I argue for an interpretation of Euclid’s postulates as principles grounding the science of measurement. Euclid’s

There is much that remains mysterious about Euclid and his seminal work

Euclid’s five postulates will be one of my main points of interest. They are as follows:^{1}

What sort of thing are these postulates, and what role are they supposed to play? They are clearly supposed to be basic principles from which the theorems of the Elements can be deduced (on some conception or other of deduction).^{2} But basic in what sense? In a paper on the nature of mathematical postulates

Because the Greeks sought truths and had decided on deductive proof, they had to obtain postulates that were themselves truths. They did find truths whose truth was self-evident to them… . Plato applied his theory of anamnesis, that we have had direct experience of truth in a period of existence as souls in another world before coming to earth, and we have but to recall this experience to know that these truths included the postulates of geometry. (1972)

In the spirit of these views, are Euclid’s postulates supposed to be a set of self-evident claims on which the discipline of geometry is then based? Or, at the other extreme, do the postulates just represent a more or less arbitrary starting point from which we may begin the mathematical business of proving the theorems of geometry, with no claim that they have any special sort of epistemic status, and no claim that they are obvious in any particular way?

There are reasons, I think, to be unhappy with both of these extremes. While some of Euclid’s postulates could perhaps be regarded as self-evident, it is far from clear that this could be said of Postulate 5 (the so-called ‘Parallel Postulate’). Moreover, in other mathematical works of Euclid—such as his ^{3}

But to give up on the idea that there is anything epistemically special at all about Euclid’s postulates also seems wrong-headed (or at least, so I shall argue.) There are many different senses, after all, in which a statement may turn out to occupy an epistemically privileged position.

The challenge then is to identify some sense in which, even though Euclid’s postulates fall short of being self-evident truths, they nevertheless represent genuine

Our investigations here will also connect (albeit somewhat loosely) with another puzzling aspect of the Elements. In modern mathematical language, a statement of something like the Pythagorean Theorem might go as follows:

^{2} + ^{2} = ^{2}.

In this statement of the theorem, it is simply presupposed that each leg of the triangle is associated with a unique real number giving its length. The main content of the theorem—that ^{2} + ^{2} = ^{2}—is then a claim that a certain mathematical relation holds between these real numbers.

Indeed, in most modern mathematical presentations of Euclidean geometry, it is simply assumed that Euclidean space is a metric space, and thus that all geometric line segments have corresponding lengths given by a real number. This metrical structure then provides us with a criterion for when two line segments have equal lengths, or when one is greater in length than another. The same sort of assumption is typically made of angles—in modern presentations of geometry, it is simply assumed that to each geometrical angle, there corresponds some real number between 0 and 2π giving the magnitude of the angle. This magnitude then similarly provides us with a criterion for the equality or inequality of angles. Likewise for areas, and so on. As in the example of the Pythagorean Theorem given above, modern presentations of geometry then tend to present their theorems as facts about the mathematical relations that hold between these quantities themselves. (Think of theorems such as that the area of a circle is given by ^{2}, or that the magnitudes of the angles of a triangle

What is interesting about Euclid, however, is that he does not present his theorems in this way. Whenever he can, Euclid states his theorems as facts about the relations that hold between geometrical objects themselves, rather than as facts about the relations that hold between mathematical quantities that may be associated with those geometrical objects. So for example, in Book I Proposition 47 of the Elements, Euclid states Pythagoras’s Theorem as follows:

The following sort of figure then accompanies the theorem:

Here, we suppose that ^{4} In this way, Euclid’s version of Pythagoras’s Theorem is then a statement with purely geometric content. At no point in the statement or proof of Pythagoras’s Theorem, or any theorem on which it depends, is it assumed that lengths, angles or areas correspond to numerical quantities.^{5}

Why does Euclid present his theorems this way? It is not that the very idea of associating physical magnitudes with numbers was somehow foreign. To the contrary, the Greeks had a well developed set of units of measurement and many sophisticated measuring instruments. In such measurements, quantities were unproblematically associated with geometrical objects in the natural world. Of course, it is certainly true that Euclid did not have the notation to write something as compact as the equation ^{2} + ^{2} = ^{2}. But it is difficult to see this as a barrier to viewing geometry as a science that first and foremost tells us the relations that hold between certain sorts of

The hypothesis I would like to advance is that the very idea that lengths, angles and areas can be compared is one that Euclid and his contemporaries took not to be given, but rather to require justification. Instead of beginning with a framework in which geometrical objects are associated with mathematical quantities from the start, Euclid deliberately works in a sparser framework in which the science of comparing lengths, angles and areas both needs to be and can be grounded. In fact, I will argue that there is a straightforward way in which Euclid’s five postulates may be seen as grounding such a science of comparisons, and that this leads to an interpretation of Euclid’s postulates as something like conditions for the possibility of comparison of lengths, angles and areas. In this way, Euclid’s postulates, while not ‘self-evident’ (or ever intended to be), are nevertheless epistemologically privileged as conditions for the possibility of a type of science of measurement.^{6}

Quite remarkably, Euclid (and perhaps his contemporaries) also notice that such a science of measurement, appropriately grounded, enables us not only to make specific claims about the equality or inequality of specific lengths, angles or areas, but also general claims about more general geometrical configurations—what I will call ‘general measurements’. While measurements of specific lengths, angles or areas can be done with traditional measuring instruments, general measurements are performed via proofs. For Euclid and his contemporaries then, proof then turns out to be first and foremost a way of making such general measurements, as opposed to a device for producing some sort of certainty about facts that are perhaps already known. This paper will largely be devoted to elaborating and arguing for the plausibility of this set of claims.

I will flesh out most of the details of this in

As a result of this investigation, we gain a richer understanding of the techniques of Greek mathematics and the motivations behind it. In addition, we gain a better understanding of the origin of proof—why, after all, did Greek mathematicians care to start

In this section, we shall focus on the question of how a geometer can compare a pair of lengths, angles or areas that have in some way been given and physically lie before him. This will give us some insight into the status of Euclid’s postulates, as we will see in

In Euclid’s definitions, postulates, and common notions, there is no suggestion that there there is anything intrinsic about line segments (or angles or areas) that allows us to immediately compare them, and say that one is greater than, equal to, or less than another in magnitude. How then over the course of

Let us begin by focusing on the case of line segments. How would a Euclidean geometer decide of two physical line segments before him which is longer, or whether they are equal in length? Nowhere in

When asked how we can compare the lengths of two line segments, perhaps the most natural response is that we can bring a ruler to one of the line segments, mark it, bring the marked ruler to the other line segment, and simply see which (if either) is the greater. A variant of this idea that does not involve marking involves bringing a (non-collapsible) compass to the first segment in such a way that its legs fall on the extremities of the interval, then bringing the compass to the second line segment and doing a comparison. Although there is in a sense nothing wrong with these sorts of solutions, we will first consider a different style of solution that will turn out to be more useful to Euclid, in a sense we will be able to articulate only later in

Consider then the problem of comparing the lengths of two line segments by a traditional straightedge and (collapsible) compass construction. Given only such resources, and given two line segments

Begin by constructing a point

As shown below, we then then draw a circle with center ^{7}

We have therefore reduced our problem to one of determining the relationship between the lengths of two line segments

In this way, we can determine the relationship between the lengths of

The reader will surely have noted that what has just been described is essentially the mathematical content of Book I, Propositions 1 through 3 of

Note also that the algorithm just given relies on nothing more than the ability to connect points (such as

Let us turn to the question of how we can compare the magnitudes of angles. Specifically, given two angles ABC and DEF, how are we to compare them?

We consider two methods for angle comparison—one very general, and the other a method that works only for specific sorts of geometric configurations. The first method piggybacks on our method for comparing lengths—specifically, it reduces the problem of comparing angles to the problem of comparing lengths.

To spell out this method, we begin with an arbitrary line segment

The lengths of ^{8}

The process of cutting off line segments equal in length to

This method for comparing angles is of great importance to Euclid. In particular, in Book I Proposition 23, its main idea is used to solve the problem of constructing on a given line and a given point on that line an angle equal to another given angle. This construction is then used repeatedly in the subsequent books of

Interestingly, Euclid does not use this method for comparing angles to

While this algorithm gives us a useful way of comparing angles, it is important to realize that it is not the only method for the comparison of angles that Euclid uses. There is no particular reason why Euclid should restrict himself to only one method for comparing angles, and in fact it turns out that a quite different approach to the comparison of angles also turns out to be useful for dealing with a large class of problems in

Suppose we have parallel lines ^{9} Suppose the transversal

In this case, we know that

I claim that we may view the equality of corresponding angles as a tool for the comparison of angles in the special case in which these angles are formed by a transversal cutting a pair of parallel lines. It is of course primarily in the context of Euclidean geometry that it makes sense to think of this as a technique for the comparison of angles. In a geometry in which there are multiple lines

The main problem is of course that angle

Of course, Euclid does not simply assume the equality of corresponding angles produced by a transversal cutting parallel lines, but instead argues for it explicitly in Proposition 29 of Book I. Given what has been said thus far, one might expect this argument to depend on the Parallel Postulate (Postulate 5), and indeed it does.

The argument is straightforward. It begins by showing that alternate interior angles produced by a transversal cutting a pair of parallel lines are equal. In the diagram below, consider the angles

Note that in this argument, in addition to assuming the Parallel Postulate we have also assumed that all straight angles are equal in magnitude. Now, Euclid tends to avoid using the term ‘angle’ to describe straight angles; he typically reserves the term ‘angle’ for what we would call an angle of size < 180°. He does, however, consistently describe straight angles as the ‘sum of two right angles’ (see, for example, Book I, Propositions 13 and 14). So in Euclidean parlance, our assumption that all straight angles are equal in magnitude amounts to the claim that all right angles are equal. This assumption is of course nothing other than Euclid’s Fourth Postulate.^{10} In this way, the specific method for comparing angles in question presupposes both Postulates 4 and 5.

To sum up, what all this shows is that Postulates 1 through 5 may be viewed as grounding a number of techniques for the comparison of angles. And so in much the same way that Postulates 1 through 3—that is, the existence of lines and circles—serve the purpose of grounding the possibility of comparisons of length, so too these postulates along with the postulates stating the equality of right angles and the existence and uniqueness of parallels serve the purpose of grounding various methods of angle comparison. However, the so-called principle of superposition must also be acknowledged as playing an important (though underappreciated) role here, and it is to this that we now turn.

The so-called method of superposition first appears in Book I, Proposition 4. Euclid supposes there that we have two triangles

Euclid’s goal is to prove that the length of

Euclid argues that because the length of ^{11} If the triangles can be made to coincide in this way, Euclid concludes that in fact all their remaining sides (and angles) must be equal.

How are we to make sense of the ‘superposing’ of one geometric object on top of another in this argument? It is difficult to see any straightforward sense in which Euclid’s postulates justify such a form of argumentation. While this form of argument is not used a great deal in

Reactions to the presence of this sort of reasoning in the Elements varies greatly. Russell was famously critical of the method of superposition, saying that ‘it has no logical validity, and strikes every intelligent child as a juggle’ (1938: 405). Others such as

There are many complicated issues surrounding this principle, and it is not my goal to discuss them in depth here. My view is that although this principle is not entailed by Euclid’s postulates, it is nevertheless a reasonable principle of geometrical reasoning. I do not know why Euclid did not include some version or other of this principle as a postulate, and will not try to speculate on the matter. Instead, I confine myself to a few remarks on the character and role of this principle in

One reason people like Russell have balked at Euclid’s use of the principle of superposition is that, insofar as it is a method that involves moving one geometric object on top of another, it involves the concept of motion. But we have been given no postulates about motions, and Euclid proves no theorems about motions in

However, I do not think that there is any reason to think of the principle of superposition as involving any sort of motion. Rather, it is a principle which says, loosely speaking, that two geometrical objects which agree in sufficiently many respects will agree in all intrinsic respects. For example, in the case of triangles, one has the following principle:

A principle of this sort is enough to prove Proposition 4 of Book I, as we can conclude from the superposition principle that, for example,

It is worth noting that in his formulation of the postulates of geometry in

As a tool for comparing quantities in ^{12}

Second, it may be argued that in

the areas of triangles

Thus, it can be argued that the principle of superposition (as thus stated) is the main tool for the comparison of areas in

At this point, I would like to make a conjecture about Euclid’s reason for choosing his five postulates.

The science of metrology is the study of how we measure things, what sorts of measuring instruments may be used for what purposes, and related concerns. The Ancient Greeks even before Euclid were presumably in possession of such a science. There is no reason to think that possession of this science requires the co-existence of a particularly rich idea of mathematical proof. One could potentially develop quite sophisticated tools for measuring areas, volumes, and angles without the idea that associated mathematical generalizations were capable of proof. One could instead develop such a set of tools and techniques proceeding instead with mostly empirical knowledge of the sorts of relationships that hold between numbers, various geometric objects, and so on.

Even absent a rich notion of mathematical proof, one may nevertheless see certain principles as playing a foundational role in metrology, acting as the grounds on which certain fundamental techniques of measurement may be based. I have argued that Euclid’s postulates can be seen as such foundational principles. Indeed, my conjecture is that Euclid chose his five postulates (and perhaps the principle of superposition should be added here) precisely because of their role as foundational principles in the science of metrology; that is to say, precisely because they are basic principles that allow one to compare lengths, angles and areas. Euclid of course did not explicitly give any reasons for choosing his postulates, and so my claim here is highly conjectural. Nevertheless, some support for this conjecture is found in the fact that, as shown in the previous sections, Euclid’s actual techniques for comparing lengths, angles and areas in

Euclid’s choice of his postulates should not be seen as the clever identification of a set of indubitable principles to act as the axiomatic basis of geometrical knowledge, allowing the geometer to dispel skeptical worries about the foundations of his subject. In fact, there is no reason to think that in Euclid’s time there was any sort of skepticism about geometric knowledge that was dispelled by his particular choice of postulates. Nor is there any reason to think that even if there were such worries, Euclid’s particular choice of postulates would have dispelled them. Rather, Euclid’s goal (I conjecture) was to show that beginning with ordinary suppositions that would have been taken for granted by any sort of scientist in possession of at least a primitive science of metrology, one could in fact

Thus far, we have been talking about measurements of particular geometrical objects—more specifically, judgments in which two physically given and completely determinate lengths, angles or areas are compared. Although the ability to make such judgments is crucial for the possibility of geometry, understanding the nature of these judgments is not Euclid’s goal. Nevertheless, some reflection on the kinds of theorems that actually appear in

Consider the fact that the sum of the angles in a triangle is 180°. In Euclid’s language, this claim is stated as ‘the sum of the three interior angles of the triangle equals two right angles’, and is part of Book I, Proposition 32. Now, the judgment that for some specific, physically given triangle and some specific, physically given straight angle (or pair of right angles) the sum of the angles in the former is equal to the later is the kind of thing that can be established by the methods of measurement discussed already. Call this a

But establishing that the sum of the angles in a specific triangle is equal to some specific pair of right angles by some sort of specific measurement is not Euclid’s goal in Book I, Proposition 32. His concern instead lies with triangles in general. We interpret this sort of generality in the following way. In the case of a specific measurement, one is interested in a comparison between two completely specified, physically given, and fully determinate objects. What Euclid wants to do, however, is compare two geometrical objects which are only partially specified; namely, the sum of the angles in an arbitrary triangle (with any angles and side lengths, located anywhere in space), and an arbitrary pair of right angles. We do not know anything about the triangle and right angles in question other than that they are triangles and right angles, and in this sense they are only ‘partially specified’ geometrical objects. We think of a comparison between partially specified geometrical objects as in effect infinitely many comparisons between fully specified, physically given and completely determinate geometrical objects compatible with the specification in question. Call this sort of comparison of only partially specified geometrical objects a

There is no a priori reason to expect that general measurements should be possible. In some cases, a general measurement will be impossible because the result of the relevant comparison is simply indeterminate. For example, is an arbitrary angle greater than an arbitrary right angle? Here, no general answer is possible—it might be greater, and it might not. In such a case, there is no possibility of a general measurement insofar as we demand that general measurements yield determinate results.

But even in cases in which there is a completely determinate answer to the question of the relationship between two partially given geometrical objects—as occurs, for example, when comparing the sum of the angles of a triangle with two right angles—it might still not be clear how exactly one is supposed to measure something

The key, however, is to note that the postulates that underlie the possibility of specific measurement—i.e., Euclid’s five postulates along with the principle of superposition—are already claims of a general nature. It will be precisely because of this that general measurements turn out to be possible. Let us consider some examples and see how this works.

First consider the simple fact that when two lines cross each other, the opposite angles are equal. In terms of the figure above, this means that angles CEA and DEB are equal. This is proved in Book I, Proposition 15. The argument is simple, and we have seen it already. The sum of angles CEA and AED is two right angles, as is the sum of angles AED and DEB. Because all right angles are equal to each other, it follows that the sum of angles CEA and AED is equal to the sum of angles AED and DEB. Eliminating the common element AED, it follows (using the common notions) that angles CEA and DEB are equal.

In this case, it is the generality of the fact that all right angles are equal (Euclid’s Postulate 4) that allows us to compare two opposite angles in the general case in which two lines intersect. In this way, a general measurement involving only a partially specified geometrical configuration is possible.

Consider next the fact that when a line crosses two parallel lines, alternate and corresponding angles are equal. In terms of the figure above, this means for example that angles AGH and GHD and CHF are all equal. This is part of the content of Book I, Proposition 29. The argument resembles one we have seen already. If angles AGH and CHF are unequal, then one is greater; suppose it is AGH. With a little work, one then argues that the sum of angles BGH and GHD is less than two right angles. But from Euclid’s Postulate 5, it then follows that the lines through AB and CD meet, and so are not parallel. Thus, angles AGH and CHF cannot be unequal, and so are equal. Because angles GHD and CHF are opposite angles it follows from the previous theorem that they too are equal, and thus all of angles AGH, GHD and CHF are equal. In this case too, it is the generality of Euclid’s postulates—and in particular, the generality of Postulate 5—that allows us to make a general comparison between angles in a geometrical configuration that is only partially specified.

Finally, we consider the fact that the sum of the angles in a triangle is equal to two right angles. Consider the triangle ABC given above. Extend line segment BC through C to a point D, and draw a line CE parallel to BA. Angles ABC and ECD are corresponding angles, and so are equal by the theorem just proved. Angles CAB and ACE are alternate angles, and so are also equal by the theorem just proved. Therefore the sum of the angles ABC, BCA and CAB is equal to the sum of the angles ECD, BCA and ACE, which is equal to two right angles.

What we have here is a more complex general measurement that builds on simpler general measurements to demonstrate a relationship of equality. In this case, it is the generality of the simpler general measurements that make possible the generality of the more complex general measurement of the sum of the angles of an arbitrary triangle.

In sum, the principles of specific measurement themselves already contain sufficient generality to serve as the basis for general measurement. In this way, the gap between specific and general measurement turns out to be quite small; the theoretical basis of the former sufficing to ground the later. Recognizing this is perhaps the crowning accomplishment of Euclid and his contemporaries.

It is not just the theorems described above that may be interpreted as general measurements. In fact, a great number of the propositions of Books 1 through 4 of

Although many of the propositions of

Although my main claim concerns the propositions of Books 1 through 4 of

This says, for example, that if two triangles have the same height, then the ratio of their areas is the same as the ratio of their bases. This is first and foremost a statement about equality between

Books 7 through 9 of ^{13}

Book 10 concerns the theory of incommensurability. There is both a geometric and a number theoretic character to the results presented there. Nevertheless, the theory of commensurability is only geometric in a superficial way, as Euclid’s original five postulates are also rarely invoked in Book 10. Given that the motivation for a theory of incommensurability and the character of such a theory are very different from that of the more purely geometrical theory developed in the earlier books of ^{14}

In Books 11 through 13 of

In sum, given our interest in the origins and nature of geometrical proof and the role that Euclid’s five postulates play therein, we focus on the results of Books 1 through 4, though much of our argument can be generalized to cover up through Book 6, and occasional proofs in later books.

Before returning to the general questions with which we began this paper, I will point out a further important way in which the proofs contained in (at least) the first 4 books of

In the propositions of first 4 books of

It will be easiest to illustrate this by means of an example. Consider again the proposition that the sum of the angles of a triangle add up to 180°. We have already seen the diagram that accompanies the proof:

Here, in addition to the triangle ABC which is given, two lines are added—BC is extended to BD, and a line CE parallel to BA is drawn. These lines are added precisely in order to allow a comparison between angles BAC and ACE, and angles ABC and ECD, using the kind of reasoning presented in

In fact, Euclid’s general proof-strategy for general measurements is to add lines and circles to a given geometrical configuration in such a way that a general measurement can be reduced to simpler general measurerements. In this way, general measurements can ultimately be reduced to the kind of canonical general measurements implicit in

In the case of Euclid’s constructions—i.e., propositions in which given some geometrical configuration, some other geometrical configuration must be constructed—typically the various circles and lines involved in the construction itself will be sufficient to then argue that the relevant equalities or inequalities hold between the lines, angles or areas constructed.

Note that in saying that Euclid reduces general measurements to the kind of canonical general measurements implicit in

At this point, the ingredients for the story I wish to tell about Euclid’s conception of mathematical proof are all in place. In fact, the story itself should already largely be clear.

I begin with the assumption that the ancient Greeks not only had a wide range of techniques for comparing specific lengths, angles and areas, but also a

My next main conjecture is that armed with these postulates, Euclid realized that not only can techniques of specific measurement be grounded, but also more general comparisons of lengths, angles, and areas between geometric objects that are only partially specified—I have called these comparisons

It is in this transition from a science of specific measurement to a science of general measurement that the notion of (geometric) proof emerges. Specific measurements are primarily done with measuring instruments, though a genuine science of specific measurement also requires some basic reasoning with the principles that make specific measurement possible in the first place. With general measurement, however, measuring instruments are of no use because the geometric object is never fully given. One cannot apply a straightedge or compass to a general triangle; one can only apply such instruments to a completely determinate and given triangle. Thus, in general measurement the only tool at one’s disposal is reasoning with basic principles. It is not obvious that this will be enough; but the genius of Euclid and his contemporaries lay in recognizing that armed only with this much, a rich theory of general measurement is in fact possible.

In this way, my suggestion is that a large part of

On this reading, one of the main motivations behind

Let us then return to the questions with which we began the paper, though their answers too should largely be clear by now. Our first question was how we should view the status of Euclid’s postulates, and in particular, the five postulates with which

The next question we posed was why Euclid did not treat lengths, angles and areas as intrinsically associated with real numbers. My answer is that to associate real numbers with lengths, angles and areas is to presuppose that it makes sense to talk about one length, angle, or area as having a definite mathematical relationship to another length, angle or area (such as being equal to, less than, greater than, twice as large as, and so on.) It is part of the job of a science of measurement, however, to

Although my main argument is complete, a final pair of more tangential remarks will also shed some further light on the issues just raised.

First, it is sometimes suggested that the purely geometrical books of the ^{15} It is sometimes even suggested that ruler and compass constructions were regarded as the most pure of geometric constructions, or perhaps even had a special religious significance (

There is however very little textual support for the idea that the Greeks around the time of Euclid thought there was something antecedently special about ruler and compass constructions, and that ^{16} Moreover, the use of other tools (such as a marked ruler) in the works of Pappus, Appolonius and Archimedes does not seem to have caused any concern that has been recorded.^{17} Indeed is not until significantly later in antiquity that one finds sustained and fully explicit discussions of ruler and compass constructions. While the mathematicians of antiquity seemed to have preferred ruler and compass constructions when available, nothing more can be inferred from this than a preference for mathematical simplicity.

My suggestion then is that we view

The second more tangential remark I wish to make concerns the comparison of line segments, and is an issue raised in

It is certainly possible to theorize abstractly about the elementary school method of line comparison just described and build a theory of general measurement around it. Perhaps the most well known approach in this rough direction is that of

What the work of Klein reinforces is that when someone like Euclid or Klein grounds geometry on the principles underlying various methods of specific measurement, there is no attempt to incorporate the principles underlying every possible method of measurement. Instead, a choice is being made as to which methods of measurement to interrogate. Some of these choices will be more fruitful than others. For Euclid, Klein’s choice was not an option, requiring as it does relatively complicated mathematics, even in the simplest case. Part of the genius of Euclid and his contemporaries lay in choosing a set of techniques of specific measurement whose underlying principles could be used to generate a rich theory of geometry using only fairly elementary argumentation. This aspect of their mathematical genius has, I think, been underappreciated.

I am very grateful for discussions with Tom Pashby, Ermioni Prokopaki, Daniel Sutherland and Anubav Vasudevan about the topics of this paper.

I work throughout with Heath’s translations, as presented in

For an excellent discussion of deduction in Greek mathematics, see

This is a point Meuller makes in a different context; see

The claim that in

The general point being made here is similar to one of the main points of

There is a further question as to whether Greek views on incommensurables also contributed to the rejection of an a priori identification of lengths, angles, and areas with numbers. A discussion of the views of ancient Greek mathematicians on incommensurables unfortunately lies outside the scope of this paper. It will suffice to note, however, that it is possible for the rejection of an a priori identification of geometrical magnitudes with numbers to have multiple causes.

The reasoning here relies on Euclid’s common notions.

This is only intended to be a way of comparing angles not greater than 180° in magnitude.

By a line parallel to a given line we mean a line which, regardless how far it is extended, does not meet the given line.

Concerning this postulate, note that Heath says, ‘this postulate … really serves as an invariable standard by which other (acute and obtuse) angles may be measured… ‘ (1908: Vol. 1, 200). I take this to be consistent with the reading I am offering here. See also

Perhaps the reasoning is that because

In this regard, it is worth noting that in proving Propositions 24 and 25 in Book I—propositions which in effect justify the method of angle comparison in question—Euclid relies on propositions proved using of the principle of superposition, such as Proposition 4 of Book I.

For a rich discussion of Euclid’s approach to number theory, see

For a thorough discussion of the Greek theory of incommensurability, see

One finds this idea, for example in

The demonstration that a whole series of constructions can be carried out with straightedge and compass is clearly one of the purposes of the early books of the Elements. Why Euclid chose precisely these means does not seem to be directly ascertainable.

However, one also finds voices of dissent. For example, see

There is of course no foundation for the idea, which has found its way into many textbooks, that “the object of the postulates is to declare that the only instruments the use of which is permitted in geometry are the rule and compass.”

The availability of searchable versions of a large number of texts of antiquity makes it relatively easy to establish this.

For a mathematical and historical discussion of the marked ruler, see