely, has no axiom corresponding to Postulate 3, or anything like it.¹⁰

Book I, Proposition 2 (Problem) seems to contain a very subtle point. The problem is: From a given point A to draw a straight line equal to a given straight line BC. Euclid's solution is quite ingenious. The novice, coming equipped with straightedge and compass, or perhaps with peg and cord, might easily be forgiven if he could not see the point: why not set the compass for drawing a circle with center B and radius BC, and then with this compass setting draw a circle with center A? Or stretch a cord from B to C, pick it up, place one end at A, and stretch the cord again? Euclid himself employs this line of reasoning in Proposition 4 (Theorem) about the congruence of two triangles, in that he "applies" one triangle to another. He therefore does not hesitate to appeal to his intuition, an intuition built up in dealing with objects, perhaps not rigid but whose shapes can be recovered.

There is a difference, however, namely the difference between a Problem and a Theorem, something to be done and something to be demonstrated. In view of the method of superposition of figures used in Proposition 4, it is difficult to see what the point of Proposition 2 is unless it is an explicit desire to abstract from actual constructions; and, indeed, this is the normal, straightforward view of the matter. Control over this process of abstraction is obtained by referring all constructions back to the first three postulates. Great meticulousness is shown in a construction that nothing more is used than was explicitly abstracted from ancient practice. But a theorem is different, and here the old intuitions are available.

Proposition 3 is also a construction problem: Given two unequal straight lines, to cut off from the greater a straight line equal to the less. Of course, there is an "error" in the proof, as there is nothing to guarantee that a line passing through a point inside a circle, even the center, will, however far produced, meet the circle. A similar "error" occurs in Proposition 2.

2. Common Notion 4 and Proposition 4

Proposition 4 is Euclid's first theorem. In it one has two triangles ABC and DEF … with two sides and the included angle of the one equal respectively to two sides and the included angle of the other, namely, AB to DE, AC to DF, and angle BAC to angle EDF; and it is asserted that the base and two remaining angles of the first are respectively equal to the base and two remaining angles of the second; and that the two triangles are equal. In the proof the points is placed on D and the straight line AB on DE, "whence the point B will also coincide with E, because AB is equal to DE". Then, it is asserted, AC will fall on DC, because angle BAC is equal to angle EDF; and C will coincide with F, because AC is equal to DF;

hence the base BC will coincide with the base EF [for if, when B coincides with E and C with F, the base BC does not coincide with the base EF, two straight lines will enclose a space, which is impossible. Therefore the base BC will coincide with EF]¹¹ and will be equal to it.

It is then concluded that the two triangles themselves are equal and that the two remaining angles of the one triangle are respectively equal to the two remaining angles of the other.

In this argument no appeal is made to any of the postulates nor to the first three propositions; for example, in saying that AB can be transferred to DE, Proposition 2 is not being used. Nor is there any appeal to the common notions, except perhaps Common Notion 4 (which says that "things which coincide with one another are equal to one another"); and above it was already remarked that there are reasons for thinking Common Notion 4 is an interpolation. Thus from an axiomatic point of view Proposition 4 is based on nothing.

The unsatisfactory nature of Euclid's proof by superposition of figures is a commonplace, but how this commonplace can be reconciled with the view that Euclid (Book I) had the axiomatic method it is difficult or, rather, impossible to say.

Bertrand Russell (Principles of Mathematics, p. 405) says that Euclid would have done better to assume I, 4 as an axiom. Yes, but Euclid couldn't have done that, for the simple reason that he had no inkling of a geometric axiom. If our present thesis is correct, Euclid wished to prove everything.

Concerning the passage in brackets cited above from I, 4, Heath writes (op. cit., p. 249):

Heiberg (Paralipomena zu Euclid in Hermes, XXXVIII, 1903, p. 56) has pointed out, as a conclusive reason for regarding these words as an early interpolation, that the text of an-Nairīzī (Codex Leidensis 399, 1, ed. Betsthorn-Heiberg) does not give the words in this place but after the conclusion Q.E.D., which shows that they constitute a scholium only. They were doubtless added by some commentator who thought it necessary to explain the immediate inference that, since B coincides with E and C with F, the straight line BC coincides with EF, an inference that really follows from the definition of a straight line and Post. 1; and no doubt the Postulate that 'Two straight lines cannot enclose a space' (afterwards, placed among the Common Notions) was interpolated at the same time.

According to Aristotle, "of the premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are included under the particular science…. It is a common principle, for instance, that if equals be subtracted from equals, the remainders are equal…."¹² If the Common Notions are common in the sense of Aristotle, then Tannery (op. cit., p. 53) was surely right in holding that Common Notion 4 should be excluded from the Common Notions, since it is incontestably geometric in character. Proclus, too, had rejected a Common Notion on the same grounds.¹³ Heath (op. cit., p. 225) agrees and adds: "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 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."

Common Notion 4 could hardly have come in except upon application of the method of superposition of figures. Euclid uses this method only in I, 4 and I, 8 (and III, 24) and Common Notion 4 in no way enters I, 8. So although I would not insist upon the quotation of a Common Notion as evidence for its authenticity, still one has to admit that there is little evidence of its application in the text itself.¹⁴

Although Proclus objected to the proliferation of the Common Notions, he also thought they should not be cut down to the minimum: he recognized Common Notion 4 and the next to be proper axioms, but mentioned that Heron only gave three (the first three).¹⁵ Simplicius, too, according to an-Nairīzī, noted that "three axioms (sententiae acceptae) only are extant in the ancient manuscripts, but the number was increased in the more recent."¹⁶

3. Postulate 4

The Common Notion that "two straight lines cannot enclose a space" is in one family of manuscripts Common Notion 9, but it also occurs in two of the best manuscripts as Postulate 6.¹⁷ Thus we conlude that at least one Postulate has been interpolated; and we are obliged to face the possibility that some of the others were, too.

Postulate 4 says that all right angles are equal. It is strange to find this postulated as it is very easy to prove, at the same time maintaining the standard of Book I. In fact, Proclus, following other commentators, gave a simple proof using the method of superposition of figures.¹⁸ Heath (op. cit., pp. 225, 249, 200) argues that Euclid was "reluctant" to use the method, as he might have used it in several places where he doesn't, and for this reason postulated the Postulate. This argument is hardly convincing, for though the scanty use by Euclid of the method is surely noteworthy, Heath himself remarks (op. cit., p. 225) that it is, for Euclid, fundamental, and that no ancient geometer expressed any doubt as to its legitimacy. Moreover, if Euclid could have postulated Postulate 4 to avoid the method, he could have postulated Proposition 4 for the same reason.

Postulate 4 is never quoted nor is it ever even tacitly invoked to say anything about right angles. The way it comes in is this: an angle is, by Definition 7 of Book I, "the inclination to one other of two lines in a plane which meet one another and do not lie in the same straight line". Thus Euclid, just like some modern authors (e.g., Hubert), did not wish to speak of straight angles. So instead he always says "two right angles". Typically, Postulate 4 enters into I, 15, which says that "if two straight lines cut one another, they make the vertical angles equal to one another", say if AB cuts CD at E … then angle AEC is equal to DEB; (the postulate first enters at I, 14, but I, 14 does not enter I, 15). For the proof, Euclid subtracts AED from the "straight angles" AEB and CED, proved equal, to get angles AED and DEB as equal remainders. It's hard to believe that Euclid considered it necessary to postulate Postulate 4 to get such a theorem.

As Hubert in effect noted (Grundlagen der Geometrie, p. 23), the statement that all right angles are equal or, what comes very much to the same thing, that supplements of equal angles are equal, can be proved from Proposition 4 without appeal to the method of superposition of figures. The proof is so simple that I am sure that Euclid, Proclus, or any other ancient could have easily found it, if only someone had told them to look. In fact, Euclid himself actually does prove it in a special case: Proposition 5 says not only that the base angles of an isosceles triangle are equal, but also the angles under the base are equal. This proof, only slightly modified, would show that the supplements of any equal angles are equal.¹⁹

That Postulate 4 enters into Book I at best tacitly can be seen from Proclus's Commentary. Speaking of Proposition 14, he says that "[here], too, we can observe an unexcelled level of precision in scientific expression". He then mentions that Postulate 2, Proposition 13, and Common Notions 1, 3, and 5 are employed; but he misses Postulate 4. Similarly, the proof of Proposition 15 depends, as he observes, on Proposition 13 and Common Notions 1 and 3; but again he misses Postulate 4. As mentioned, Proclus did take note of Postulate 4, in fact, he proved it, but he seems unaware that Euclid ever applied it. It is possible, of course, that he never did.

Proclus does mention Postulate 4 in discussing Proposition 13 (that "if a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles") and may be saying that the postulate enters the proposition. If so, then Proclus sees the Postulate coming in where it doesn't and doesn't see it come in where it does. This surely indicates that Euclid wasn't very clear on the point.

According to Heath (op. cit., p. 201), the "raison d'être" of Postulate 4 is Postulate 5 (though this conflicts with his view previously mentioned). Postulate 5 says that "if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles", and Heath argues that the condition in the latter would be useless unless it were first made clear that right angles are angles of determinate and invariable magnitude". From the point of view of simple logic, I cannot agree with Heath, since even if right angles were not all equal, Postulate 5 as stated would still make sense. It is, however, conceivable that Euclid thought about Postulate 5 in just the way Heath did. How, then, are we to imagine Euclid's insertion of Postulate 4? Did he realize all along, from an early point in his plans to write an Elements, the significance of Postulate 4? Or did he, so to speak, put it in at the last minute? Now if he had had the statement of Postulate 4 in view from the beginning, surely he would have given a proof, since this is easy and he, like any other geometer, would not have been content to assume something he could prove; or at least not leave it at that. Even if he felt that he had to give all the postulates at the beginning, in particular Postulate 5 (the significance of which the reader can anyway not grasp till after Proposition 28, which says that if the interior angles on one side make two right angles, then the lines do not meet), and therefore also Postulate 4 because Postulate 5 wouldn't make sense without it, still he could have proved Postulate 4; or more simply, he could have avoided any reference to right angles at the beginning by phrasing Postulate 5 in the form that "if a straight line falling on two straight lines make the interior angles on one side of it less than the interior angles on the other, the two lines, if produced indefinitely, meet", necessarily on the side for which the angle sum is less. Surely the geometer who had the virtuosity to get the Theorem of Pythagoras (I, 47) into Book I, thus bypassing the geometric algebra of Book II and the theory of similar triangles of Book VI, also had the resourcefulness to make such a trifling verbal reformulation.²⁰ It looks then as if the most we can concede to Heath is that at the last minute Euclid, out of a logical scruple, inserted Postulate 4. Thus Postulate 4 would be a patch. Whether this patch was inserted by Euclid or by someone else is of no moment to my thesis.^21,22

Concerning the fifth Postulate, Heath (op. cit., p. 202) writes: "When we consider the countless successive attempts made through more than twenty centuries to prove the Postulate, many of them by geometers of ability, we cannot but admire the genius of the man [namely, Euclid] who concluded that such a hypothesis, which he found necessary to the validity of his whole system of geometry, was really indemonstrable."

To me this seems little more than hero-worship. It may be that Euclid could see no way out of using the Pos tulate—if so, we can still admire his genius—but to credit him with the realization that the Postulate was "really indemonstrable" is, in view of the known history of the Postulate, a clear distortion.

Heiberg rejected some four Common Notions and one Postulate (Postulate 6, otherwise known as Common Notion 9) as not genuine, Heath thought two others were probably interpolations, and I have cast doubt on Postulate 4, as did Tannery. To me it seems (for reasons I hope to make plain) that Postulate 5 was also interpolated, i.e., it was not in The Elements when the work left Euclid's hand; that it is the doing of some commentator who, not convinced of the argumentation of Proposition 29, placed a sentence of the proof at the beginning of The Elements as something unproved. If this seems too iconoclastic, one could save the phe nomenon by stretching the word "interpolate" just a little and saying that Euclid himself interpolated Postulate 5.

But however that may be one could avoid the issue of the parallel postulate altogether, for this postulate does not enter until Proposition 29, and if Euclid really had had the axiomatic method in mind, surely this would have revealed itself in the first twenty eight propositions. Thus we can easily whittle the postulates down to the three "postulates of construction".

4. Postulates 1 and 2

As to these postulates, even they are not stated very carefully for postulational purposes. Postulate 1 says: "Let it be granted that a straight line may be drawn from any one point to any other point". Those who, like Heath, would have it that Euclid had an axiom schema in mind have to make out that with his postulate Euclid was saying that there is only one straight line on two given points, something that is clearly not being said.²³

It has been said that Postulate 1 gives the existence of a line on any two points.²⁴ Well, yes, it does: no construction could be successful unless the object to be constructed exists. This is a constraint on any construction. But the assertion of existence could hardly have been Euclid's intention. He nowhere posits the existence of points, so why should be posit the existence of lines? In Postulate 1 Euclid is surely saying nothing about uniqueness, nor anything about existence, either, in intention. What he is saying is that one can draw a line.

The effect of the first three postulates is to restrict the constructions to "ruler and compass". This statement is just a way of formulating what one actually finds in The Elements, and to see it, one need make no historical studies (beyond, of course, examining ccThe Elements itself), e.g., by comparing with what Aristotle, Proclus, et alii, said. The Greeks did indeed have other types of constructions, for example, the neusis construction, in which one places a given segment AB so that A falls on a given curve Γ, B on a given curve Δ, and in such way that line AB passes through a given point P. But we are not talking about the Greeks in general, just about The Elements. The cause of Postulates 1, 2, and 3 is a different matter: the effect cannot be the cause. As to what the cause may have been, this is a subject I have discussed at length elsewhere, and beyond remarking that I think the cause was an actual restriction on ritual constructions, I refer to a previous work²⁵ for the development of this side of the argument. For the present I confine myself to the effects as we see them in The Elements.

If one were to describe the game of Chess I suppose one would inadvertently say something about space. It is the same with the game of Straightedge and Compasses. Postulates 1, 2, and 3 say nothing explicitly about space, and they tell us nothing about space except incidentally, that there is a line on any two points. Thus (setting aside Postulate 4 for reasons given) the first 28 propositions of Book I are based on no explicitly made assumptions about space, unless there are some such in the Definitions or Common Notions.

5. Common Notions 1, 2, and 3

As to the definitions, Definition 17 does include the statement that a diameter bisects a circle; but this is nowhere used in Book I and creates no difficulty for us.

Thus we are left with the Common Notions, especially Common Notions 1, 2, and 3. As to these, it is reported that Apollonius had attempted to demonstrate the Common Notions. Proclus criticizes Apollonius and in the course of doing so cites Apollonius's proof of Common Notion 1, presumably word for word, as follows:²⁶

Let A be equal to B and the latter to C. I say that A is also equal to C. Since A, being equal to B, occupies the same space as it, and since B, being equal to C, occupies the same space as it, A occupies the same space as C. Therefore they are equal.

Proclus, following Geminus, objects that this argument "involves a premise not better known than the conclusion, if indeed it is not more doubtful", but it seems to me that Proclus's objections are not well-taken. In the manuscript there is a figure showing A, B, C as three line segments, and though it was understood that the assertion was not just about line segments, let us consider this case first. Then using the method of superposition of figures, we place A on B, whence A will coincide with B. Then we place C on B. whence C will coincide with B. Now A and C not only fill the same space, but even the same place, namely, that of B, i.e., they coincide, whence they are equal. One may object to the method of superposition of figures as lacking clarity (or sense), but Proclus accepts this method, as shown by his acceptance of Common Notion 4; and if he accepts this method, I cannot see that he would be justified in rejecting the above proof for segments.

In Book I, Proposition 1, then, one will not have to use Common Notion 1. In that proposition, one has a given segment AB and has constructed a point C such that AB equals AC and BA equals BC; and to prove that ABC is equilateral, one still has to prove AC equals BC. Euclid proves this by appealing to Common Notion 1. However, as explained, if he had been willing to use the method of superposition of figures, which he uses in Proposition 4, he could have proved this last point by placing AC and BC on AB. Then AC and BC would coincide and hence be equal.²⁷

In Proposition 2, Euclid has segments DAL and DBG with DL equal DG and DA equal DB … ; and, he concludes, "therefore the remainder AL is equal to the remainder BG" presumably appealing to Common Notion 3. He could also have argued as follows: Place DL along DG. Then L falls on G, since DL equals DG; and A will fall on B since DA equal DB. Then AL coincides with BG, and so is equal to it.

In this way, using the method of superposition of figures, one can get through the first 28 propositions without using the first three Common Notions. Or, otherwise said, without diminishing the rigor of the Book, one can eliminate the use of the first three Common Notions in the first 28 propositions. This leaves Common Notions 4 and 5. Apollonius himself in the cited passage does not mention superposition, but if our reconstruction is granted, he presumably would not have tried to eliminate Common Notion 4; or if he had, I don't see how he would have succeeded. We also do not know that he attempted Common Notion 5, but if he had, I can't imagine what he did, even for line segments. Anyway, as already mentioned, Heath is willing to regard these two Common Notions as interpolated. So this would leave nothing.²⁸

Though the above remarks would suffice to eliminate Common Notions 1, 2, and 3 from the first 28 propositions, let us still pursue these notions a bit for finite unions of line segments. We will consider finite unions of segments, disjoint except that two segments may share an endpoint (though the disjointness assumption is easily removed and the segments could even be counted with multiplicity). Consider two such unions A and B and define them to be equal if the segments of A and those of B can be finitely subdivided to yield sets A' and B', respectively, in such way that the segments of A' can be put in one to one correspondence with those of B' and with corresponding elements congruent. By congruent we mean equal in the sense of Euclid. Now we can state the following theorem:

Theorem. Let A be a finite union of segments, disjoint except that a pair of segments may share an endpoint, and let B, C also be such unions. Then: if A is equal to B and B to C, then A is equal to C.

For the proof, we consider subdivisions of the segments of A and of B showing that A equals B, and by a change of notation may suppose the segments of A can be put into one to one correspondence with those of B in such way that corresponding segments are congruent. Now let B', C" be subdivisions of B, C showing that B equals C. Since B' is a subdivision of B, each segment of B is divided into a finite number of segments. Let b be one such segment of B and let a be its correspondent in A. Then the subdivision of b yields a subdivision of a such that the segments of the subdivision of a are in one to one correspondence with those of b and with corresponding segments equal. Applying this to all the b in B we get a subdivision A' of A such that the segments of A' are in one to one correspondence with those of B' and with corresponding segments congruent. Combining the stated correspondences of A' with B' and of B' with C' we get a correspondence between the segments of A' and those of C' showing that A equals C. Q.E.D.²⁹

In brief, quite generally, at least for unions of line segments, "things equal to the same thing are equal to each other". And similar considerations yield Common Notions 2 and 3 in complete generality (for unions of line segments).

Common Notions 1, 2, and 3 are also applied in Book I to areas, though this is not done in the first 28 propositions. Book I can be divided into three parts. The first part, Propositions 1 to 26, takes up the basic constructions and the relations to each other of sides and angles of triangles.³⁰ To get to area, Euclid first needs to know some things about parallels; and this is taken up in Propositions 27 to 33. Then Propositions 34 to 45 consider area; and, after constructing the square in Proposition 46, Book I culminates with the Theorem of Pythagoras and its converse, I, 47 and 48. Area is mentioned only twice in the first part: first in Proposition 4 where two triangles are said to be equal under certain circumstances; this assertion is placed there in preparation for its use in Proposition 34; second, in Proposition 6, which says that if the base angles of a triangle are equal, then the triangle is isosceles. The use of area here is surely inessential. Euclid never applies this Proposition in Book I and the Proposition is a corollary of Proposition 18, that "in any triangle the greater side subtends the greater angle". Or Euclid could have gotten the result quite directly without appealing to area and simply arguing in the like way relative to angles. The appeal to area may reveal something of Euclid's thinking, but otherwise is insignificant. Anyway, Common Notions 1, 2, and 3 do not enter into the consideration on area even in Propositions 4 and 6.

Proposition 35 says that "parallelograms which are on the same base and in the same parallels are equal to one another". Euclid has parallelograms ABCD and EBCF on the common base BC and with A, D, E, F on a line parallel to BC. One may consider two cases: (1) AD and EF overlap, and (2) AD and EF are separated…. Triangle ABE is congruent to DCF; and in case (1) the parallelograms are each split into two polygons which are in one to one correspondence in such way that corresponding polygons are congruent, namely, ABE to DCF and EBCD to EBCD. In case (2), which is the only one taken up by Euclid, such a splitting into mutually congruent polygons is not possible;³¹ but if DC and BE meet in G, then after adding triangle DGE to both parallelograms, the resulting polygons can be so split, namely, into ABE and GBC and into DCF and GBC. Hilbert has defined two polygons to be equal in area if they can be decomposed into a finite number of triangles which are respectively congruent in pairs; and to be equal in content if it is possible by the addition of other polygons of equal area to obtain two resulting polygons of equal area (so that with these technical terms, Euclid's theorems refer to content rather than area).³² The proof that polygons P1, P2 equal in area to polygon P3 are themselves equal in area is very much like the proof given above concerning unions of line segments; and the same can be said for polygons equal in content. In a similar way, one can establish "Common Notions 2 and 3" for the content of polygons.

Hilbert's considerations are extremely ingenious and illuminating. One may well hesitate to ascribe such virtuosity to Apollonius; and I would not wish to give credit where credit is not due. However, we have at least to recognize that Common Notions 1, 2, and 3, as far as Book I is concerned, can be proved and that Apollonius thought so.

Proclus has several objections to Apollonius's proof.³³ I think that those that concern Book I can be met, but there is one that cannot be met: namely, that the Common Notions do not refer just to geometrical magnitude, but to magnitudes in general. This seems to be a tacit reference to their use in Book V. In Book V the Common Notions 1, 2, and 3 play a true and valid role; and there they really cannot be proved.

In Book V the term magnitude is not explicitly stated to be undefined; but from some brief remarks of Aristotle,³⁴ from those or Proclus just referred to, and from the Book itself, it is clear that not just geometrical magnitude, but magnitude in general, is the subject, and this understanding eo ipso renders the term undefined. For the same reason, the adding together of magnitudes could hardly be but undefined. Under such circumstances, the "Common Notion" that if equals be added to equals, the sums are equal can clearly not be proved and is truly an axiom in our sense of the word. Book V is not specific enough to fix the author's intention with complete clarity, but I think it is a fair judgement to say that the axiomatic method as understood today, was employed by the originator of the theory.³⁵

We may grant, then, that Common Notions 1, 2, and 3 play a valid role in Book V, but is it true, as Aristotle says, that these notions may be applied without more ado to the special sciences, in particular geometry?³⁶ In Book I we may allow that line segments have magnitude and that magnitude is not to be defined. In Proposition 1, then, Euclid's argument that AC equals BC is correct and follows by a correct application of Common Notion 1. In Proposition 2, however, there is already a difficulty, for here we must know more specifically how to get the difference of two magnitudes; and in particular that if RST are collinear with S between R and T, then the difference of the magnitudes associated with RT and RS is the magnitude associated with ST. Now Aristotle does say that "there need be no assumption as to the meaning of terms if it is clear: just as in the common (axioms) there is no assumption as to what is the meaning of subtracting equals from equals, because it is well-known".³⁷ Thus Aristotle explicitly allows tacit understandings. Now we have no desire to be pedantic, but it appears that with Euclid everything is tacit, unless it appears accidentally, as in the conclusion from Postulates 1 and 2 that there is at least one line on any two points.

Moreover, is it true that only a definition is lacking? For simplicity let us consider the definition of adding together two segments, say AB and CD. One presumably considers three collinear points E, F, G with F between E and G and with AB equal to EF and CD to FG; and then defines the sum of AB and CD to be EG. The question arises, however, whether this "definition" yields a unique sum, for if it doesn't, one does not have a definition: more specifically, if E', F', G' are collinear with F' between E' and G' and with AB equal to E' F' and CD to F' G', the question is whether EG equals E' G'. Of course, if we had the definition, then the equality of EG and E' G' would follow from Common Notion 2; but apparently we first need this equality to get a definition. So what is lacking is not a definition but rather an assumption that sounds like Common Notion 2 and is analogous to it but does not follow from it.

It may be objected that it is pointless and historigraphically unjustified to expect the needed precision from the Greeks, but the originator of the theory of Book V—I'll call him Eudoxus—knew perfectly well how to make the distinctions involved. He starts with magnitude: and let us assume he has made appropriate assumptions governing the notion that magnitude A equals magnitude B and the notion that magnitude A is greater than magnitude B. He then defines the notion that the ratio of A to B is the same as the ratio of C to D and the notion that the ratio of A to B is greater than the ratio of C to D. Then in V, 11 he proves that "ratios which are the same with the same ratio are also the same with one another". For the proof, he does not appeal directly to Common Notion 1, which one might expect of someone lost in the words, but goes back to the definition. Similarly in V, 13 he shows that if A:B=C:D and C:D>E:F, then A:B>E:F. Thus it is clear that he realizes that although two magnitudes determine a ratio, the ratio they determine is not just the pair (A,B), but, as we would say, the ratio is represented by the pair. Now if Eudoxus could see that the "Common Notions" do not suffice here, he could equally well see that they do not suffice in Book I. This would require no greater logical acumen.

Van der Waerden (op. cit., p. 183), after citing Proclus's rather vague statement that Eudoxus "increased the number of so-called general theorems", wonders whether this may not be a reference to Book V; and recalling in this connection the "general understandings" (i.e., Common Notions), adds:

Aristotle already knows the 'so-called general axioms' which form the foundations for all demonstrative sciences and must necessarily be accepted by anyone who wants to gain knowledge. As an example, he always quotes the third of the Euclidean axioms which have just been cited. It is therefore quite possible that Euclid took these axioms, used constantly in the theory of ratios in Book V, from Eudoxus.

This is a most appealing hypothesis, at least as far as Common Notions 1, 2, and 3 are concerned, and I propose to adopt it. According to this hypothesis, then, Common Notions 1, 2, and 3 were formulated in connection with the theory of Book V. This certainly fits with my remark that these Notions play a valid role in Book V but have a rather unassimilated appearance in Book I.

Van der Waerden is surely right when he says that Common Notions 1, 2, and 3 are used in Book V. For example, Common Notion 1 is used in Proposition 11, Common Notion 2 in Proposition 1 and Common Notion 3 in Proposition 6. As to Common Notion 4, this is certainly not used in Book V. However, in this connection one may wonder whether Common Notion 4 is not a garbled version of the statement: Anything is equal to itself, which could very well serve as an axiom in Book V. Thus from A=B and C=D we get, using Common Notion 2, A+C=B+D. If we want similarly to get A+C=B+C, we need that C=C. Common Notion 5 appears nowhere to be used in Book V, but it, too, might very well have been an axiom in a general theory of magnitudes.

Some of the Common Notions were, of course, used long before Eudoxus: for example, it would be impossible to understand the fragment on the lunules of Hippocrates without appeal to the notion that if equals be added to equals, the wholes are equal.³⁸ Historically, then, the Common Notions do really come from the kind of considerations found in book I: the suggested hypothesis merely says that these notions were not formulated as Common Notions before the creation of the theory of Book V. Common Notions 1, 2, and 3, then, would have formed an integral part of Book V. They were then displaced (without due analysis) to the beginning of The Elements, where they appear to serve a bona fide role; but where they also serve to camouflage the lack of an axiomatic method.

6. Postulate 5

We have now disposed of the Common Notions and of Postulates 1 to 4, and there remains Postulate 5; or more precisely said, the Parallel Postulate, for there are two families of manuscripts and the Postulate occurs in different positions in the various manuscripts. Up till 1814, all Greek editions placed Postulates 4 and 5 amongst the Common Notions,³⁹ usually in positions 10 and 11. These editions derive from Theon's recension of The Elements.⁴⁰ Then Peyrard, just before completing the publication of a French translation (having already brought out Books 1, 2, 3, 4, 6, 11, 12), decided to compare the Oxford edition with the manuscripts of the Bibliothèque Imperiale in Paris, and in the course of doing so, he came upon the Vatican Codex 190, now also called P.⁴¹ Altering his plans somewhat, Peyrard brought out a translation based principally on Codex 190 (but also on the other manu-scripts). Here Postulates 4 and 5 are found in their now familiar positions; and Common Notion 9 (in Heiberg's edition) appears as Postulate 6.

Codex 190 was for a long time, and is even now, considered more genuine than Theon's version; but there have been some second thoughts.⁴² In this connection, we would like to ponder the question of how Postulates 4 and 5 got into the positions in which we find them.

Let us start with the idea that some of the Common Notions were interpolated. This seems rather certain, as one of the Common Notions says that the doubles of equals (literally, of the same) are equal, which is just a special case of Common Notion 2. And Proclus complained of the proliferation of the Common Notions. So we can imagine someone joining this activity and interpolating Postulates 4 and 5 amongst the Common Notions, as we find them in Theon's version. Then, however, someone could argue that they are obviously not common notions, but specifically geometric in character, and so displace them to the postulates. On the other hand, if they were originally amongst the postulates, how would they have gotten over into the Common Notions? Geminus, indeed, held that Postulate 4 should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property of right angles.⁴³ But how could Geminus have gotten the notion that a postulate should assert the possibility of a construction? There is nothing in the etymology of the word postulate or in its everyday usage that would suggest such a limitation. What does suggest itself, rather, is that when Geminus learned the subject, a postulate was a construction: his argument is, then, not an argument for displacing a postulate, but rather an objection to having the assumption placed amongst the postulates.

The notion that a postulate must refer to a construction has had a remarkable tenacity. Dehn argued that Postulate 5 asserts the possibility of a construction (namely, of the intersection of certain pairs of lines).⁴⁴ To me such suggestions seem far-fetched and futile efforts to save the phenomenon.

There was in ancient times a diversity of opinion on the meaning of the words postulate, common notion, axiom.⁴⁵ It is well to know these opinions in order to gauge the a priori possibilities; but even if, for example, one knew with precision how Aristotle used them, we could make no certain conclusions about The Elements. We don't know with certainty that The Elements had any Common Notions, or if it did, whether they were called Common Notions. Postulates 1, 2, and 3, the postulates of construction, form an integral part of the method in Book I and their presence at the beginning of Book I under the heading of Postulates has never been challenged. Common Notions 1, 2, and 3 may have been first formulated in reference to Book V, but their presence at the beginning of Book I is intelligible. The Parallel Postulate is of a different character. We may, or even must, grant that Euclid or his predecessors realized that there was a difficulty in the theory of parallels;⁴⁶ Aristotle even mentions a petitio principii committed by "those who draw parallels", but it is not certain he was referring to the postulate (or, say, to the proof of Proposition 29).⁴⁷ But whether they knew what to do about the difficulty is a different matter. Geometry already had two categories of assumptions and somebody, perhaps Euclid, decided to put the bothersome phrase amongst the assumptions. The clear fact is, however, that the Postulate never found a haven in either category.

7. Euclid on the Circumference of a Circle

So much for the positive evidence that the writer of Book I had no axiomatic method in mind: there is also some negative evidence, namely, the absence from The Elements of the theorem that the circumferences of two circles are to each other as their diameters or, what in view of V, 16 on the interchange of means is equivalent to it, that the ratio of the circumference to the diameter is the same for any two circles. It may be difficult to prove that Euclid had heard of this theorem (if he hadn't, there is nothing to explain), but it is not far-fetched to suppose he did. The Babylonians long before Euclid had operated as though they knew it;⁴⁸ and it has been suggested that the Old-Babylonian mathematics influenced the Greek. An analysis of the squaring of the circle with the quadratrix appears to involve knowing the theorem, and there are traditions that Hippias of Elis and Dinostratus, the brother of Menaechmus, both predecessors of Euclid, worked with the quadratrix. Since these traditions are obtained from late authors, it is surely in order to scrutinize them, and one may question them;⁴⁹ but there are no sufficient grounds for rejecting them. Moreover, from some passages in the De Caelo and the Physics of Aristotle and from some in the Mechanica, which is included in his writings though not, indeed, his, one may be certain that the theorem was known to the school of Aristotle and nearly certain that Aristotle himself knew it.⁵⁰

If, then, Euclid had heard of this theorem, how is it that he did not prove it? The simple reason, I believe, is that he did not know how. In fact, even to speak of the ratio of the circumference to the diameter, he would have had to know that these were magnitudes of the same kind. How, for example, would Euclid have shown that an arc of a circle is greater than its chord? Besides the notion of invariance under congruence, Euclid knows no principles of comparison except such as are comprised in "Common Notions": the whole is greater than the part; if equals be added to equals, the wholes are equal; if equals be subtracted from equals, the remainders are equal; there is also the "Axiom of Archimedes" corresponding to V, Definition 4. These serve for comparing areas, but the case of arclength requires some further principle, since there is no way to make a line segment coincide with a circular arc. There is no way to establish from The Elements that the arc is greater than the chord. Euclid, in I, 22, proved that the sum of two sides of a triangle is greater than the third and was ridiculed for his efforts, since even a donkey knows this. But a donkey knows equally well that the arc of a circle is greater than its chord, something Euclid did not know.

From the lack of the theorem one can get a better idea of Euclid's standard of rigor than from his accomplishments, which, naturally, contain errors and "gaps".

8. Aristotle on the Circumference of a Circle

Aristotle frequently makes passing remarks on mathematics. These remarks are often tantalizing in their vagueness, but still one may hope to glean some information from them of the state of mathematics at his time. As to the circumference of a circle in the De Caelo (II.4.287^a 27-28) he says:

Of lines starting from a point and returning to the same, the circle is the shortest.

Here, presumably, he left out the condition that the "lines" should enclose a given area. In any case, the length of a circumference is considered and compared with other lines, including, presumably, straight lines. Again in the De Caelo (II.8.289^b 15) he says:

It is not at all strange, nay it is inevitable, that the speeds of circles should be in proportion to their sizes.

This appears to say that the circumference of circles are to each other as their diameters.

Now, however, we come to a passage in which Aristotle denies that a circular arc and a straight line are even comparable. No reason is given for this and it appears to be taken for granted as well-known. Thus in Physics (VII.4.248^a 10-13; 18-b 17, b10-12)⁵¹ he writes:

The question may be raised whether every motion is comparable with every other or not. If all motions are comparable … we are met by the difficulty … that we shall have a straight line equal to a circle. But these are not comparable; therefore neither are the motions comparable….

Thus Aristotle unquestionably held that the arc of a circle and a straight line are not comparable (though he also appears to say the opposite). If this view was shared by the mathematicians, then our conjectures on the status of the foundations in Euclid's time are confirmed.

9. Archimedes on the Circumference of a Circle

It took a genius like Archimedes to see that some assumptions are needed. In On the Sphere and Cylinder I,⁵² Archimedes makes the "assumption":

1. Of all lines which have the same extremities the straight line is the least.

Euclid would have been thunderstruck! It would never have occurred to him that to prove a theorem ("the arc is greater than the chord"), it is all right to generalize it, and then assume the generalization. In fact, though with the Parallel Postulate he may have admitted he was stumped, there is no clear evidence that he thought it was all right to make any geometrical assumption whatever.

Archimedes himself seems to have been uneasy about "his" axiom (cf. Elements V, Definition 4; cf. also X, 1).⁵³ In the Quadrature of the Parabola, in a prefatory letter to Dositheus, after stating the main result, he adds that in the proof "the following lemma is assumed: that the excess by which the greater of (two) unequal areas exceeds the less can, by being added to itself, be made to exceed any finite area". And he continues:

The earlier geometers have also used this lemma; for it is by the use of this same lemma that they have shown that circles are to one another in the duplicate ratio of their diameters, and that spheres are to one another in the triplicate ratio of their diameters, and further that every pyramid is one third part of the prism which has the same base with the pyramid and equal height; also that every cone is one third part of the cylinder having the same base as the cone and equal height they proved using a certain lemma similar to that aforesaid. And, in the result, each of the aforesaid theorems has been accepted no less than those proved without the lemma. As therefore my work now published has satisfied the same test as the proposition referred to, 1 have written out the proof and send it to you, first as investigated by mechanics, and afterwards too as demonstrated by geometry….

In this passage Archimedes appears to be taking a de fensive attitude, as though he feared some possible criticism. If this were criticism by his contemporaries (external criticism), his position—the ancients did it, so I can do it, too—is surely unworthy of a mathematician. I therefore interpret it as internal criticism: he was really worried about the assumption, but resolved his doubts by accepting it.

A subsequent work, On the Sphere and Cylinder I, also has a prefatory letter to Dositheus. In this, after stating the main results, Archimedes adds:

Now these properties were all along naturally inherent in the figures referred to, but remained unknown to those who were before my time engaged in the study of geometry. Having, however, now discovered that the properties are true of these figures, I cannot feel any hesitation in setting them side by side with my former investigations and with those of the theorems of Eudoxus on solids which are held to be the most irrefragably established, namely, that any pyramid is one third part of the prism which has the same base with the pyramid and equal height, and that any cone is one third part of the cylinder which has the same base with the cone and equal height. For, though these properties also were naturally inherent in the figures all along, yet they were in fact unknown to the many geometers who lived before Eudoxus, and had not been observed by anyone.⁵⁴ Now, however, it will be open to those who possess the requisite ability to examine these discoveries of mine. They ought to have been published while Conon was still alive, for I should conceive that he would best have been able to grasp them and pronounce upon them the appropriate verdict; but as I judge it well to communicate them to those who are conversant with mathematics, I send them to you with the proofs written out, which it will be open to mathematicians to examine. Farewell….

The work itself then begins with six definitions and five assumptions, of which the fifth is the "Archimedean" one.

On this preface one may make two observations: first, that no mention is made in it of assumptions, though four new ones will follow. Second, that (though it seems a bit jocular) there is a clearly avowed Platonism and it is not clear what the relevance of this could be.

Though it's harder to see what Archimedes is doing here, I think he is doing the same thing, namely, worrying about the nature of his assumptions. He invokes the name of Eudoxus, really in reference to Assumption 5, but this is veiled by a reference only to some theorems in the proofs of which it was used. The four new assumptions deserved a remark, but Archimedes could not bring himself to make one.⁵⁵

Thus we see with what doubts Archimedes used the axiomatic method.⁵⁶

10. Deductive Mathematics in the Ancient East

The thesis that Euclid developed geometry axiomatically is part of the broader one that deductive mathematics started in Greece; and this is (or was) part of the view that mathematics as a body of knowledge worthy of the name of Science did not exist in the ancient oriental civilizations. At the beginning of this century, there was, indeed, little known about the mathematics of (Old-) Babylonia and Egypt. From Babylonia one had a table of squares up to 60 χ 60 and an astronomical text giving the magnitude of the illuminated portion of the moon for every day from new to full moon.⁵⁷ From Egypt one already had the Rhind mathematical papyrus, but paleographers could argue whether the area of a triangle was correctly computed in it. So, W. W. R. Ball could easily bring himself to write:⁵⁸

The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks.

The monumental work of O. Neugebauer published in the thirties quite transformed our notions of ancient oriental mathematics, at least for Babylonia.⁵⁹ A magnificent mathematics, on a high level indeed, was disclosed.

There are two types of mathematical texts from Old-Babylonia: the table texts and the problem texts. In a typical problem, one is given some relations between the dimensions of a geometrical figure and is required to find the dimensions. The solutions show step by step how to get the answers. For example. Text AO 8862 begins as follows:⁶⁰

Length, width. I have multiplied length and width, thus obtaining the area. Then I added to the area, the excess of length over width: 3,3 (i.e. 183 was the result).⁶¹ Moreover, I have added length and width: 27. Required length, width, and area.

(given:) 27 and 3,3 the sums

(result:) 15 length 3,0 area
12 width

One follows this method:
27+3,3=3, 30
2+27=29.

Take one half of 29 (this gives 14; 30).
14; 30x14; 30=3, 30; 15
3, 30; 15-3, 30=0; 15.

The square root of 0; 15 is 0; 30.
14; 30+0; 30=15 length
14; 30-0; 30=14 width

Subtract 2, which has been added to 27, from 14, the width. 12 is the actual width.

I have multiplied 15 length by 12 width:
15×12=3, 0 area
15-12=3
3,0+3=3,3.

Some people like to adopt a superior attitude to this way of conveying mathematics: no general theorems are stated. Yet the text itself speaks of a "method" and clearly is conveying a general method via an example. Analysis of the problem shows that considerable ingenuity was employed in getting the answers.⁶² But quite aside from the ingenuity we do have a general statement ("I have multiplied length by width, thus obtaining the area"). It is, or should be, clear that someone had to do some thinking to get this formulation. We can hardly suppose that length was multiplied by width just because this seemed to be a nice thing to do with two numbers; and the result called "area" for lack of a better term.

Still, one has to admit, unless one wishes to argue that a verification (as in the last three lines, above) is a proof, that the Babylonian mathematics shows no evidence of proof. The Babylonians knew that the area of a trapezoid is computed as one half the sum of the parallel sides times the width, but the texts contain no proof of this. As Neugebauer (Exact Sciences, p. 45) says: "It must also be underlined that we have not the faintest idea about anything amounting to a 'proof concerning the relations between geometrical magnitudes." One may argue if one wishes that the Babylonians (or predecessors) must have had some "proof of their formulae (as, for example, for the area of a trapezoid), but if so it has not made its appearance in the surviving evidence.

The advent of Greek mathematics used to be explained by appeal to Greek "genius" or looked upon as a miracle. Since, however, we are not inclined to believe in miracles, the disclosure of the Babylonian mathematics came as something of a relief. Though the lines of transmission were far from clear—van der Waerden (op. cit., p. 89) called the Old-Babylonian mathematics "dead wisdom" in the 6^th century B.C.—the Greek mathematics was now viewed as a derivative of the Babylonian: or, more presisely said, the latter was regarded as the basis of the former.

Basing himself on the work of Zeuthen and Neugebauer, van der Waerden (op. cit., pp. 118-125) expounded this view with great clarity and cogency. According to this exposition, the discovery of incommensurables prevented the Greeks from dealing with magnitude via number; and the Babylonian algebra of numbers was replaced by an algebra of segments, areas, and volumes ("geometric algebra"), the elements of which are given in The Elements, Book II. In this, traces of the Babylonian method of solving quadratics can be found. This algebra was further developed, as in Euclid's Data, but remained cumbersome as a result of the notion that a product of more than three line segments is meaningless.

E. T. Bell (Development of Mathematics, p. 31) at one point briefly expresses skepticism:

Historically the most remarkable thing about this rapid progress [of the Babylonians] in the subjugation of number is that it appears to have been ignored by the Greeks of the sixth century B.C. For what now seems to us the simplest, most natural development of mathematics this was a calamity. The fact that it happened casts a slight shadow of doubt on the vaunted intelligence of the early Greek mind. But since to press this point would be tantamount to historical blasphemy, we merely suggest that the mathematically informed observer examine the evidence and reach his own conclusions, even at the risk of upsetting sacrosanct tradition.

Still Bell (ibid., p. 46) concludes:

Unless, if ever, evidence is uncovered proving that the Greeks were anticipated in their conception of mathematics as a deductive science, the greatest contribution of the Babylonians and the Egyptians must remain their unconscious part in helping to make possible the golden ages of Eudoxus and Archimedes. It was enough, and should preserve their memory as long as mathematics lasts.

Meanwhile, the Sulvasutras, an ancient Indian sacred work on altar constructions, had been all but forgotten. Van der Waerden and Bell do not so much as even mention it, though its contents are briefly described in Cantor's History, which both cite. This ancient work contains some geometric algebra (for example, a rectangle is converted into a square in a typical geometro-algebraic way).⁶³ It follows that the Indian geometry is a derivative of the Greek or that the explanations of Neugebauer and van der Waerden cannot very well be correct.

The Sulvasutras were translated by G. Thibaut in 1875 and showed that the Indian priests possessed no little mathematical knowledge: in particular, the Theorem of Pythagoras is stated there in complete generality. Thibaut's work⁶⁴ received considerable attention at the time, because of the implications it might have had for Greek studies. It was also of interest from the point of view of the history of mathematics, since nothing remotely approaching the level of the Sulvasutras was known from the ancient world outside of Greece.

The dating of the Sulvasutras was, of course, vital. Thibaut himself, though he fixed their position relative to the other sacred literature, did not, at first, assign them an absolute date. In 1877 Cantor, realizing the importance of Thibaut's work, began a comparative study of Greek and Indian mathematics. He concluded that the Indian geometry was a derivative of Alexandrian knowlege, an opinion he held for some twenty-five years before finally renouncing it.⁶⁵

There was no general agreement on the date: those who favored Greece tended to put the Sulvasutras late. The attitude of the Greek scholars may, perhaps, be judged from a remark of John Burnet, who in his Greek Philosophy, p. 9, n.l, wrote: "It is a pity that M. [G.] Milhaud has been persuaded to accept an early date for the Sulvasutras in his Nouvelles études (1911), pp. 109 sqq." The Sanskrit scholars had objected to the late date (as, for example, the 100 B.C. of Cantor) and in 1899 Thibaut ventured to assign the fourth or the third century B.C. as the latest possible date for the composition of the Sulvasutras (it being understood that this refers to a codification of far older materials).⁶⁶ But there wasn't much else that the Sanskrit scholars could do about it, except to let their objections gather dust on the shelves.

In 1928 Neugebauer published a paper on the history of the Pythagorean Theorem.⁶⁷ He was already busy with the translation of Old-Babylonian mathematical texts. The main point of the paper is to disclose the existence of the Theorem of Pythagoras well over a thousand years before Pythagoras. In this connection he mentions the Sulvasutras and writes that "the difficulties [involved in the view] of a direct borrowing by the Greeks from India fall away on the assumption of a common origin in Babylonia." This would surely be true if the common elements in the Greek and Indian geometry originated in, or at least passed through, Babylonia. While that is the case for the Theorem of Pythagoras, it was not Neugebauer's view for other common elements, especially the geometric algebra, either in 1928 or later.

In 1957 Neugebauer Exact Sciences, p. 35) again briefly mentioned the Sulvasutras. Here he considered that its present form might be dated to the third or fourth century B.C. NO argument or reference is given.

In my paper on the Ritual Origin of Geometry I tried to go over the old battlegrounds. The Sanskrit scholars on the whole put the Sutra period between 500 and 200 B.C. But the Brahmanas, which according to these scholars belong to an older stratum of literature, already give the rules for the sizes and variation in shapes of the altars. According to Thibaut, "the earliest enumeration [of the different shapes] we find in the Taittiriya Samhita V, 4, 11." Bürk⁶⁸ placed the Samhita period no later than the eighth century B.C., whereas Keith, who brought out an English translation of the Taittiriya Samhita placed it no later than the sixth. There are even vague references to altar construction in the Vedic hymns, which are vastly older according to these scholars.

Where the argument depends on linguistic or literary evaluations I cannot check these estimates directly, since I know no Sanskrit. However, there are other arguments. I have no intention of repeating them, even briefly, and refer the reader to the cited place; there I have concluded that not only are the Sulvasutras pre-Greek, but that even the Old-Babylonian mathematics derives from a system of practices much like those disclosed in the Indian sacred works. These conclusions are not really at issue for the moment, however, since I merely wish to examine the question of whether the Sulvasutras show the notion of proof.

We have already cited Neugebauer on the absence of proof in pre-Greek mathematics. This is the standard opinion. The closest to a contrasting opinion is that of O. Becker, who does, indeed, agree that there are no proofs in the Sulvasutras, but at least he observes that it contains theorems formulated with complete generality.⁶⁹ This he considers "an essential, even decisive step, beyond the Babylonian mathematics". This may not be exactly put because of the chronological connotation, but unquestionably Becker has correctly indicated a difference in the two mathematics. Becker accepts, though somewhat hesitatingly, an early date for the Indian geometry (the 8^th century B.C.) Since the Sulvasutras have the Theorem of Pythagoras, Becker (op. cit., p. 41) looks to Old-Babylonia for the source of Indian geometry. He has, however, not apprehended the difficulties of this view. The Sulvasutras convert a rectangle (say a by b) in a typical geometro-algebraic way; the Old-Babylonian would simply multiply a by b and take the square root. The Indian priest constructs the side of the required square; the Old-Babylonian computes it. So the geometric algebra of the Indian could not very well have come from the Old-Babylonians. And it couldn't have very well come from the Greeks, at least not in the 8^th centuryB.C., as Greek mathematics is only supposed to have started about 600 B.C. And if the Greeks got its geometric algebra from the Indians, then the history of Greek mathematics has to be rewritten.⁷⁰…

[I]f one sticks to the doctrine that deductive mathematics started in Greece, one will have to reject an early date for Indian geometry, or at least the part of it that contains proofs. One will conclude that the notion of proof got to India from Greece, and then one will be stuck with the problem of how such a sophisticated notion could be transmitted and such simple notions as triangle or angle (except for right angle) were not transmitted. On the other hand (and third), if one accepts the early date for India and if one considers that proof is a sophisticated notion not likely to be generated spontaneously, but to be the product of certain special circumstances, one will conclude that proof came from India to Greece; or from some third, unknown, possibly pre-Old-Babylonian source. Of course, there is the possibility that the notion was independently invented, once in Greece and once in India, the order being immaterial.

Since the origin of geometric algebra is crucial, we allow ourselves to add a remark on this. Let x, y be the sides of a rectangle with x>y. The passage from the Sulvasutras on converting a rectangle into a square (cited above) can be translated in a simple, direct way into algebra to yield the formula:

or, after a trifling reformulation,

The formula (2) has three terms, so that if two are known, the third is determined, as the sum or difference of the other two. Now let us imagine that a clever Old-Babylonian, skilled in computation, were shown this formula. Then it is not much harder to imagine him saying to himself: "So, if I have the product of two numbers and their sum, I can find their difference; or if I have the product and the difference, I can find the sum. Now I can invent a new class of problems." If things really went that way, or somewhat that way, the Old-Babylonians were well on their way to handling quadratic equations.

So, we have used geometric algebra, in particular the problem of converting a rectangle into a square, to motivate the Old-Babylonian problem: given the product of two numbers and their sum or difference, to find the numbers (the geometric algebra can in turn be motivated by the theology of altar construction; see the Ritual Origin paper cited). Van der Waerden has proceeded in just the opposite direction. He starts from the Old-Babylonian problem and their solution of it, presumably
from the observation that; the
Pythagoreans, however, because of the discovery of incommensurables, converted this into geometric algebra and in particular formulated Euclid's Elements II, 5 and a variant II, 6; the procedure is completed by II, 14, the conversion of a rectangle into a square, which is the counterpart of taking a square root.

But what is the motivation of the Old-Babylonian problem? Van der Waerden says nothing on this, nor does Neugebauer. S. Gandz⁷¹ considers it obvious that the problems xy=a, x+y=b and xy=a, x−y=b started from the rectangle, and thinks he can see a development from there to their handling of quadratic equations (and he may well be right). But the problems, except for terminology, have nothing to do with geometry and can be phrased in purely arithmetic terms. Then why drag in the rectangle? My answer is: It was not dragged in. It was already there.

The above remarks accord with my thesis that the elements of geometry as found in the ancient civilizations, in Greece, Babylonia, Egypt, India, and China, are a derivative of a system of ritual practices as disclosed in the Sulvasutras. They also suggest that the ritualists knew some deductive mathematics. Did they, then, also have the axiomatic method? Of course not, but they were concerned with exact thought. The ritual in general was to be carried out exactly; for example, "the laws of phonetics were investigated because the wrath of the gods followed the wrong pronunciation of a single letter of the sacrificial formulas."⁷² Why the ritualists should want so much to be right is hard to say, unless it is that they were concerned with symbolic action and that there is not much point to symbolic action unless it is right.

11. The Axiomatic Method in Modern Times

To complete our account of the axiomatic method, especially with reference to geometry, let us still sketch its history in modern times. The axiomatic method as understood today was initiated by Moritz Pasch in his book "Vorlesungen ūber neuere Geometrie" (Leipzig 1882, 2nd ed. Berlin 1926). This method consists of isolating from a given study certain notions that are left undefined, and are expressly declared to be such (the so-called Kernbegriffe in Pasch's terminology of 1916), and certain theorems that are accepted without proof (the so-called Kernsätze, i.e., axioms); from this initial fund of notions and theorems, the other notions are to be defined and the theorems proved using only logical relations, and without appeal to experience or intuition. The resulting theory takes the form of purely logical relations between undefined concepts.

The nature of the initial theorems had been an issue since the time of Aristotle. Pasch always prefaces his choice of an axiom with an informal explanation, and in doing so a definite philosophy for choosing the axioms is disclosed. According to Pasch, the initial notions and theorems should be founded on observations. Thus the notion of point is allowed, but not that of line, as no one has ever observed a complete (straight) line; rather the notion of segment is taken as primitive. Similarly, a planar surface, but not a plane, is primitive.⁷³

Pasch's analysis relating to the order of points on a line and in the plane is quite striking and instructive. Every student draws diagrams and sees that if a point B is between A and C, then C is not between A and B; or that every line divides the plane into two parts. But no one before Pasch had put down a basis for dealing logically with such observations. Perhaps these matters were considered to be too obvious to waste words about. The result, however, of such neglect is that one must be constantly referring to one's intuition, so that the logical status of what is being done cannot become clear. According to Pasch, the appeal to intuition formally ceases once the Kernbegriffe and Kernsätze are put down.

According to Aristotle a science is a body of knowledge and one way to know something is to prove it (Analytica Posteriori I. 2). But then the question is: how can we know the first propositions, the axioms? According to Aristotle, their truth is apprehended by an infallible intuition (ibid., II. 29, 100^b6). Pasch instead draws a sharp line between the intuitive background and the deductive system itself. The first propositions, the axioms or Kernsätze, may correspond to transparent truths, but they are themselves not truths, transparent or otherwise: they are merely starting points of strictly logical deductions. Nor do the propositions, the later as well as the first, have any meaning, since the primitive terms are undefined. The resulting axiom system is therefore also not a body of knowledge.

Although deduction is a prominent feature of The Elements, its contents and the history of the parallel postulate show that geometry was conceived of as the study of a certain definite object (namely, "external space"). With the invention of non-Euclidean geometry around 1800, it began to dawn on mathematicians that their concern is with deduction, and not with some supposed external reality. Pasch (as we said) drew a sharp line separating the empirical background from the axiomatic set-up; but he still thought the goal was a knowledge of external space. External space remains too close for comfort. With Fano's miniature projective plane of just seven points and seven lines, of 1892, the revolution can be said to have been completed. Hilbert consolidated it with his many ingenious considerations. Except for a quotation from Kant ("All human knowledge begins with intuitions, thence passes to concepts, and ends with ideas"), Hilbert has scarcely a word to say about external space.

12. Summary

Our main thesis is that Euclid did not have an axiom system in mind, and did not develop geometry axiomatically, in Book I of The Elements. The main impediment to establishing this thesis is that the Book does have explicitly stated assumptions, indeed, several sets of them: there are the Postulates and the Common Notions. The Postulates divide into two groups: the postulates of construction; and Postulate 4 and 5. The construction postulates are bona fide and are axioms in a sense: they serve to control the straightedge and compass constructions. But they are not axioms for a development of geometry and, indeed, tell us nothing about space, except incidentally that there is a line on any two points. As to Postulates 4 and 5, there is the tendency to attribute profound significance to them: and whoever made Postulate 5 into an explicit assumption, if he did not do it as a simple comment, deserves great credit; but Postulate 4 is not needed, as already observed by the ancients, and its inclusion as an assumption is inept. The first part of Book I, say the first 23 propositions, appears to convey in systematic form an older theory of straightedge and compass constructions. To do this, some argumentation not directly of a constructive nature is needed, and this argumentation, not counting the constructive part, rests on no explicitly made geometric assumption whatever (except possibly Common Notion 4).

The Common Notions, too, separate into two groups. Common Notion 4 is geometric in nature, and hence not really a common notion. This casts doubt on its authenticity; and some ancients (e.g., Heron) accepted only the first three Common Notions. As to these, they appear really to be part of Book V and to have been inserted into Book I without due analysis.

Thus we are left with only one bona fide geometric assumption, namely, the Parallel Postulate. The definitions, Postulate 4, and the Common Notions could all go, and, indeed, omitting them would improve the Book.

If Book I was supposed to have been founded axiomatically, we can only conclude that this was most ineptly done. On the other hand, it is clear that the Book itself is masterfully conceived. The author, beginning with Proposition 1, wastes no words, goes directly to his goal, keeps to essentials, and does not allow himself to get sidetracked. The first 26 propositions give us the basic constructions and basic facts about the mutual relations of the sides and angles of triangles. Then come 7 propositions establishing the theory of parallels, which is needed, first, for the theory of area, and then in Proposition 46 for constructing a square. The author then still adroitly includes the famous Theorem of Pythagoras and its converse. The Book sans Common Notions and Postulate 4 (and the definitions) is worthy of a great mathematician. We see then how unjust it is to attribute an axiomatic method to the author.

Bolyai, writing to his father about his work on the theory of parallels, said: "From nothing I have created another wholly new world." Euclid might very well have taken this proud declaration as his motto.

Notes

¹ Thales 585, Pythagoras 550, Oenopides 450, Hippocrates of Chios 430, Democritus 430, Hippias 420, Plato 380, Eudoxus 370, Menaechmus 350, Dinostratus 350, Aristotle 340, Alexander the Great 333, Euclid 300, Archimedes 250, Apollonius 210—all B.C.; Heron 60 A.D., Proclus 450 A.D. These dates are taken from B. L. van der Waerden's Science Awakening, pp. 82, 129, 146.

² See T. L. Heath, The Thirteen Books of Euclid's Elements, Vol. 1, Chap. V. J. L. Heiberg's edition has recently been edited by E. S. Stamatis.

³ Heath, op. cit., vol. 1, pp. 223, 249.

⁴ Heath, op. cit., p. 154f, gives these as follows: "Postulates. Let the following be postulated: 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 centre and distance. 4. That all right angles are equal to one another. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles. Common Notions. 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part." These Common Notions are Heiberg's 1, 2, 3, 7, and 8.

⁵ See Glenn R. Morrow's translation of Proclus's Commentary on the First Book of Euclid's Elements. In giving the page references in this book, I will also parenthetically include the corresponding ones in Friedlen's text. Thus: for the postulates and Common Notions see Proclus, op. cit., pp. 140-155 (F. 178-198).

⁶ Heath, op. cit., pp. 225, 232. Paul Tannery, "Sur l'authenticité des axiomes d'Euclide" in Mémoires Scientifiques, vol. 2, pp. 48-63.

⁷Op. cit., pp. 199, 195, 124.

⁸ Proclus, op. cit., p. 63f (F. 77f).

⁹ As far as I know. Any such negative statement will usually have to be accompanied (perhaps tacitly) by such a qualification.

¹⁰ The reason Hilbert avoids circles is that he does not wish to put down any axiom giving a condition for two circles to meet.

¹¹ The brackets will be commented on below.

¹²Analytica Posteriora, I. 10, 76a31-77a 4; cf. Heath, op. cit., p. 117.

¹³ Proclus, op. cit., p. 154 (F. 196) in reference to the Common Notion (9 in Heiberg) that two straight lines do not enclose (or contain) a space. Cf. Heath, op. cit., p. 232.

¹⁴ The lack of a citation can hardly be an argument for the absence of a Common Notion (or Postulate), as Euclid seldom cites a reason.

¹⁵ Proclus, op. cit., p. 154 (F. 196). Cf. Heath, op. cit., p. 222, 225.

¹⁶ Heath, op. cit., p. 223.

¹⁷ Heath, op. cit., p. 232. See also F. Peyrard, Les Oevres d'Euclid, especially p. 5.

¹⁸ Proclus, op. cit., p. 147F (F. 188f); Cf. Heath, op. cit., p. 200. Curiously, Proclus considers that the Postulate is a common notion and that it can be proved.

¹⁹ It was early noted that the second half of Proposition 5 is never applied in The Elements. Strictly, this is true, but in Proposition 7 (that on a given side of AB there cannot be constructed distinct triangles ACB, ADB with AC = AD, BC = BD) Euclid omitted one case, and the second half of Proposition 5 is needed for this missing case. Proclus has put the matter in somewhat apologetic terms. (Op. cit., p. 192f (F. 247f).)

²⁰ Let ABC be a right triangle with hypotenuse BC and let AD be the perpendicular dropped from A to BC. The Theorem of Pythagoras is proved twice in The Elements, first in I, 47 and then, in generalized form, in VI, 31. The leading idea of both proofs is that the square on AB is equal to the rectangle on BD and BC; but in VI, 31 this equality is transparent from the similarity of triangles ABC and DBA, whereas in I, 47 the equality is rather ad hoc. One might be prone, then, to prefer the second proof; but the first proof, from a logical point of view, is better, as the second uses the "Axiom of Archimedes" for line segments (via V, Definition 4) and the first doesn't.

²¹ Tannery (op. cit., p. 55) makes the same point, admitting that what he considers to be the bungling emendation of Euclid's work may realy be Euclid's emendation of a predecessor's work.

²² Professor Freudenthal thinks he knows why Euclid included Postulate 4 and has promised to tell us why (Nieuw Archief voor Wiskunde, vol. 5 (1957), p. 122), but I have not been able to locate the promised disclosure. Eighteen years have elapsed since 1957. Perhaps Professor Freudenthal has found reason not to publish his conjectures.

²³ Heath, op. cit., p. 195.

²⁴ In referring to Postulate 1 we shall sometimes (as here) tacitly include a reference to Postulate 2.

²⁵ A. Seidenberg, "The Ritual Origin of Geometry", Archive for History of Exact Sciences, vol. 5 (1962), pp. 488-527.

²⁶ Proclus, op. cit., p. 152f (F. 194f); cf. Heath, op. cit., p. 222.

²⁷ It is, to be sure, not a logical mistake to use unnecessary but explicitly made assumptions.

²⁸ Of course, Heath's acceptance of the contention, that Common Notions 4 and 5 are interpolated, does not establish this contention as a definite fact; but we may accept it temporarily for polemical purposes.

²⁹ This is an adaptation of a proof of Hilbert on which we will comment in a moment.

³⁰ Proposition 12 is to drop a perpendicular and Proposition 23 is to transfer an angle. Both of these constructions have been attributed to Oenopides. Heath (Greek Mathematics, vol. 1, p. 1751) thinks that "the geometrical reputation of Oenopides could not have rested on the mere solution of such simple problems as these", and hence that "Oenopides's significance [may lie] in improvements of method from the point of view of theory." K. von Pritz (Pauly-Wissowa-Kroll Real-encyclopädie der Classischen Altertumswissenschaft XVII, 2267 ff) has expressed similar views. In The Elements all the propositions before 23, except 6, 12, 14, 17, and 21, come into play for 23. Proposition 6 (already commented on above) is nowhere used in Book I and is placed after 5 merely because it is the converse of that proposition. Proposition 12 is the other construction attributed to Oenopides. Proposition 17 is a variant of 16. Proposition 21, like the second half of Proposition 5 (see note 19 above), is nowhere applied in Book I but may have been intended for a case omitted in Proposition 24 (see Heath's Elements, vol. 1, p. 297). Of these propositions, only 14 is explicitly applied in Book I (namely, in 45 and 47). So it would be no exaggeration to say that the first part of Book I (1-23) is organized around the two constructions attributed to Oenopides. One might well conjecture that this first part—its organization, that is—is due to Oenopides.

³¹ In case 2 if AD≥DE, or more generally, if nAD≥DE for some integer n, then by repeating the argument given in case 1, one could still get the desired splitting. Thus one can get this splitting if the so-called "Axiom of Archimedes" holds for line segments; but otherwise not. Thus Book I manages without the "Axiom of Archimedes". It is tantalizing to ponder the question whether the absence of this axiom was purposefully arranged; or whether it might not be merely a side result of the desire to give simply expressed proofs.

³² Hilbert, op. cit., p. 70. The translations equal in area and equal in content for Hilbert's zerlegungsgleich and ergänzungsgleich, respectively, are taken from E. J. Townsend's translation of the first edition of Hilbert's work.

³³ Proclus, op. cit., 152ff (F. 194ff); cf. Heath, Euclid's Elements, vol. 1, p. 222f.

³⁴Op. cit. 15; cf. van der Waerden, op. cit., p. 177.

³⁵ Actually, there are two "errors" in Book V that go against the idea that Book is dealing with magnitude in general. First, in Proposition 5, which essentially asserts that, for any positive integer m, ma−mb=m(a−b), the proof divides ma−mb into m equal parts, something that cannot be done to an arbitrary magnitude; it should, rather, have started with a−b and multiplied by m (Heath, op. cit., vol. 2, p. 146f). Then in Proposition 18, which essentially asserts that if a:b=c:d, then (a+b) : b=(c+d):d, the proof proceeds by reductio ad absurdum and says to take d±x as a fourth proportional to a+b, b, and c+d, again something that cannot be done for magnitudes in general (cf., loc, cit., p. 170ff). Both errors are easily fixed. One can hardly attribute these "errors" to the originator of the theory, since they lose sight of the immediate goal (namely, a theory of magnitudes in general); so the errors are considered (quite plausibly) to be due to poor editing. (There is a third error, in Proposition 10, where it is tacitly assumed that A:C>B:C and B:C>A:C cannot hold simultaneously.)

³⁶ See the passage cited at note 12, above.

³⁷ Analytica Posteriora, I. 10, 76a 31-77a 4; cf. Heath, op. cit., vol. 1, p. 118.

³⁸Cf. van der Waerden, op. cit., p. 132f.

³⁹Cf. F. Peyrard, Les Oeuvres d'Euclide, p. xix.

⁴⁰ Heath, Greek Mathematics, vol. 1, p. 360.

⁴¹ Peyrard, op. cit., p. xiii.

⁴² Heath, Euclid's Elements, vol. 1, p. 52f.

⁴³ According to Heath (Greek Mathematics, vol. 2, p. 223), the view most generally accepted is that Geminus wrote about 73-67 B.C.

⁴⁴ M. Pasch, Vorlesunge n über neuere Geometrie (2^nd ed.), with an appendix Die Grundlegung der Geometrie in historischer Entwicklung. by M. Dehn, pp. 189, 249.

⁴⁵ For a study on the Greek terminology, see K. von Fritz, "Die APXAI in der griechischen Mathematik", Archiv fur Begriffsgeschichte I (1955), p. 13f.

⁴⁶ There are some indications in Book I that Euclid (i.e., the author of Book I) held a definite view in regard to the Parallel Postulate. Proposition 17, which says that any two angles of a triangle together make less than two right angles, though it is a simple vari ant of Proposition 16, which is applied, is itself nowhere applied in the Book. Since the Proposition would be an obvious corollary of Proposition 32, that the angles of a triangle make two right angles, it would appear that the author wished to emphasize that Proposition 17 does not depend on the Parallel Postulate (or, say, on Proposition 29). Proposition 26 yields a similar indication. This proposition says that "if two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will have the remaining sides equal to the remaining sides and the remaining angle equal to the remaining angle." Here one may observe, first, that Proposition 26 really embraces two propositions; this is a little unusual. Now the first part of the Proposition, for organizational reasons, falls quite naturally where it is; but the second part would again be an obvious corollary of Proposition 32 (and the first part). So here, too, it is conceivable that Euclid wished to emphasize the independence of the second half of Proposition 26 from the Parallel Postulate.

⁴⁷ Heath, Euclid's Elements, vol. 1, p. 191f.

⁴⁸ See A. Seidenberg, "On the area of a semi-circle", Archive for History of Exact Sciences, vol. 9 (1972), p. 185ff.

⁴⁹ See A. J. E. M. Smeur, "On the value equivalent to π in ancient mathematical texts", Archive for History of Exact Sciences, vol. 6 (1970), pp. 249-270.

⁵⁰See below; and cf. Seidenberg, op. cit., p. 199ff.

⁵¹ Or see Heath, Mathematics in Aristotle, for this and the above passages.

⁵² See T. L. Heath, The Works of Archimedes.

⁵³ Definition 4 of Book V says that: "Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another." This looks like, is, and functions as, a definition in Book V, but it is the source of hidden assumptions for the later Books. Thus Euclid tacitly assumes that line segments, areas, etc. have a ratio: this is an assumption, not merely a definition. Archimedes (in On the Sphere and Cylinder) explicitly formulated the assumption as follows: "The larger of unequal lines, areas, or solids exceeds the smaller in such a way that the difference, added to itself, can exceed any individual of the type to which the two mutually compared magnitudes belong." One can refer (as often) to Definition 4 loosely as "Archimedes' Axiom", but it is not quite the same. (Cf. Van der Waerden, op. cit., p. 186.)

⁵⁴ Archimedes corrects this assertion in The Method, where Democritus is credited with asserting the propositions.

⁵⁵ The first and fifth assumptions have already been cited. Assumption 2 says: "Of other lines in a plane having the same extremities, [any two] such are unequal whenever both are concave in the same direction and one of them is either wholy included between the other and the straight line which has the same extremities with it, or is partly included by, and is partly common with, the other; and that [line] which is included is the lesser." Assumptions 3 and 4 are the analogues of 1 and 2 for surfaces.

⁵⁶ J. Hjemslev ("Eudoxus' Axiom and Archimedes' Lemma". Centaurus, vol. 1 (1950). pp. 1-11) has insisted on a difference between "Archimedes' lemma" and Definition 4 of Book V; but unfortunately he calls Definition 4 "Eudoxus' Axiom", whereas it is not an axiom. Hjemslev sees an essential difference between Archimedes and his predecessors and considers that this difference resides in Archimedes' understanding of his "lemma" (Assumption 5 of On the Sphere and Cylinder). I would say, rather, that the essential difference lies in his willingness to posit the first four assumptions (of the Sphere and Cylinder), assumptions that allow him to compare curved lines and curved surfaces. In this way, to be sure, Definition 4 can be applied to magnitudes whose comparison was possibly not contemplated by Eudoxus.

(Archimedes' treatment is (I believe) not quite complete. He tacitly assumes, for example, that if ACB is a circular arc, then the magnitude of ACB is the sum of the magnitudes of AC and of CB, whereas this should have been proved.)

⁵⁷ F. Cajori, A History of Mathematics, p. 4ff. There were, however, plenty of mathematical astronomical texts from the Seleucid (post-Alexandrian) period; see O. Neugebauer, The Exact Sciences in Antiquity, Chap. V. especially p. 103. See also Cajori, op. cit., p. 8.

⁵⁸A Short Account of the History of Mathematics, 3^rd ed. (1901), p. 1.

⁵⁹ "Mathematische Keilschrifttexte", Quellen und Studien zu Geschichte der Math., Astro., und Physik, Abt. A, vol. 3 (1935).

⁶⁰Cf. Van der Waerden, op. cit., p. 63.

⁶¹ The Babylonians had a 60-system, which it is convenient to keep in translation: thus 3,3 means 3x60+3, i.e., 183.

⁶² See van der Waerden, op. cit., p. 63ff. The line: "Subtract 2… from 14 the width. 12 is the actual width", shows that a "new" width=(old) width+2 has been introduced. The Old-Babylonian, however, made a mistake and lost a solution, for though, of course, (actual) width ≤ length, it is not necessarily the case that "new" width ≤ length.

⁶³ See Seidenberg, "Ritual Origin of Geometry", p. 524.

⁶⁴ "On the Śulvasútras", J. Asiatic Society of Bengal, vol. 44:1 (1875). For reference to the translation, see Seidenberg, op. cit., p. 527.

⁶⁵Archiv der Mathematik und Physik, vol. 8 (1904).

⁶⁶ A. A. Macdonell, A History of Sanskrit Literature, pp. 244, 259, 424. Thibaut, "Astronomie, Astrologie, und Mathematik". Grundriss der Indio-Arischen Philologie und Altertumskunde, vol. 3:9 (1899), p. 78, Cf. Seidenberg, op. cit., p. 505.

⁶⁷ "Zur Geschichte des Pythagoräischen Lehrsatzes", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (Math.-Phys. Klasse), 1928, p. 47.

⁶⁸ "Das Āpastamba Śulva-Sūtra", Zeit. d. deutschen Morgenländischen Gesellschaft, vol. 55 (1901), p. 553.

⁶⁹Geschichte der Mathematik (with J. E. Hoffman), pp. 39, 41.

⁷⁰ In his review of Becker's work (Centaurus, vol. 2 (1953), p. 364), Neugebauer has no comment on Becker's acceptance of the eighth century B.C. for the Sulvasutras, though he takes exception to a number of other points.

⁷¹Osiris, vol. 3 (1938), p. 405.

⁷² Thibaut, On the Sulvasutras, p. 227.

⁷³ I don't quite understand Pasch's position, since no one has ever observed a point either.

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