Abstract
In this paper, I present an interpretation of the use of constructions in both the problems and theorems of Elements I–VI, in light of the concept of given as developed in the Data, that makes a distinction between the way that constructions are used in problems, problem-constructions, and the way that they are used in theorems and in the proofs of problems, proof-constructions. I begin by showing that the general structure of a problem is slightly different from that stated by Proclus in his commentary on the Elements. I then give a reading of all five postulates, Elem. I.post.1–5, in terms of the concept of given. This is followed by a detailed exhibition of the syntax of problem-constructions, which shows that these are not practical instructions for using a straightedge and compass, but rather demonstrations of the existence of an effective procedure for introducing geometric objects, which procedure is reducible to operations of the postulates but not directly stated in terms of the postulates. Finally, I argue that theorems and the proofs of problems employ a wider range of constructive and semi- and non-constructive assumptions that those made possible by problems.
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Notes
Of course, there are no explicit constructions in Elements V. Nevertheless, I take Elements V to be necessary to the development of Elements VI, which includes a number of important problems and involves considerably use of construction. Furthermore, it is possible that construction is implicitly involved in the essential definitions of ratio and proportion, Elem. V.def.4 and V.def.5.
In fact, the Data is more of a compilation than the Elements and it is really only the first half to three quarters of the text that can be read as articulating a single program.
These semi-constructive, or non-constructive introduction assumptions have the same status as the analytical assumption of an ancient geometrical analysis–synthesis pair; see Sidoli (2018, Sects. 2, 4).
For example, for the construction of Elem. III.33—to describe a segment of a circle on a given line admitting a given angle—Heiberg (1883, I.251–255) makes reference to only one problem, Elem. I.23, and Heath (1908, I.67–70) cites no problems, while Vitrac (1990–2001, I.455–457), Joyce (Online, III Proposition 33), and Fitzpatrick (2008, 101–103) cite all of the problems that are employed, omitting only the postulates—presumably because they are taken to be obvious. (Note that Joyce incorrectly cites Elem. I.12 in place of I.11.) Finally, Vitrac (1990–2001, I.514–517) gives the full application of all of the postulates in his tables for Elements I–IV.
I have worked through all of the problems for Elements I with paper and pencil and all of the problems for Elements I–VI using Alain Matthes’ tkz-euclide, which allows us to emulate constructions according to the methods stipulated in the text (see Sect. 3, below). If we move through the text in order, we see that all of the problem-constructions are fully specified by the text, with the following exceptions: (1) Elem. III.16.corol. states that the line drawn at right angles to the diameter of a circle is tangent to the circle. This would allow us to produce a tangent to a circle at a given point with Elem. III.1, I.post.1, and I.11. Such a construction is implicitly used in Elem. III.34, IV.2, IV.3, IV.7, and IV.12. (2) Elem. IV.1 and IV.6 begin by producing the diameter to a given circle. This can be done with Elem. III.1, I.post.1 and I.post.2. (3) In Elem. IV.16 an equilateral triangle is inscribed in a given circle. In order to do this, we will have to apply Elem. I.1 to an arbitrary line before we can apply IV.2. The model for this is Elem. IV.10 and IV.11. (4) In Elem. VI.13—to find a mean proportional between two given lines—the given lines first have to be set up in the necessary configuration. This can be done with Elem. I.post.2 and I.3. Also, in order to draw the semicircle in this problem, a right angle must be produced, Elem. I.11, before the semicircle can be drawn with Elem. III.33. In these four cases as well, however, we can easily see how these constructions can be reduced to the postulates and previously established problems, as I have made clear.
Elem. IV.10, which does not assert any object to be given in the enunciation, appears to be an exception, but a line is set out assumed as given in the beginning of the construction and the proposition could be rewritten along these lines. See note 29, below, and Sidoli (2018, n. 9).
Greek philosophers such as Proclus, on the other hand, speaking rather more loosely, do refer to the objects that we start with in a theorem as being given (for example, Friedlein 1873, 203–205).
In fact, the difference between (\(\mathbf G _{\mathbf{1}}\)) and (\(\mathbf G _{\mathbf{1}}\)) can only be stated locally, as is discussed in detail in (Sidoli 2018).
For the details of this type of argument and the use of Data for this purpose, see Sidoli (2018, Sects. 2–4).
Proclus states that given “in ratio” is a mode of being given, whereas Data Def. 2 simply defines “given ratio.”
The claim that the endpoints are given in position may sound pedantic, but the Data also deals with lines given in position and magnitude whose endpoints are not themselves given (consider the implication of Data 27).
See Sidoli and Isahaya (2018, 16–17) for a reconstruction of such an argument. If we only assume that the line is given in magnitude it is only possible to show that the other two lines of the equilateral triangle are given in magnitude, but not that the point which completes the triangle is given.
Furthermore, given in magnitude simply is a more geometrically involved constraint for a segment than given in position and in magnitude. To be given in position and in magnitude involves the two endpoints being given in position, but being given in magnitude simply involves something like the segment of a line falling at a given angle between two parallel lines given in position, or being the radius of a given circle. That is, the description of a segment given in magnitude only involves objects that are themselves not elements of the segment.
Since there is no segment given on this line, the only alternative to the claim that this line is given in position would be the idea that given has a non-technical meaning in the Elements that is unrelated to its detailed treatment in the Data. But such an ad hoc assumption is doubtful and unnecessary.
It may be helpful to think of an analogy from analytical geometry. When we consider a Cartesian plane we tacitly assume the origin, (0, 0), as the reference point by which all other points will be determined. In Euclid’s sense of the term we are assuming this point as given in position—it does not undergo any transformation. That is, the position of everything that comes later in the discourse will be set out against this as the framework.
Again, it may be helpful to consider the analogy with analytical geometry. Euclid’s concept of a given point refers both to a particular point, as say (2, 3), and also to any point that may be taken as given, as say (a, b).
In the later Imperial period, as I have argued, Greco-Roman mathematicians utilized related strategies to make general arguments about computational procedures (Sidoli 2018, Sect. 5).
See Sect. 4.3 for the full argument for this claim.
That points can be set out as given may be seen from, for example, Elem. I.11, I.23, Data 32, 33, 37, 38. The case for lines can be seen in Elem. I.22, Data 39–43. That the points and lines set out in the problem-constructions of the Elements must be considered as given can be shown from the fact that they then serve as the basis for further constructions that are performed through problems that themselves assume as given these very points and lines. Of course, the assumption of a given line can generally be reduced to the assumption of two given points and an application of Elem. I.post.1 or I.post.2 (see Sect. 4.1).
The fact that the auxiliary objects do not appear in the diagram is not simply an accident of the manuscript transmission. The absence of such auxiliary objects is a characteristic of all problem-constructions in Elements I–VI in all of the primary manuscripts of the Greek, Arabic and Latin transmissions. As I will argue below, this a characteristic of the algorithmic practice of problem-constructions.
We will see below \(dg_{m,p}\) must be understood to be given.
Line \(dg_{m,p}\) is given for the same reasons.
See Acerbi (2011a, 57–65) for a discussion of the stylistic format of the two types of specification.
The definition of the circle, Elem. I.def.15, rules against the possibility of considering points outside the circle in the indirect argument—although the argument will work for any other point in the plane.
This counterfactual assumption is discussed again in Sect. 5.2.
Other exceptions in Elements I–VI are in Elem. II.14, III.30, IV.10, IV.12, VI.9, VI.10, VI.13, VI.28 and VI.29.
An exception is Elem. IV.10, which does not mention any given objects in the enunciation, as remarked upon by Proclus (Friedlein 1873, 204–205), who incorrectly believed that there are no given objects and that this forced the exposition and specification to be missing. In fact, however, as the argument makes clear, line AB is taken as given, and the proposition could be rewritten such that this given line is mentioned in the enunciation and set out in the exposition. This proposition is probably abbreviated as it is because it simply serves as lemma to the following problem, Elem. IV.11.
This is very often abbreviated or absent.
The details of this proposition will be taken up in Sect. 3.1, below.
My approach differs from the recent reading of Schneider (2015, 20), who interprets the postulates as extensions of the definitions and as making assertions about the fundamental nature of the objects involved. My thinking about the Euclidean postulates had been much clarified by the work of modern logicians, such as Mäenpää and von Plato (1990) and Beeson (2010). Whereas these scholars have used the Elements as a motivation for producing a well-founded logic of construction, I have been interested in using ideas from modern logical studies to explicate the received text of the Elements.
The word that I have translated with “to produce” (
) literally means “to lead,” which is how Vitrac (1990–2001, I.167) translates it (mener). I use “to produce,” following Fitzpatrick (2008, 7), because “to produce” has an overlapping meaning with “to lead” as a kind of abstraction. In any case, it does not mean “to draw,” as it is often translated.The word translated by “any” (
) literally means “every,” but it is often used in Greek to imply a certain generality which in English we more naturally convey with “any” (Heath 1908, I.195). The same applies to Elem. I.post.3, below.I formulate such functions using Martin-Löf’s intuitionistic type theory, following Mäenpää and von Plato (1990).
Following Heiberg (1883, I.77), this use of Elem. I.post.2 is generally not noted in modern translations (Heath 1908, 316; Vitrac 1990–2001, I.254; Fitzpatrick 2008, 34). Notice, however, that its use in Elem. I.31 is included in the table provided by Vitrac (1990–2001, I.514) and in the text of Joyce (Online, Book I, Proposition 31).
It is perhaps also implicitly used in setting up Elem. VI.13. See note 6, above.
We can use Data 26 to show that where
is an arbitrary given length, then \(c_p\) will also be given, but the postulate is not used in this way. It is simply used to make lines as long as we like, while points along this extended line are determined by further constructions.If we read through the constructions provided by Martin (1998, Chapter 1)—and especially if we do the exercises to translate these into the notation that he develops for constructions—we see that all of the constructions can be carried out using lines and intersections. For example, if we compare Martin’s account of the constructions for Elem. I.2, II.11 and II.14 with those in the Elements, we see that the applications of Elem. I.post.2 are avoided by simply considering the intersections of previously introduced objects (Martin 1998, 8, 12–13).
This is the reason why my formulation of the function below differs from that of Mäenpää and von Plato (1990, 285), since they view the function as operating on a point and a line.
It would also be possible to argue that Elem. I.post.3 defines a function that takes a given point and a given line having the given point as one endpoint, and then claiming that Elem. I.12 simply neglects to join this line. But this requires us to read material into Elem. I.12 and is not the most straightforward explanation of the text.
Drafters use, or used to use, a compass with two pins, known as a divider or a drafting compass, to carry length more accurately than a compass with a drawing end, but the design is essentially the same—in fact, many modern designs accommodate both functions (Martin 1998, 6).
For my interpretation of Euclid’s proofs by superposition, see Sect. 5.3, below.
The constructive aspect of the notion of given in magnitude is explicit in the articulation of Data Def. 1. See also the discussion in Sidoli (2018, Sect. 3).
In its extant articulation, the demonstration for Elem. I.46—to construct a square on a given line—requires Elem. I.post.4, because one angle is constructed equal to a right angle, say \(\mathbf {rAngle}_1\), and then that angle with an equal angle is shown to be equal to two right angles, say \(2\mathbf {rAngle}_2\). But if we do not have Elem. I.post.4 to assure us that \(\mathbf {rAngle}_1=\mathbf {rAngle}_2\), the proof will not follow. This argument depending on Elem. I.post.4, however, can be changed to one depending on Elem. I.def.10 alone by producing the external right angle and then arguing by Elem. I.29, so that it is not enough, on its own to require the formulation of Elem. I.post.4. (The uses of Elem. I.post.4 in Elem. I.13–15 can be reduced to the definition because the right angles in question are at a single point.)
This is especially clear when we compare it to many of the alternative postulates that make direct existence claims (Heath 1908, 220).
If the center of a circle falls on the circumference of another, it will clearly have some points inside and some points outside the other circle. Hence, the two circles will intersect.
This case can be seen constructively by producing
, which will be given in position by Data 28.For the sake of comparing the figure given here with that in the Greek text, we would have
, 
, 
and 
(Heiberg 1883, I.102–103).That is, the segment joining the centers will meet one circle or both circles as a radius, as per Elem. I.def.15—which determines whether it is entirely inside or also has some points outside the circle.
A near exception is Elem. IV.15.
Indeed, if they should happen to coincide, the algorithm for Elem. I.9 will fail, and the simplest way to avoid this is to construct the triangle facing away from the given angle.
The issue of the uniqueness of the triangle produced in Elem. I.1, and generally of the objects produced by a problem, has been discussed by Manders (2008, 100–103). Although my reading of the text has been influenced by his approach, I believe the ancient geometers would have addressed uniqueness through the concept of given (Taisbak 2003, 95; Acerbi 2011b, 146–148), where what it means for a point to be given is fundamentally that its position is unique.
Beeson (2010, 8) gives a similar account of Elem. I.2.
Indeed, in the manuscript diagrams, line G generally appears as a vertical line and AD is drawn as an equal, randomly skew line whose inclination does not depend on an operation of Elem. I.2, so that AD must have been produced by purely graphical techniques that are unrelated to the specifications of the problem-constructions (Saito 2006, 99). This is the first indication of a general pattern. The diagrams produced in the manuscripts—and, indeed, the diagrams produced by any normal human geometer, diverge more and more from the algorithms laid out in the problems as we progress through the text.
There are two issues here. The first is that the manuscript diagrams appear to have been produced by individuals, or at least by graphic techniques, that were not much concerned with the underlying mathematics to be depicted. This is an accident of transmission. The second issue is more fundamental. Human geometers construct diagrams with drafting tools, or with the postulates themselves, so that they will rarely follow the algorithms set out in the text, and almost never for the later problems, as we will see below.
In fact, in Elem. I.12 the given point that is introduced is on a certain side of a given line, so that there must be some constructive procedure for deciding this. Perhaps we could join the two given points and check whether or not there is an intersection with the given line.
There are a few exceptions to the use of full circles, such as the use of a semicircle in Elem. VI.13, Data 43 and 90. These can be regarded as the production of a segment containing a right angle, Elem. I.11, III.33. I exclude cases where the text describes the production of a circle, but the manuscript figures show only an arc—these exhibit graphical choices that are related to the production of material objects and tell us little about the mathematical intention.
Catton and Montelle (2012, 35–37) claim that this is not the case, but this is because the “actions” that they use to produce the construction do not agree with those given in the text.
This emphasis on the construction as providing starting points for the deduction should be contrasted with accounts that place these starting points in the diagram itself, such as that provided by Netz (1999a, Chapter 1).
As is often the case in problem-constructions, Heiberg (1883, I.153) did not attempt to provide a justification for every step, mentioning only Elem. I.46, once. In this he is followed by Heath (1908, 402). Vitrac (1990–2001, 353, 515) mentions no construction steps in his translation, although they are all listed in his table. Joyce (Online, Proposition II.11) and Fitzpatrick (2008, 63) give a fuller, although still incomplete, set of justifications in the text itself.
It should be noted that some of the steps of the subroutine, for example, in applications of Elem. I.3, will be degenerate, so that we will not actually produce all 80 circles and lines, but a somewhat smaller subset. (See the discussion of Elem. II.11 [6], below.)
Having worked through Elements I following the full constructions as stipulated in the text using a straightedge and compass, I found that proceeding through the full sequence of operations for Elem. I.46 has a tendency to produce a misshapen figure.
Again, in this case a number of the operations of Elem. I.3 are degenerate, and, hence, all of the applications of the postulates in the full algorithm will not appear as objects in the diagram. (See the discussion of Elem. II.11 [6], below.)
See notes 63 and 65, above.
Heron’s reworking of Elements II is preserved only in al-Nayrīzī’s commentary, in both Arabic and Latin (Besthorn and Heiberg 1897–1905, II.1.4–79; Tummers 1994, 73–89). For recent work in the long literature on Heron’s rewriting of Elements II, see Corry (2013, 133–139) and Sialaros and Christianidis (2016, 652–654). Acerbi and Vitrac (2014, 31–39) give an overview of Heron’s commentary and discuss Heron’s more abstract proofs on p. 36. For a discussion of Euclid’s use of this abstract approach, see note 69, below.
For example, Acerbi (2007, 272) mentions that the constructions in Elem. II.1–8 serve such a purely illustrative function.
It should be noted that this was probably a stylistic choice on Euclid’s part, not a conceptual limitation. As the arguments in Elem. III.35, III.36, IV.10, and Data 86 make clear, Euclid himself was capable of applying theorems of Elements II in the abstract, operational way that we find in the work of Archimedes and Apollonius, known as the application of areas, or geometrical algebra (Zeuthen 1917, 313–316(115–118); Taisbak 1996, 2003, 211–224), and there is no reason why he could not have written Elements II in the same abstracted vein as we find in Heron’s commentary.
Of course, many of the applications of Elem. I.3 will be degenerate, as discussed above, but since this is not discussed in the text, and the only way to know which are degenerate and which are not is to go through the construction and count, I simply count the operations of the algorithm as presented by the text. The diagram shows only the lines and circles that would actually be produced—which is significantly less than 183, but still far more than a human geometer would require.
For example, although for the sake of an argument we might suppose that two numbers that are equal might be different and subtract one from the other, no effective procedure carried out on numbers would ever produce this difference as a natural number.
This example was discussed in Sect. 2.1, above.
There is an extensive literature on superposition in the Elements, which I will not pretend to survey. Axworthy (2018, 6–9) provides a recent overview. Vitrac (2005, 49–52) treats the ancient and medieval discussions of superposition. Acerbi (2010) discusses the evidence for homeomeric lines in Greek mathematical authors, which makes clear the extent to which the technique of superposition was accepted and used by Greek mathematicians.
Here, I refer to scholarly work aiming to explain the ancient position, not mathematical work meant to criticize it and produce a new, more complete, formulation.
Without attempting to be exhaustive it is sufficient to point out that Levi (2003, 103–109), Wagner (1983), and Saito (2009, 807–809) argue for construction from the postulates, while Heath (1908, 225–228), Mueller (1981, 21), Vitrac (1990–2001, I.293–299), and Panza (2012, 92, n. 71) speak of a rigid displacement. Alvarez (2003) treats constructions by postulates as a sort of movement, or transformation. Finally, Dean and Mumma (2009, 725) argue that the assumption made in Elem. I.4 is essentially hypothetical.
My position is close to that put forward by Vitrac (2005, 49–52), except that I do not think we need to invoke the notion of motion in anything like the normal meaning of the word. Furthermore, I put greater emphasis on the notion of given to explicate the ancient and medieval understanding of superposition, and I agree with Dean and Mumma (2009, 725) that the supposition of superposition must be regarded as purely hypothetical. The case for the purely hypothetical status of superposition was clearly articulated by Commandinus (Axworthy 2018, 25–33).
I have translated the genitive absolute using when, to emphasize the difference between this type of hypothesis and that setting out the conditions of the theorem, which is introduced with
.I have translated both the transitive
and the intransitive 
with “to fit,” as opposed to the usual convention of using two different verbs, such as “to apply” and “to coincide” (Heath 1908, 224–225; Vitrac 1990–2001, I.181, n. 13). The reason for this is that we have a long tradition of reading “to apply” and “to coincide” as technical terms that we think we understand. For example, Heath (1908, 225) claims that the linguistic expression used to introduce superposition must be read to mean that one figure is “actually moved and placed upon the other.” But, nothing in the text is so explicit. In fact, both the text and the argument are vague and subject to a range of interpretations. Hence, I use a literal translation, with an English verb that can be both transitive and intransitive.See the discussion of the verb
by Taisbak (2003, 93–94), who also argues that it denotes an abstract type of change. This point was also made by Russell (1938, 405–406), but note that he also claims that a “point of space is a position”—which is contrary to Euclid’s view. An important goal of the Data and of the language of givens is to develop a meaningful way to differentiate between points which are positioned and points that have various degrees of freedom.The latter assumption is a sort of converse of Elem. I.c.n.7 (or 4). This is also all that is required for Elem. III.24, because the points and the lines are assumed to fit, from which the segment is shown to fit.
One could argue that this assumption is mathematically equivalent to the assumption of Hilbert’s sixth axiom of congruency, III-6.
) literally means “to lead,” which is how Vitrac (
) literally means “every,” but it is often used in Greek to imply a certain generality which in English we more naturally convey with “any” (Heath 
is an arbitrary given length, then 
, which will be given in position by Data 28.
, 
, 
and 
(Heiberg 
.
and the intransitive 
with “to fit,” as opposed to the usual convention of using two different verbs, such as “to apply” and “to coincide” (Heath 
by Taisbak (References
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Acknowledgements
The diagrams in this paper were created with Alain Matthes’ tkz-euclide package. This package gives us tools for starting with a set of arbitrary points and producing new points as the intersections of objects produced with lines and circles, as is often done in modern treatments of Euclid’s constructions. Of course, the underlying engine of this package assumes a Cartesian plane, and the location of each object is computed using modern formulas. Nevertheless, in this way, it is possible to emulate Euclid’s actual constructions by drawing in only those lines and circles called for by each of his problems and building up a set of routines that can act as subroutines in subsequent constructions. The ideas in this paper were presented at the PhilMath InterSem, Paris, February 2016, at a Workshop in History of Greek Mathematics, Stanford University, October 2017, and the French-Japanese Workshop in Philosophy of Logic and Mathematics, Keio University, January 2018. I thank the organizers of these workshops for the chance to present my ideas. I received a number of useful comments and questions following my talks. I discussed my ideas about Elem. I.4 informally with Daryn Lehoux, and many times with Ken Saito and Marco Panza. Earlier drafts of this paper were read by Victor Pambuccian, Len Berggren, Marco Panza, Michael Fried, and Bernard Vitrac—all of whom made useful comments that helped me clarify my ideas. Bernard Vitrac will no doubt still disagree with my methodology, but his comments have nevertheless been helpful for me. I have greatly benefited from discussions with Marco Panza, who has helped me to sharpen my thinking on a number of fundamental points.
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Communicated by Jeremy Gray.
Ancient Mathematical Works Referenced by Title
Ancient Mathematical Works Referenced by Title
Commentary on Euclid’s Elements I, Proclus: Greek text edited by Friedlein (1873). English translation by Morrow (1970).
Data, Euclid: Greek text edited with Latin translation by Menge (1896), German translation by Thaer (1962), English translation by McDowell and Sololik (1993), Greek text reprinted with English translation and commentary by Taisbak (2003), Greek text reprinted with Italian translation by Acerbi (2007, 1860–2113), Greek text reprinted with English translation by Fitzpatrick (2008).
Elements (Elem.), Euclid: Greek text edited with Latin translation by Heiberg (1883), English translation with commentary by Heath (1908), French translation and commentary by Vitrac (1990–2001), Greek text reprinted with Italian translation by Acerbi (2007, 778–1857).
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Sidoli, N. Uses of construction in problems and theorems in Euclid’s Elements I–VI. Arch. Hist. Exact Sci. 72, 403–452 (2018). https://doi.org/10.1007/s00407-018-0212-4
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DOI: https://doi.org/10.1007/s00407-018-0212-4