On the Oxyrhynchus papyrus fragment P. OXY. 29, and Euclid’s Proposition 5, Book II of the Elements.

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Dimitrios S. Dendrinos, Ph.D.

Emeritus Professor, The University of Kansas, Lawrence, Kansas, USA

In Residence at Ormond Beach, Florida, USA

Contact: cbf-jf@earthlink.net

©Dimitrios S. Dendrinos

June 13, 2022

 

Figure 1. Fragment of a papyrus, discovered in Oxyrhynchus, designated as “P. OXY. 29”, and dated thus far to approximately 75 – 125 AD. It is a reproduction of Euclid’s Proposition 5, Book II, of his Elements. Source of image: The photograph is in the public domain.

 

Table of Contents

Abstract

Brief Introduction

Analysis of the fragment and its contents

Alternative statements of Proposition 5

The spatial-temporal context

Propositions 1- 4, 6 – 8

The impossibility to generalize Proposition 5

Temporal considerations and the framework of P. OXY. 29

The Proposition’s writing and drawing styles

On the Epistemology of the Elements, and Proposition 5

What is: definition, given (known), axiom, proposition, theorem, proof, and the sought after (unknown)

Alternative proofs

Using areas

The algebraic version of Proposition 5

Imperfections in the diagram, their meaning and context

The diagram’s maker

Two persons involved

More analysis needed

Linguistics, diacritic elements and dating the fragment

Style of writing

On the term “gnomon”

Conclusions

Appendix: Proposition 6

References

Copyright Statement

Abstract

The paper’s objective is to discuss and critically analyze the papyrus fragment P. OXY. 29 that contains what is considered to be Proposition 5, from Book II, of Euclid’s Elements. This particular fragment contains enough information on it to draw some preliminary conclusions regarding its making. It is suggested that P. OXY. 29 contains a scribe’s personal drawing, inserted at some later date than when the text was written. Hence, it is inferred that possibly two different scribes are involved in the making of this artifact; or the same scribe writing on the papyrus at two different points in time, relatively within close temporal proximity. The text of course was not directly copied from the original Euclid text, which was very likely written in capital (uppercase) Greek letters, but rather from a prior copy (or the end point from a series of multiple copies) of the original and after the transition from Greek koine to a proto-Byzantine lowercase cum uppercase writing. Analysis of the fragment’s context, form, as well as its content, and especially what is omitted from the diagram as drawn that also provides clues as to its dating, are attempted. The spatial-temporal paths of both fragment and content are sketched out. It is suggested that the writing took place, possibly, slightly later than the currently prevailing view, which holds that the artifact was made in the 75 – 125 AD time frame.

Analysis of the fragment’s contents identifies imperfections associated with the manner the figure was drawn by the scribe, and ventures into the copier’ and the scribes’ underlying motivations to write the text in the manner written, and draw the diagram the way it was drawn correspondingly. It is suggested that the drawing was in part an attempt to both memorize and in a grosso modo prove the theorem embedded in the statement of the Proposition. The paper also attempts to place the papyrus’ geometric contents to a period somewhat later than the currently prevailing dating, based on this analysis.

Epistemological issues associated with the very nature of the Euclidean Proposition 5 as stated in the Elements are brought up, related to axioms, theorems, propositions subject to a statement of givens (knowns) and sought after (unknowns), and their connections to proofs. Moreover, the paper analyzes slightly differing alternative statements of the original Proposition, given by the various analysts that have studied this artifact as well as have translated the various versions of Euclid’s Elements. It also discusses a set of proofs that have been suggested associated with it, as well as the Algebra related equivalents of the geometric statement of the theorem (Proposition 5) and its purely geometric proof. Commentary is supplied, placing in context the “Geometry-Algebra equivalence” of the period in question.

The inherent impossibility to generalize this particular Proposition is examined, in terms of Analytical Geometry. In addition, preceding Propositions (1 – 4) as well as subsequent Propositions (6 – 8), as found in Book II of the Elements, are addressed and the import of Proposition 5 is stressed within the context of the Elements. By doing so, not only the spatial-temporal framework suggested for the fragment is further documented; but also, some observations regarding certain mathematical (arithmetic, geometric) and epistemological aspects behind the Elements in general, and this part of Book II (Propositions 1 – 8) are drawn.

Brief introduction

As mentioned in ref. [1], the papyrus fragment, designated as P. OXY. 29, shown on the cover page of this paper, was found in the period 1896-7 through an excavation by B. F. Grenfell and A. S. Hunt (both from Oxford University) at a site in the old city of Oxyrhynchus, (in Greek, Οξυρρυγχος) modern day Behnesa, about 160 kilometers (roughly 100 miles) South-West of Cairo. Note, that in ref. [1] it is (erroneously) mentioned that Oxyrhynchus lies about “100 miles up the Nile”.

A few introductory comments about the text imprinted on this specific papyrus-artifact are in order. It is not known, not only when exactly this piece of papyrus was produced, but most importantly when the text was written, and from what specific text of the Elements this particular text was copied. The entire spatiotemporal path of the fragment, from the time produced to the specific point in space-time it was found can’t be traced with any degree of certainty. Hence, matters associated with these questions can only afford speculative answers.

In reference to the Euclid Elements related manuscripts (copies) now extant, the oldest surviving texts are two: one is that produced by mathematician Theon of Alexandria and his daughter Hypatia (4th and early 5th century AD), which was copied from an earlier text that has not survived, and to which commentary by Theon was added, see ref. [16]; and the so-called Vatican text (it does not contain Theon’s commentary), which is speculated to be a 9th century AD Byzantine Era copy of an earlier than the Theon copy, see ref. [17]. Of course, it is not known if the two extant copied sources had a common ancestor or not.

Moreover, it is not known what style (whether Attic, koine with or without any diacritic elements, the specific dialect or dialects, phraseology and Linguistics, let alone the material on which these copied specimens were produced) were used in the two prior ancestral copies (if they were not one and the same). Hence, the literary lineage of the Elements, and its entire genealogy up to the 9th century AD remain uncertain. One thing seems to be quite clear: the Linguistics of, cum the material included in, and the medium used to write, the Elements in the forms that they have survived today are not exactly those of Euclid’s (such exactness is simply impossible). The writing style, dialect, characters, and in general the syntax and grammar of the Alexandrian 300 BC Attic or koine (not to mention the medium used to record documents) are not the same as the 4th century AD equivalents (only to address the Linguistics of the matter and not the implied Mathematics of it, let alone the medium).

Writing imprinted on the fragment is a mix of upper and lowercase (for example, ω not Ω) Greek proto-Byzantine (koine) to Byzantine (with diacritic elements) style. The fact that nothing more than the statement of Proposition 5 is mentioned, along with the type of writing found on the papyrus fragment, as well as the drawing (of what it does and does not contain) and the peculiar symbol written right next to the diagram, seem in combination to suggest that this artifact (part of a lengthy papyrus roll) was made very early into the Byzantine period for or by someone probably interested just in writing down Euclid’s theorems, possibly only from the Elements.

Subsequently, and maybe not much longer after this copy of Proposition 5 was produced, a scribe (possibly different than the original writer of the text, or the same scribe but later, this being a topic explored in some detail in this paper), drew the figure on the papyrus roll. Intentionally or unintentionally the fragment of the roll, which contained both the text and the drawing, survived, and this fact may be related to the import of the fragment’s content. Although the fragment’s specific path in space-time to the specific place, leading to the condition it was found in late 19th century, remain largely unknown, there is some information imprinted on the artifact that offers hints as to its tumultuous dynamical path.

More specifically, with regards to the writing, one observes that there is no space between words, and especially no diacritic elements present in the text or onto the figure attached to it. It is now universally accepted that what is written on this papyrus fragment corresponds to Proposition 5, of Book II, of Euclid’s seminal work on Geometry Elements (ΣΤΟΙΧΕΙΑ). A compendium of all thirteen Books of this work (which does not exclusively address subjects in Geometry) is found in refs. [2] and [10], sources heavily drawn upon in the writing of this paper, which should be read in conjunction with ref. [3]. Much of the analysis in this paper draws from these references, i.e., refs. [1], [2], as well as ref. [3], the latter heavily relying in its treatment of Euclid on the Thomas Little Heath translation of J. L. Heiberg’s translation in Greek of Euclid’s Elements, from a series of reproductions all based on the Vatican edition, see ref. [10]. It must be stressed that both the Heiberg and Heath Greek texts are in Byzantine (not Attic) lettering.

The papyrus fragment is estimated to belong to the 75 – 125 AD time period, by papyrologist Eric Turner, see ref. [1]. However, it is argued here that the papyrus fragment belongs to a text written a bit later, possibly a century later. However, it does suggest that not much modification from the original (and unknown) source copy (which was possibly directly drawn from Euclid’s original early 3rd century BC text) has taken place following this fragment; and if any modification of the original text did occur, it must have happened in the approximately half of a millennium time span following the writing of the original text by Euclid (c. 300 BC) and the time of this copy (c. 200 AD). The paper supplies the spatial and temporal context this papyrus segment was created, but not in its entirety the spatial-temporal path of the artifact since its creation. Moreover, the paper is not a treatise on the evolution of the Linguistics (in form and structure) of either the Elements in general, or Proposition 5 in specific. In ref. [20] a sample of Hellenistic period koine version of Greek is shown; it contains uppercase (capital) letters. Juxtapose that with the pre-Byzantine Greek with lowercase letters containing diacritic elements of the same reference. P. OXY. 29 writing style is much closer to pre-Byzantine than koine (Attic version) of Greek. It is recalled that the koine Greek on the Rosetta Stone, an artifact of the 196 BC time period, is all in upper case Greek lettering style writing. In addition, right next to the figure drawn on the papyrus fragment, a symbol resembling a Latin “n” is drawn.

For the reader who might be interested in the entire context in which the Oxyrhynchus (P. OXY. 29) papyrus fragment was found in late 19th century, which was inside a pile of discarded rubbish containing hundreds (if not thousands) of papyri fragments, source [18] offers a good account.

Analysis of the fragment and its contents

Alternative statements of Proposition 5

The fragment’s contents (Greek narrative of a Proposition and a drawing associated with it) represent a variation of Proposition 5 from Book II, of Euclid’s Elements. In ref. [2], the complete work of the thirteen Books of the Elements is offered, in English and in a manner to be understood by present day readers. The text in ref. [2] contains the full proof of the Propositions as supplied by Euclid according to Heiberg, ref. [13], with extensive additional commentary and clarifying diagrams. Many among them are not directly needed, and not necessary for the exposition of the Proposition’s details, although somewhat essential for obtaining a more complete understanding of Euclid’s proof. That material includes Algebra-related expositions, along with the purely Geometry-related aspect of the Proposition. It does not include, however, Analytical Geometry. But what it does include, is some work related to the Euclidian use of the term “gnomon”, absent from the Proposition as either stated or its proof drawn in the fragment, but apparently present in Euclid’s original proof; this is a subject to be discussed later in this paper.

What is of import here, besides the issue of the “gnomon”, is that the description of the Proposition in ref. [2] does not exactly match the description as shown in the fragment of Figure 1 (cover page). It does so only (but accurately) in the overall spirit of the problem statement. Thomas Little Heath, in ref. [10], pp: 40 and 41, offers a translation of the source Heiberg Greek text and proof. The translation goes as follows: “If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square of the half”.

Specifically, in ref. [2], the description is mentioned as (translated into English from Greek): “If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole with the square on the straight line between the points of section equals the square on the half”; see reference [2.1]. On the other hand, the description of the Proposition according to the papyrus’ fragment, as provided by B.P Grenfell and A.S. Hunt, as well as by J.L. Heiberg, see ref. [1], is slightly terser and laconic, i.e., slightly more efficient, than the Proposition as supplied above (from ref. [2]). In the Grenfell-Hunt and Heiberg reconstructions (in Byzantine Greek) of the fragment, the statement of the Proposition shown in Figure 1 (as translated into English by the author of this paper) goes as follows:

“If a straight line is cut into equal and unequal (segments), the rectangle defined by the unequal (segments’ lengths) of the whole (line), plus the between cuts square, equal the half (line’s) square”, where in parenthesis and the commas are this author’s additions to more clearly present the meaning of the statement (Proposition). It is also noted that in the (characterized as “modern Greek” in ref. [1]) version of J. L. Heiberg, the term “ορθογώνιν” should read “ορθογώνιον” (in an obviously spelling error). This otherwise inconsequential discrepancy, although meaningless in itself, is indicative of how errors (as well as additions and deletions) occur in the process of copying documents.

More substantially however, it can be asserted that the more the differences between the original Euclid text and the copy at hand, the more the time space between them. It is noted, that variations in text, even slight ones, are critical in determining lineages from the original to current versions of the text. “Doxography”, a method coined by philologist Herman Alexander Diels, is of interest at this point, as it applies specifically to ancient Greek writers, see ref. [14]. Doxography has been used by T. L. Heath in his classical set of references on Ancient Greek Mathematics and Astronomy, see ref. [3] for more details on this topic.

Stripped from any redundancy the Proposition reads verbatim (as translated by this author from the Grenfell-Hunt reconstruction, and not accounting for differences between Greek and English grammar, such as the existence of plurals on adjectives in Greek): “If a straight line is equally and unequally divided by two points, the rectangle derived by the unequal segments, plus the square of the distance between the two points, equals the square of the half.” This narrative (in Greek of course) could be how Euclid originally wrote it. It is not possible to exactly translate this description into Euclid’s space-time applicable Greek, since: first, it is not known what version from all of the above mentioned he used to write this Proposition; or what type of Greek was in use at the time of Euclid in Alexandria, since it is not known with certainty when precisely Euclid wrote this work, i.e., this specific part (Book II) of the Elements, and most importantly what exact version of Alexandrian (Attic, Doric, or any other) Greek Euclid spoke and wrote.

The spatial-temporal context

It is generally assumed that the thirteen-book volume of the Elements was written in Alexandria. However, on what (parchment or papyrus), when and even by whom exactly is unclear, see ref. [3]. It is likely, as argued in ref. [3], that Euclid was a historical person, and that the Elements were written very likely by him (and having been educated quite likely in Athens, he wrote in the Attic dialect) during the first half of the 3rd century BC. It is known with some certainty that Alexandria was founded by Alexander III in c. 331 BC, although it all depends by what is meant by “founded”, see ref. [15]. That of course does not necessarily mean that no one lived there prior to 331 BC. Moreover, being an imperial city and a growth pole in the system of cities founded by Alexander III, it attracted individuals from all over the Mediterranean Basin, and possibly beyond, ref. [3].

More specifically, it is presumed that Euclid composed his Elements in the period c 300 – 275 BC, i.e., about half a century after Alexander’s death, from what is mentioned in the Introduction of ref. [10]. Maybe, one of the numerous reasons why Alexandria became the major center of intellectual (specifically, mathematical and astronomical) developments was the ability to widely use papyrus to write on, as opposed to parchment, which was the material that Pergamon was well known for. It is recalled that by comparison, the Method, Archimedes’ recently discovered manuscript, was written on parchment, and in the Doric dialect.

The key names associated with Euclid’s Elements and used to calibrate the approximate time period Euclid worked and published his manuscript are the following five: Eudoxus of Knidos (c. 375 BC), as apparently Euclid used some of his work (without attribution), but well known then to belong to Eudoxus, and it is thought that Euclid streamlined it (according to mathematician Proclus, who lived in the 410 – 485 AD period, see ref. [10], p. xx); Archimedes of Syracuse (c. 287 BC), who apparently made a reference to Euclid, which some contemporaries argue it was added, or “interpolated”, see ref. [10], p. xix,  subsequently (by some unnamed individual at some unnamed time period); Apollonius of Perga (c. 200 BC), Pappus of Alexandria (c. 320 AD), and of course Proclus, the last three making explicit references to Euclid, see ref. [10], pages xix and xx. It is on the works of these individuals, and that by Heath, that ref. [3] draws its conclusions about Euclid and the Elements.

Now, regarding the spatial-temporal context of the Oxyrhynchus papyrus (fragment). Turner’s estimate, that the papyrus was written in the 75 – 125 AD period, would place this fragment before Pappus, and hence far before Proclus too. In ref. [3] a detailed historiography and timeline of major ancient Greek mathematicians (and astronomers) is supplied, in where the following first, second and third centuries AD mathematicians, and commentators of prior mathematical works, are mentioned and their work looked at: Heron of Alexandria (c. 10 – 75 AD), Menelaus of Alexandria (c. 70 – 130 AD), Claudius Ptolemy of Alexandria (c. 85 – 165 AD), and Sporus of Nicaea (c. 240 – 300 AD). Hence, this artifact was produced at a time that Greek Mathematics were still briskly active, and the presence of a major figure in this list (Claudius Ptolemy) was dominant, at a place of close proximity to Oxyrhynchus (Alexandria). It is here that a historical perspective is of import, and the reportage that goes under the name “doxography” (a term that can be loosely translated as the ‘writing of, or about, opinions’) becomes of essence.

Then, and in addition, one must analyze the context in which Proposition 5 appears within the Elements. Linguistically, a number of different ways Proposition 5 (one of Euclid’s most imaginative and consequential Propositions) could be slightly restated, by either adding explanatory words to the Proposition’s statement; or, by reformatting the statement to add some (necessary) redundancy for minimal clarification and clearer syntax. For example, by following the exact steps of the Euclid statement, the theorem could be stated as follows: “if a straight line is cut into equal and unequal parts, then the total area of the rectangle formed by the unequal parts’ lengths, plus the area of the square formed by the two cuts’ length, equals the area of the square formed by a length equal to half of the line’s length.”

Methodologically, and again placing the Proposition into perspective, the tools used by Euclid in proving the Proposition is the application of the law of Proportions; it is not the application of either overlapping of areas or Algebra. Use of the terms “area” and “length”, both are not used by Euclid; instead, “figure” and “segments” are. Issues of consistency in the use of terms as defined by Euclid himself, as well as the precise meaning of certain basic notions (such as “point”, “line”, etc.) in the Elements, present issues touching the Epistemology of not only Euclid but Mathematics proper, and these issues are to some extent addressed later in this paper.

The point of the above set of statements of Euclid’s Proposition 5, in Book II of his Elements, involving various degrees of redundancy in them is to indicate that (quite likely) the more redundancy in the statement, the later the statement was reproduced. Similarly, the more material was added to the text, or subtracted from it, the later the copy was very likely made. The intervening temporal distance between redundancy levels (or more broadly, statement alterations, modifications) corresponds (likely) to differing levels of socialization between the statement of the Proposition and its apparent or intended by the modifier audience.

Propositions 1 - 4 and 6 - 8

To place Proposition 5 in the context of the Elements, one must review what the preceding and succeeding Proposition are. In ref. [3] more elaboration of this set of Propositions is supplied. Here, very briefly it will be mentioned that Propositions 1 to 4 (the introductory Propositions of Book II) are a prelude to Proposition 5, supplying the necessary Geometry background to set up the arguments for proving Proposition 5. Specifically, Proposition 4 is an essential theorem on the basis of which Proposition 5 is based for its proof, setting aside the prior Propositions from statements and definitions. Proposition 4 is how to locate square roots, see ref. [11], p. lxxv.

On the other hand, Proposition 6 is a theorem which, although it does not generalize Proposition 5, extends it. Proposition 6 reads as follows (see ref. [10], p. 41): “If a straight line be bisected and a straight line be added to it in a straight line, the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half is equal to the square on the straight line made up of the half and the added straight line.” What exactly all that means is shown in the Appendix, where a simple Algebra-based proof is supplied.

Undoubtedly, Proposition 5 is one of the most interesting and fundamental Propositions in the entire Book II, and possibly one of the most fertile Propositions of the Elements. It allows for numerous extensions, Propositions 6, 7, and 8 being cases in point. It is of interest as well because of the implications it holds from the impossibility to generalize it, as it will be discussed next.

The impossibility to generalize Proposition 5

It must be noted that the Proposition could not be stated in generalized form, but only extended in the manner Proposition 6 states. Propositions 7 and 8 are also ways to extend Proposition 5, in the same manner that Proposition 5 is an extension of Propositions 1 to 4 in Book II. The matter of extensions is a topic of algebraic (and in general of mathematical) interest.

To fully appreciate the reasons why Proposition 5 was not generalized, is to look for the Definitions supplied in Book II, the effort to avoid negative numbers and only seek positive roots to quadratic (and also to cubic) equations (when intersections of spheres, conics and cylinders are discussed) through the application of the Theory of Proportions (a way also for the Greeks to deal with irrationals and incommensurables). In all Greek Mathematics of antiquity, negative numbers do not appear; and so is the case with the number zero. The Greeks of that time either did not want (or wish) to deal with negative numbers (and the notion of zero, let alone the number zero), for possibly numerous reasons to be discussed in the Epistemology section of the paper; or they did not know about negative numbers (and of zero), see for more ref. [3]. The Algebra (and broader mathematical) part of the argument will be addressed later in the paper.

The reader must distinguish (notwithstanding its fuzziness) between “extending” a theorem and “generalizing” it. Proposition 5 is related to Algebra and the solution of quadratic equations, thus constraining the possibility to generalize, as only positive solutions were sought. On this see ref. [12], pp:100-5. The generalization of the Proposition could be stated as follows, where now some letters have been added (following ref. [2]) for clarification: “Consider an arbitrary in length line where two points A and B are placed (point A being at left of point B on the line); designate the AB segment’s middle point by C; consider an arbitrary point on the line as follows: either (a) within the line’s segment AB, but off C; or (b) on the line’s extensions in either direction, and designate it by D.  Without any loss of generality, assume that the point is at right of C (as is the case in Figure 1). If the point falls within the segment AB, then of course, length CD is smaller than CB, as is the case in Figure 1, and apparently implied in the original Euclidean Proposition 5, Book II statement; otherwise, it is greater, CD > CB, and this possibility’s implications is what the Proposition’s generalization is now addressing, for reasons that will be explained in turn.

The restatement of the Proposition, would that extension been possible, it would read as it reads under the proviso that point D is located within the segment AB, but off point C. However, this statement of the Proposition can’t be proven, when the segment CD is greater than AC (or CB). This can be easily checked by either the algebraic equivalent of the Proposition, as it will be also discussed shortly; or as a part of Analytical Geometry, whereby a function F(x) is defined on a Cartesian coordinate, where variable x varies between zero and some arbitrary value B on the x-axis (corresponding to point B as designated above), and where now point A is the origin on the Cartesian coordinate, the y-axis depicting the value of the function F(x), as variable x varies between zero and B (the case of the original Proposition). In this segment, as x varies in the range between zero and B, the value of F(x) is zero. It is not so off that range (domain). The message from the impossibility to generalize is that Proposition 5 has a limited domain of applicability, the (0, B) space on the x-axis for the function F(x) to have real positive values. Beyond this (positive or negative) domain, the Proposition does not apply. In fact, as Proposition 6 indicates, a different and unavoidable extension of Proposition 5 (namely, Proposition 6) must be considered. Of course, it is not known (and neither can it be known) if Euclid was aware of both, the algebraic and geometric expressions of his Propositions; and whether he had some inkling about the Analytical Geometry part of Proposition 5 and its strict domain of applicability.

A geometer the caliber of Euclid must have thought of the implications the choice of location D on the AB line must have had. The very fact that he did not generalize the Proposition must have meant something to him. Thus, by stating, as the immediate extension of Proposition 5, Proposition 6 (and a number of other slightly more complex Propositions, namely Propositions 7 and 8), it means that he must have carefully thought the issue through. However, maybe he did not have yet either the interest, means, or time to get into this issue more thoroughly.

All one can do at this point, more than 23 centuries later, is to carefully and methodically speculate as to what he knew and understood his Proposition to mean and imply. The fact that Euclid in his Proposition 5 does not even hint as to the position of the cut, i.e., the location of D on the foundational for his statement of the Proposition’s line (γραμμή), or the depiction of points A, B, on a line that could be of indefinite length, allows room for an analyst to speculate on whether Euclid was aware of the limitations of his Proposition and the impossibility to generalize its statement. That realization, had he expressed it then, would have held significant implications for the course of evolution in the field of Geometry. For one, it would have opened the road to Cartesian Analytical Geometry long before Descartes.

The algebraic formulation of the Proposition will be examined shortly, but the question lingers. Why Euclid did not include the above-mentioned extension (and more precisely, the inability to so generalize by a reductio ad absurdum) in the Proposition’s statement? Merely pointing out this inability is by itself of interest as a negative result, with theoretical underpinnings and importance. Maybe the Algebra-Geometry connection was not there yet; maybe the extension was to be shown as an exercise to a student or scribe. Which leads to speculating as to the very meaning and use of the fragment, and the copy from which it was likely copied itself. Moreover, one may ask, in reference not only to the original from which this artifact (the Oxyrhynchus papyrus fragment) was produced, for what purpose and for whom the original Euclid Elements manuscript was composed, as well as this specific papyrus copy from which this fragment remains extant. From what is included in the papyrus fragment one can surmise that the copier in this case was just interested in the theorem(s), and not the proof(s). But the question as to the intended by Euclid audience of the Elements remains open. Was it made to be used by researchers as a reference manual? Was it a research report addressed to the community of geometers then? Was it an attempt to write a compendium on Geometry? Or, was it just an instruction-oriented students’ textbook? Possibly all of the above, and then some more (possibly commissioned work by the very Librarian of the Library of Alexandria and its financial backers).

Returning to the matter at hand, instead of generalizing Proposition 5, Propositions 6, 7, and 8 were produced by Euclid, and shown at the very beginning of Book II. The fact remains, that a generalized version of the Proposition and especially its consequences do not appear, not only in Book II but in the Elements as a whole. Its absence is significant, as to the implications it holds, beyond its opportunity cost. The presence of Propositions 5, 6, 7, and 8, in combination with the absence of a generalized version of Proposition 5 (even as a negative result, in the form of “prove that it can’t hold, if D falls outside the finite AB line”) is itself informative. Besides its theoretical value, its mere absence from Book II could possibly be used as a temporal marker, to indicate that this part was written at the very early stages of Euclid’s preoccupation with Geometry, thus pushing further back the date the Elements were composed.

Temporal considerations and framework for P. OXY. 29

The fragment is heralded as: “one of the oldest and extant diagrams” (title) and “one of the oldest and most complete diagrams from Euclid’s ‘Elements’” (first line) in ref. [1]. If the expressions: “extant” and “oldest and most complete diagram” imply that this fragment is claimed to be part of some “original” manuscript (presumably written by Euclid himself), then the claim is very doubtful. As it will be argued, it is not at all sure that the same person who wrote the text was the person who drew the diagram (figure) on the papyrus (either as a fragment or as a roll). If the claim “oldest” is attached to another claim, namely that Euclid was a contemporary to the papyrus writing style, albeit this may not be Euclid’s writing; or that this fragment belongs to a lineage of copies going back to a first copy directly obtained from the original Euclid manuscript; then one might confidently reject both of these claims as very unlikely, too, for reasons that will become apparent in the analysis that follows.

Moreover, whatever the meaning of the two above-mentioned expressions might be, both inferences are very doubtful under the supposition that Euclid was not a person who lived in the 75-125 AD period, a period to which the dating of the papyrus fragment is currently attributed or even a bit later as this paper will argue. Euclid lived and worked significantly earlier, possibly in the 4th century BC, see ref. [3] for a more detailed reference to (and elaboration on) this subject. However, it should also be noted, given the uncertainty that is always present in dating the lives and works of ancient writers, the fragment’s writing per se does not directly and explicitly exclude this possibility. There are other, far more important and basic, factors that determine Euclid’s life span and story.

The Proposition’s writing and drawing styles

What is striking about this fragment’s text is not only the style of writing, but also the fact that it does not contain Euclid’s proof. Instead, the artifact contains a hand-drawn figure (diagram, schema) by an apparent scribe. The person who produced the text on this artifact (papyrus roll or fragment), and the reason(s) for doing so, is a set of questions that very likely is not resolvable. In omitting the proof, the writer may be directly confirming that the proof is redundant to the statement of the Proposition, i.e., the proof is included somehow in the predicate of the statement. To what extent this is so, is an epistemological issue to be addressed momentarily.

The writer behind this copy may have considered that in setting up this particular predicate, or the theorem’s statement (Proposition 5), that very statement of the problem contains within it the answer (as an identity, equality or tautology) to the question which it implicitly states. Or, alternatively, the proof is trivial, or at least too easy to expend resources on it. Or, that the proof is easily memorized and thus it does not need to be written down. Or, the copier of the Proposition(s) was just interested in the Propositions not their proof, taking for granted that once these Propositions have been proven, there is no need to keep proving them. Of course, all these possibilities are related to the copier’s intent behind copying the Proposition from some prior copy of Euclid’s Elements, or some other source. At present, one has no grounds to guess and speculate as to what that intent might have been. The papyrus fragment’s drawing is simply a graphic confirmation of the problem’s verbally expressed statement (theorem or Proposition). One might think of the diagram as offering the “proof”, except that it contains an unnecessary line (the diagonal). There are no special designations or writing on the diagram. Evidence seems to suggest that the person who drew the diagram is different than the person who wrote (copied) the text. That duality will be further expanded later in the paper. It also does not contain the gnomon, an element contained in the Elements in the proof of Proposition 5 as offered by Euclid.

The diagram is drafted in a manner that has the problem’s line (την γραμμη) cut almost in one quarter from its right end, and hence in three quarters from the left end, these segments constituting the two unequal parts. Consequently, the four corresponding rectangles, as drawn at the right-hand side of the diagram, are optically very close to four squares, hence diminishing the intensity of the theorem’s optical message conveyed. The optics of this fragment point to a scribe who was not particularly concerned about the aesthetic qualities of the diagram’s optics.

In Greek Geometry, optics played a major role, and it is not totally a random event that the Geometry and Algebra of the irrational numbers named “Golden” and “Silver” Ratios, designated as ϕ = 1.618033…, and δ = 2.414213…, correspondingly, aside the irrational number π, were all analyzed by Greek mathematicians and geometers not only because of their inherent mathematical (associated with the philosophical implications of “incommensurability” and the manner by which they can be approached) properties, but also their innate aesthetic appeal. The Golden Ratio is Euclid’s golden section (the “extreme and mean ratio”, ακρος και μεσος λογος). Arithmetic, geometric and harmonic means were studied for both mathematical and aesthetic reasons. A number of Classical Era architectonic structures were built embedding these ratios, such as (among the numerous other edifices) the Parthenon, where an approximation to the Golden Ratio is embedded on its Eastern and Western side elevations (facades), see ref. [4]; and an approximation to the Silver Ratio is encountered on the floor plan of the Temple of Apollo Epicurius at Bassae, at the mountains of the Arcadia region of Central Peloponnese, see ref. [5].

Moreover, the irrationals and incommensurability were topics that ancient Greek Mathematics dealt with extensively and over a long time period; the quadrature of a circle, doubling of the cube, and trisecting an angle were geometric problems that preoccupied mathematicians and philosophers of the Helladic Space over centuries, from Pythagoras to Plato and Aristotle, to Euclid and Archimedes, down to Apollonius and Pappus. The manner and methods of approximating the square root of prime numbers were at the core of Greek mathematical analysis in Geometry, as well as in Greek Epistemology. The Law of Proportions and the method of exhaustion were the basic tools employed by Ancient Greek mathematicians, through Geometry, to do what became later Algebra and Calculus. They used spheres, cones, cylinders, pyramids, and tori, together with various spheroids and conoids and their nonlinear intersections to solve quadratic and cubic equations. Through Geometry, they approached the study of ellipses, parabolas and hyperbolas. At the time of P. OXY. 29, assuming it was someplace in the 1st, 2nd or 3rd century AD, the state of the art in Greek Mathematics had advanced since Euclid.

Two, far more graphically appealing, examples of this theorem as stated, in terms of the difference between the lengths of the two unequal parts, i.e., the manner the foundational line (γραμμή) is partitioned, and hence their lengths’ ratio being significantly greater than 1, are drawn by the authors of both ref. [1] and ref. [2]. No skillful geometer would draw a line partitioning schema, as shown in the diagram of Figure 1, to prove a theorem involving squares of the unequal line segments, let alone a geometer the caliber of Euclid. This realization points to a scribe behind the drawing on this papyrus fragment, a topic to be elaborated later.

On the Epistemology of the Elements, and Proposition 5

What is: definition, given (known), axiom, proposition, theorem, proof, and the sought after (unknown)

Now, the Epistemology related aspects of Proposition 5 will be examined, and generalized to cover the epistemological angle of the entire Euclid’s Elements. Euclid, according to the surviving document as shown in refs. [2] and [10], follows up each Proposition (theorem) general statement with a rephrasing of it, that contains the explanation of the theorem as assigned to a figure (diagram) with lettering identifying what is given (usually with the preamble “let”) and what is sought after (starting with the expression: “I say”). Both sound like widely used Byzantine liturgical terms. It is not known whether these are the original Euclid terms, and not simply what has survived by the numerous copies and modifications the original document has been subjected to over the centuries since its composition by Euclid c. 300 BC. Since the original Euclid document in no longer extant, no one can be sure what it originally contained, its specific Linguistics and underlying Logic, and the form it expressed its statements and provided its proofs.

Proving Proposition 5, as stated, is extremely simple, once the person attempting to prove the statement transitions (through one-to-one correspondence) from the foundational (theorem stating) enunciation to drawing the operative (three in this case) rectangles and squares involved. If one designates the line as AB, with C at its center, and D the point of unequal division, see diagram in ref. [2], then the proposition becomes: prove that the rectangle formed by the lines AD and DB, plus the square formed by the segment CD, equal in area the square formed with sides equal to the line AC (or CB). Thus, drawing the rectangles and the squares automatically becomes part of the Proposition’s proof. Hence, this is to some extent in contrast to what was earlier remarked that the papyrus fragment “does not contain the proof”, which is also what the author of ref. [1] states.

But this is only apparently accurate. It is argued here that within the problem statement, in effect the proof lies. There is a fuzzy distinction as to what constitutes “theorem” and what constitutes “proof”. Proclus, see ref. [10], p. xxiii, recognizes six distinct components of any Proposition (written in Modern Greek): enunciation (πρόταση), setting-out (έκθεση), definition (διορισμός), construction (κατασκευή), proof (απόδειξη), conclusion (συμπέρασμα). Noticeable is that the term “theorem” (θεώρημα) does not appear in the above list.

All of these terms are subject to fuzzy, fluid, and ambiguous definitions a priori, see ref. [6]. They have overlapping borders, and this is the case here as well. “Enunciation”, “setting-out”, “definition” and “construction” express more or less the same thing and in effect they appear simultaneously within the Proposition’s statement and partly within the proof itself. Even in the above exposition of terms, linguistically the term “proposition” in modern Greek is (or “means”), “πρόταση” (i.e., “enunciation”). But more importantly, it is what is involved in the “proof” in reference to what is involved in the statement of the theorem, or “Proposition” (or “enunciation”) that matters most.

In the “construction” of the figure (i.e., in the drawing of the diagram, the image, the schema) part of both the “theorem” and the “proof” are included. More exactly, when one draws the square of the half-line in Proposition 5, the rectangles in question and the square of the half line overlap. So, here one has visual overlap of both the proposition statement and its proof.

This overlap directs the problem solver to the solution, unambiguously. Hence, the statement (Proposition) is a one-way, dictated, means (or road) to obtaining the solution. Noted is also the fact that no additional lines are needed to prove the Proposition. In effect, the very statement of the Proposition outlines a landscape in which a road (at times numerous roads) is (are) shown to lead one from the origin (the statement of the Proposition) to the destination (the proof and the conclusion). Parenthetically, often but not always, a “conclusion” in Geometry is more or less a re-statement of the Proposition, with the addition that what it was asked, “was proved”.

Hence, in all of the above-mentioned terms, significant redundancies are present. This realization does two things: it makes the distinction between what is proof and what is Proposition (or statement of a theorem) ambiguous; and hence it also shows that the author of ref. [1] is partly erroneous and partly correct in strictly affirming that “no proof” is shown in the diagram of Figure 1. The issue is a basic one, so much so that one wonders whether figures (diagrams) are needed at all and whether all geometric actions can take place mentally; and even question whether figures were at all present in the proofs of the original Elements. The ability to visualize and memorize, it seems, was valued and of import then.

Moreover, the statement of the Proposition is a statement of a Theorem, in which the implied axioms are embedded. Notice that in the Proclus exposition of the six terms present in any statement of a theorem in Geometry, no mention of the term “axiom” (αξίωμα) is made. Axiomatic statement of theorems in Mathematics (and in Symbolic Logic, as well as all of the Natural, as well as some of the Social, Sciences and Linguistics) is a 19th century development on which much of modern-day Geometry, Algebra, Arithmetic and other branches of Mathematics are based. For more on this angle of analysis, see ref. [6]. In addition to the blurry borders between Proposition statement and proof, i.e., where one ends and the other starts, interpretation of the fuzzy border directly hinges on the fundamental question what is a “proposition” in terms of the “knowns” it contains and the “unknowns” to be found within it, as both become entangled entities imprinted in the very statement of the problem to be solved. Similarly, one has significant difficulties exactly identifying borders between elements of an axiom and elements of a theorem. Or, whether enough axioms are mentioned, or whether they are complete. Along the same lines, regarding primordial and completeness, interpretation of what constitutes “knowns” and “unknowns” also renders the very definition of these two terms fuzzy and imperfect, not at all clear-cut and unambiguous. This is especially so when new lines are needed to be drawn in order to prove a Proposition (theorem). The epistemological aspects of this angle of looking at the Proposition, aspects that obviously transcend this specific Proposition 5, or the Elements, or even Euclid, have been revisited in the 19th century by logicians and mathematicians; see ref. [6] for more elaboration on this topic and associated references.

The reader is reminded that Epicurean philosopher and mathematician Zeno of Sidon (c. 150 – 70 BC) was the one who first raised issues (in so far as written documented evidence suggests) about the epistemology of Euclid’s Elements and the meaning of his “axioms”. See ref. [19].

This ambiguity among key notions in not only Proposition 5 but in the entire thirteen Books of the Elements becomes apparent when one looks at or draws the figure (diagram). Once the scribe forms the rectangles, as shown in either ref. [2], more precisely, or ref. [1], it becomes immediately apparent that the proposition holds, since in it (from the diagram in ref. [2]) the rectangle ACLK is equal to the rectangle BDGF (BM=BD=CL, and BF=AC). Drawing the diagonal line EHB in the diagram of ref. [2]; or in the (undesignated) diagram directly corresponding to the schema drawn in the papyrus fragment in Figure 1, by the writer in ref. [1]; or the diagonal in the papyrus fragment itself; is wholly unnecessary. Unnecessary is also the threequarters-circle NOP (Euclid’s “gnomon”, a topic to be discussed later in the paper) drawn on the figure corresponding to this Proposition in ref. [2].

Moreover, the elaborate proof supplied by the author of ref. [2], which is an approximate replica of Euclid’s proof (as it has survived to this day) apparently is (was) done to add commentary with extensions regarding a more comprehensive approach to the entire, not only set of propositions in Book II, but the thirteen Books manuscript of Euclid’s Elements, as they emerge from this theorem (i.e., as found in Proposition 5, Book II). However, it is unnecessary if one wishes (as Euclid apparently did not wish) to just prove Proposition 5 efficiently. They were obviously other reasons (one can only currently guess) Euclid wished to serve, including an attempt towards completion, self-consistency, streamlining and progression, all aims towards offering a comprehensive (for the time) reference manual and a compendium on Geometry, for mostly (albeit not exclusively) educational purposes.

Construction of the relevant rectangles and squares is far simpler than described in ref. [2], however. The only condition needed is to know how to draw vertical lines to a foundational line; this is an easy task, done by using a compass and a ruler, that is by drawing circles with centers on the line and enough radii in length so that they appropriately intersect, and by linking the two points of the intersecting circles. Hence, only vertical lines to the original (foundational) γραμμή at points A, C, D, B are needed; along with a line parallel to the original γραμμή from a point at any of these vertical lines, at a distance equal to the smaller segment of the γραμμή where the cut (τομή) was taken to be; and another parallel line to these two at a point on any of the vertical lines (although only three of the four possible intersections are of the essence, designated as L, H, M in the diagram of ref. [2]) and at a distance equal to the length CD. This is all that is needed to both conceptualize the Proposition and most importantly commence proving it.

The transition from the original foundational line (γραμμή) to the 2-d drawing is what matters. It is again emphasized that no diagonals are needed to be drawn, as they add absolutely nothing to either the problem statement or its proof. It must be remarked that the Proposition as stated directly implies the drawing of the diagram, hence the very proof of the Proposition (which now can be viewed as a theorem) itself.

A transition from the problem (theorem) statement, although not necessarily automatic and inevitable, since some minimum creative thought is required by the problem solver (the scribe in this instance), directly implies the drawing of the schema as shown in the papyrus segment of Figure 1. The only other way to solving this problem, or prove the theorem (Proposition) is algebraically, as it will be shown in subsequent sections of the paper.

 

Alternative proofs

Using areas

A much simpler proof of the purely Geometry based statement of (and found in an apparent version of Euclid’s original) Proposition 5, Book II, is the following representation and designation involving five areas: consider the area (again, referring to the notation of ref. [2]), formed by rectangle ACLK as x; by rectangle CDHL as y; by rectangle DBMH as w; by the square LHGE as z; and by rectangle HMFG as u. Then, one has Proposition 5 stated as follows:

prove that: x + y + z = z + y + w + u.

Cancelling terms from both sides, and by simplifying, the above equation directly becomes: prove that, x = w + u, which holds as an identity since the smaller in length (as drawn) side AK of the rectangle with area x is equal to the side DB of the rectangle DBFG; and the length of the side BF of this rectangle (DBFG) is half of the original foundational line (γραμμή) AB (AC=CB=BF), since DB=BM, and MF=EL=EG=LH=CD.

It is at this point where it becomes apparent that the proposition could not be generalized along the lines suggested earlier. The areas simply do not allow for the statement to go forward. The Algebra to clearly show this inability will be provided momentarily, so that the above statement can be directly checked.

The algebraic version of the Proposition

As noted in ref. [1], the algebraic expression associated with this problem statement, Proposition (or theorem) is: {(ab) + [(a – b)^2]/4} = [(a + b)^2]/4, which is a variation of a rather simple Algebra problem, involving expansions, from elementary quadratic equations, as in the expression:

(a + b)^2 = a^2 + 2ab + b^2.

A hybrid statement, containing both Algebra- and Geometry-based expression, of the verbally stated problem (theorem or Proposition), as directly emerging from the above statement (as given in refs. [1], [2], or this author’s restatement of J.L. Heiberg translation into “Modern Greek” of the Proposition as shown on the papyrus fragment in Figure 1) is, using the letter-based notation as already discussed earlier from ref. [2]: (AD)(DB) + CD^2 = [(AB)/2]^2. In pure Algebra-based expression, where the original line’s two segments are designated as: AD=a, DC=b, the problem becomes as stated above (and in ref. [2]).

In the case of the generalization, suggested earlier and shown not to be possible, the Algebra version of it is that: when the point D falls off the segment AB, at some extension of that line, then the corresponding Algebra of the Proposition becomes: {(ab) + [(a + b)^2]/4} = [(a – b)^2]/4, since CD = (a – b)/2 + b = (a + b)/2, and consequently the above condition, as stated in Proposition 5, does not hold. It is a case of reductio ad absurdum.

Noteworthy is that, the Algebra of the Proposition as provided by a current day analyst omits certain intermediate steps in the algebraic expression/equality shown above; in ref. [1], the analyst skipped two steps that enter between the left- and the right-hand sides of the equation, namely the expressions (in terms of equality): 4ab + a^2 + b^2 - 2ab = a^2 +2ab +b^2 (as well as certain, more elementary, intermediate steps). These steps are not explicitly shown, they really do not need be there. They are presumably assumed to be in the mind of the person who does the necessary Algebra. It is a reasonable assumption, albeit still an assumption.

It is reasonable to expect that a similar type of omission and expectations were in place back at the time the papyrus was used. At the time of Euclid, no matter what that time was, as long as it falls in the 300 BC (and possibly earlier) to the 100 BC range (and possibly a bit later), see ref. [3], this level of Algebra (not using numbers, but symbols in the form of letters, corresponding to Arabic numerals) was well known among Greek mathematicians. For sure, at the time period this papyrus fragment is dated (75-125 BC), Algebra was well developed to accommodate simple operations of the type involved in Proposition 5 (and not only), and certainly the intermediate steps involved in the solution of the quadratic equation involving part of this Proposition’s elementary Algebra.

Algebra in the Helladic Space, including Alexandria of the Hellenistic period and the Roman Era, was developed by Diophantus of Alexandria (in the 3rd century AD). The argument that P. OXY. 29 was made prior to Diophantus (as suggested by E. Turner) is hard to accept. The fact that the gnomon is not present in the diagram of the extant papyrus fragment (as it was on Euclid’s alleged original proof) can be attributed to the realization that Geometry would be enough to solve the Proposition, and that Algebra could provide an alternative way to do so; hence, we are observing in P. OXY. 29 initial developmental stages of the eventual split, possibly a split that was underway then. And that time was much closer to Diophantus than the “125 AD upper bound” of the Turner hypothesis.

 

Imperfections in the diagram, their meaning and context

The diagram’s maker

In summary, in so far as the Geometry of the Proposition is concerned, the critical points with regards to apparent imperfections about the diagram as drawn in the papyrus fragment and shown in Figure 1 are: (a) the drawing of the unequal line’s segments so that they look like one quarter and three quarters, respectively, of the original foundational line (γραμμή); and (b) the diagram’s diagonally drawn line. These two elements of Figure 1 provide two critical clues as to why this is not a professional geometer’s work (let alone Euclid’s), but a scribe’s rendition of Proposition 5. More regarding the fragment’s imperfections and how they may be related to the maker (writer, copier) or user (scribe) of the papyrus fragment in a subsequent subsection of the paper.

It is obvious that the person, if not a scribe very likely someone just above a scribe’s level in the educational hierarchy of the time, who wrote the problem statement (as reproduced in ref. [1]) and shown in Figure 1, either copied it from some other master source; or wrote it down from someone who orally dictated the Proposition to the writer. Then the scribe entered the picture, and either inadvertently and in a hurry drew the diagram, so that the line (γραμμή) was almost partitioned in quarters, and also drew the (unnecessary) diagonal line possibly to facilitate the drawing of the Figure’s diagram (in absence of a ruler and a compass). Judging from this papyrus fragment’s upper side, where writing can still be clearly discerned, and (see ref. [1]) possibly referring to the previous Proposition 4 (of book II) of Euclid’ Elements, this fragment is very likely part of a greater document possibly exclusively related to the Elements.

The author of ref. [1] mentions that this fragment was likely part of a 30-feet long roll of papyrus. If so, it almost certainly contained more theorems and problem statements (although not necessarily figures). Very likely, the section of the papyrus that didn’t survive (if it did not contain Proposition 4 of Book II), contained material directly related to this particular theorem.

The problem (statement, theorem, Proposition) as presented is self-contained on the fragment’s part that survived. However, this is only the case if one assumes that the user of the section of the Book II was aware of the previous Propositions, or at least of the Definitions offered by Euclid before Book I. A careful examination of the fragment’s photograph reveals that some ineligible symbol(s) is(are) drawn next to (immediate upper right-hand side of) the figure. No letters (or other designations, or diacritical marks) are put on the diagram, and no equations are directly indicated to be associated with the problem’s possible algebraic expression.

Two persons involved

The diagram shown in Figure 1 contains eight lines. A current day analyst can’t be sure as to (a) the exact sequence the scribe followed in drawing these eight lines, hence, which rectangles or squares were drawn and appeared first in the drawing; and (b) what tools (of the required set, i.e., ruler and compass) the scribe had at hand while drawing the diagram in Figure 1. Possibly, none of the above two tools (used by Greek geometers then to do Geometry), were at the scribe’s disposal at the particular point in time; very likely, the scribe drew the diagram unaided by hand and in a hurry. The manner the lines intersect is rough, not especially sharp and precise. Moreover, the lines are not exactly parallel.

These imperfections are indicative of work most likely done by a scribe who was not particularly fond of exactness, and definitely indifferent to the aesthetic appeal of the drawn diagram. In combination, these numerous imperfections are quite informative, since they act (among other things) as time markers and stamps in the transmission of message from the maker to the observer. Such attributes however stand in sharp contrast to the very careful, almost calligraphic and exact writing style of the person who wrote the fragment’s text. The same calligraphy and the ink’s similar tone is also found in the unclear symbol drawn (upper right-hand side), which looks like a Latin “n”, next to the figure. The imperfections cited with regards to the figure signal to the current observer that: (a) the papyrus fragment drawing was likely done by a scribe and not a professional geometer; (b) the conditions under which the second scribe likely drew the diagram show that the scribe was not only in a hurry but also careless; and finally (c) the papyrus fragment was intended to be used for learning, and not for presentation purposes or other more formal proceedings. Most importantly, a careful examination of the text and the figure reveals that: (a) two different types of writings are involved in the fragment; one is the relatively thin writing of the text, juxtaposed to the relatively thick writing of the drawing’s lines. (b) the ink used in the two forms of writing (text and drawing) have different tones and absorption rates by the papyrus; the shade of the drawing’s writing is considerably darker than that of the text.  All the above-mentioned factors point to a preliminary, but very likely, scenario: two different individuals’ writing are found on the papyrus fragment; or that the same scribe wrote twice at close temporal proximity.

However, the overall context of the fragment P. OXY. 29 and the writing style seem to indicate that the writing of the text (and the symbol next to the figure) and the drawing of the figure happened in relatively very close temporal proximity, possibly within a year of each other.

More analysis needed

In addition, to a geometer and analyst that looks at the diagram, deducing the sequence in the drawing of the diagram’s lines is important, as it could reveal how much background knowledge was involved on the part of the scribe. It could also possibly reveal the approximate amount of time it took for the scribe to arrive at the problem’s solution. It could also reveal what was the intended use of the papyrus, either of this particular section or of the whole roll.  Thus, the entire archeological matrix of the papyrus segment need be examined, including some forensic analysis, something however that falls outside the purview of this paper and the capacity of this author to carry out, in order to reach firmer conclusions.

Potential use of the papyrus is essential in understanding its making. Did the scribe use the papyrus as an exercise implement? Or, alternatively, did the scribe use the Proposition(s) containing part of the papyrus roll as a textbook? That is, the scribe used the material on it not as something to be proven, but rather something to be learned, i.e., to be taken as a given and well accepted and to retain in memory type set of theorems (Proposition 5 being one of them), and accept it (them) as a matter of course, memorize it (them), without any proof of it (them) needed (at least immediately) at the time of reading it (them). This is not idle speculation type set of questions. Important epistemological issues hang in the balance, as already discussed in the previous section of the paper. Issues involved have to do thus, not only with definitions of what axioms, theorems, proofs, deduction, etc., are.

But also, questions of inquiry versus statement of facts, i.e., absolute a priori “truths” a scribe was to know about Geometry; versus the scribe learning on how to prove the validity of geometric statements, i.e., derive and supply the proof of theorems (Propositions, in the case of the Elements).

It is very likely that this papyrus (in its entirety) was for the private use of the scribe, and it was not meant for further use by any other person. This is attested, at least, by the rough drawing, which is not of presentable quality.

Since Algebra is not present in the above fragment, it can’t be known what was in the scribe’s mind, or to the mind of the person who dictated that problem (theorem, Proposition) to the scribe. Or what was the original copier’s intent and knowledge base regarding both Geometry and Algebra. That some type of possession and retention of prior knowledge in terms of Geometry and/or possibly Algebra was undoubtedly required for the scribe (and the original writer/copier) to have; this is rather safe to assume.

As discussed in ref. [3], where the Egyptian Rhind and Moscow papyri are reviewed in some detail, and where also certain Babylonian tablets were analyzed, such strong retention was required for scribes in both Ancient Egypt as well as Mesopotamia. It is Egypt and about 180 miles SE of Alexandria that this papyrus fragment (P. OXY. 29) has been found. It is within the broader spatial framework, inclusive of these places and the Helladic Space, that the fragment and its contents must be reviewed and analyzed. That framework is outlined in ref. [3] by this author.

Two additional points need be made: (a) The diagram is right-hand side drawn, as the τομή (point D, from the previous analysis) is at the right-hand side of the foundational line’s center; it could have been drawn in a left-hand manner, i.e., with the “cut” (point D of ref. [2]), left of center C; and (b) this fragment, fortuitously, survived in such a manner that the entire statement of the problem, Proposition 5 (or theorem) could be reconstructed. Although not much more can be directly inferred by these two observations, neither of these two conditions is totally random.

 

Linguistics, diacritic elements and dating the fragment

Style of writing

The total absence of any Algebra in the presentation of either the Proposition, or the drawing shown in the papyrus fragment of Figure 1 may indicate that this papyrus roll, and whatever else it contained, was purely Geometry related, and not Algebra. Maybe the scribe used this fragment in a course only on Geometry, and possibly another fragment (or roll) in a course on Algebra. Because it is rather certain that by the time this papyrus fragment came to light Algebra was developed well past elementary quadratic forms and equations. In this case, this papyrus fragment could be of an even later date, when a possible complete differentiation of courses, between topics exclusively related to Algebra and material exclusively connected to Geometry was in effect. 

Maybe, the manuscript (i.e., the original handwritten by Euclid manuscript, the Volume titled Elements), was put together at a critical time, a temporal junction when either this bifurcation between Geometry and Algebra, or the merger of the two branches of Mathematics occurred. This is another critical point of ambiguity that a contemporary analyst encounters and has to resolve (if at all possible, to currently resolve), as the analyst reviews not only the content but the entire context surrounding this particular fragment of the Egyptian papyrus.

However, it is not the absence of an algebraic expression that is of interest so much in the above fragment. It is the potential lack of extreme efficiency in the Proposition’s statement. What is of interest along these lines is the convoluted Linguistics used, albeit still efficiently stated since there is no excess verbiage in it or unneeded repetition. But the statement is only one among the numerous ways (more or less efficient) the theorem, problem or Proposition, could have been stated, no matter when beginning with the original statement by Euclid or even its antecendent statements by prior geometers. This possibility was pointed out earlier with this author’s additions in parenthesis inserted to make the exposition of the Proposition clearer.

As it currently stands, it is purely a dry Geometry-based statement, i.e., its verbal articulation is merely Geometry, although easily perceived in algebraic terms as well. And in the drawing attached to the Proposition’s statement, Figure 1, one finds a naked diagram, void of any explanation or diacritic signs. Any potential inefficiencies one detects in the Proposition statement constitute an additional cause to doubt the temporal proximity between the papyrus segment and Euclid’ hand written original document.

But where does that decrease in temporal proximity, or increase in temporal distance, lead one is unclear. It is to some extent documented that Archimedes mentions Euclid, see refs. [3] and [10] for more extensive coverage and potential issues with this reference. Hence, if one assumes that Euclid’s Elements was a single person’s work and not a set of individuals’ mostly (albeit not exclusively) Geometry related output, included in a Volume titled Elements; then one must assume that this papyrus maybe of a later date, rather than Euclid lived earlier than the time usually assigned to him (late 4th to middle 3rd century BC, well into the Hellenistic Era).

Noteworthy is that the original finders of this papyrus fragment, see ref. [1], placed this manuscript to circa 300 AD. Potentially, dating the artifact between the 125 AD and 300 AD range could be much closer to a very likely range. The writing style tends to point at such a range, as it resembles writing of the pre- or proto-Byzantine handwriting, hence pulling the lower bound closer to Byzantine times. Furthermore, as indicated at the Introduction, Greek koine writing, even at the time of the (see Figure below) 196 BC Rosetta Stone time period (well beyond the Euclid, c. 300 BC period when the original set of thirteen Books of the Elements are thought to have been composed) was in uppercase (capital) letters, and so was the Greek numbering system prior to the Byzantine style of lower and uppercase writing, see ref. [21]. Moreover, key omissions pointed out from the P. OXY. 29 papyrus fragment text (i.e., the lack of any proof-related text material, thus deletion of key elements from the original Euclid manuscript) would tend to indicate a greater temporal distance from the original than contained in the 125 AD limit.


Figure 2. The Rosetta Stone. The bottom writing imprinted on this artifact of 196 BC is in koine Greek. The writing contains exclusively capital letters. Source of image and credit: https://discoveringegypt.com/egyptian-video-documentaries/mystery-of-the-rosetta-stone/

 

On the term “gnomon”

Another key issue that must be raised is the use of the word “gnomon” by Euclid, see ref. [2]. The term “gnomon” does not appear on this version of Proposition 5, or on its accompanying figure. Its absence from the P. OXY. 29 fragment is instrumental in dating this artifact. As noted in ref. [2.2], Euclid used it to imply three quarters of a circle: it is explicitly defined by Euclid at the very beginning of Book II, under “definitions” (it is the second one), see also ref. [10], p. 37. The term is also used by Archimedes in his Book: On Conoids and Spheroids, see ref. [11], p. 144.

The subject of the “gnomon” is addressed by Thomas L. Heath in his classical treatise A Manual of Greek Mathematics, see ref. [12]. Usually, a “gnomon”, in Greek Geometry and Astronomy, is associated not only with right angles, but also (and mainly) with sundials.

It is the shadows cast by gnomons on sundials’ surfaces that are used to gauge not only time of day, but also day of the year, hence seasons, through the use of sundials.

Maybe, its appearance in the Elements (or subsequent versions of it) is a highly idiosyncratic use of the term by Euclid (and also by Archimedes). Or, maybe, it was a term with a wider use in Alexandria during its early formative time period, at the aftermath of its founding by Alexander III in c. 331 BC. It is one additional element in the cloud of fuzziness that surrounds issues under discussion, a central point in this writer’s approach to matters of Archeology and History, see ref. [7], [8] and for the general theoretical framework see ref. [9]. Be that as it may, the absence of this gnomon from the papyrus fragment P. OXY. 29 may be instrumental in placing a date on the artifact’s making, at least in so far as the point in time the drawing was sketched out. It could have been the beginning of the appearance of Algebra and the Diophantine era in Mathematics.

 

Conclusions

In concluding this paper, it is noted again that no specific symbols or diacritic elements of any type appear on the fragment’s text or figure, with the possible exception of the ineligible symbol at the upper right-hand side of and almost immediately next to the drawing. This symbol has the writing and ink tone and absorption characteristics of the main text and not of the drawing. A possible explanation could be that this is a mark of approval by the teacher for the scribe, once the scribe produced the answer (in terms of proof of the Proposition) to the teacher.

A number of preliminary conclusions have emerged from this analysis, which focused on papyrus fragment labeled “P. OXY. 29”. First and foremost is that much of the archeological matrix surrounding the papyrus fragment that contains a copy of Euclid’s Proposition 5 from Book II of his Elements remains still clouded in mystery. Although the statement of the proposition as recorded on the fragment is unambiguous (as is the original statement by Euclid) in its Geometry, and rather accurately reflects its original content, the question as to whether the papyrus’ fragment was intended to simply be a remainder of the embedded Theorem or the beginning of a proof employing exclusively Geometry remains unanswered.

Numerous issues were touched in this paper: (a) the time frame within which the original Euclid proposition was stated; (b) who were behind this specific papyrus fragment; (c) the writing by possibly two persons in firstly copying from a prior source and writing text on the papyrus roll, and secondly using the papyrus to both memorize and drawing a figure to prove/solve the embedded Proposition (theorem); (d) the writing details of the text and the drawing of the associated figure; (e) the possible epistemological aspects in regards to both, Proposition 5 from Book II of Euclid’s Elements, and the manuscript in general.

In addressing these topics, the analysis presented suggests (i.e., preliminarily concludes) that: (a) the manuscript P. OXY. 29 was probably, at the time, the end point in a line of a continuous string (chain) of copies from the original Euclid’s Proposition 5 from his Elements, and that it is possibly younger than 125 AD and older than 300 AD; (b) that behind this papyrus fragment are two persons, an original copier who wrote the text, and a scribe who composed the fragment’s figure in a hurry and in a clumsy sort of way; (c) that the method used to either memorize the theorem or solve it, exclusively concentrated on Geometry, as in the statement of Proposition 5 itself there was no hint as to an equivalent Algebra related problem, or to one that could be stated in terms of Analytical Geometry.

Moreover, in the original statement by Euclid of the Proposition, to the extent that one could tell by the copies available today, there is no hint of Algebra or Analytical Geometry, or to the implications of a possible generalization of Proposition 5, although a number of the Proposition’s extensions are listed (Propositions 6, 7, and 8 in Book II); (d) the fragment’s contents have a style of writing that corresponds to pre- or early Byzantine writing, but still placing this extant copy earlier than both Pappus and Proclus.

Finally, mention was made to a symbol, placed right next to the diagram, at the right-hand side, analysis of which has escaped the analysts’ attention thus far. It was written by the first scribe.

 

Appendix: Proposition 6

In plain English, and using simple Algebra, Proposition 6 states the following: Given a straight line, AB (of length a), and an extension of it (on the side of point B, without any loss of generality); consider point C, at half-point on the segment AB, and point D on the extension, such that length AC = CB = c = a/2, and length BD = b. Prove that: (b + c)^2 = (a + b)b + c^2.

Expanding, one obtains on the left-hand side: b^2 + 2bc + c^2; on the right-hand side, one has: ab + b^2 + c^2. What is left to prove is that ab = 2bc. Since c = a/2, by substitution one obtains directly what is asked, {or QED (quod erat demonstradum), as it is usually employed in current translations from the latest Greek text, whereas Euclid used the expression (if he originally did) ΟΕΔ (ΟΠΕΡ ΕΔΕΙ ΔΕΙΞΕ)}. It ought to be remarked that Proposition 6 is instrumental in solving the problem of doubling a cube, in the context of Archimedean ΝΕΥΣΙΣ (ΝΕΥΣΕΙΣ plural), see Thomas L. Heath, ref. [11], Chapter V, §4, p. cx. This theorem is instrumental, too, in the problem of trisecting an angle, as Heath points out in his next section, where the issue becomes whether plane vs solids (conics) are needed to solve it.

In terms of Geometry, just the drawing of the rectangles and the squares as asked by using the ruler and the compass, and without the need of any diagonals or any additional lines (or the “gnomon” shown in references [2] and [10]), would enable one directly to obtain the sought-out result.

 

 

 

References

[1] https://personal.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html

[2] http://aleph0.clarku.edu/~djoyce/java/elements/Euclid.html

[2.1] http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII5.html

[2.2] http://aleph0.clarku.edu/~djoyce/java/elements/bookII/defII.html

[3] Dimitrios S. Dendrinos, 2022, “Eratosthenes, Egyptians, Mesopotamians, and the length of Earth’s polar circumference: historical evidence and fuzziness”, academia.edu (forthcoming).

[4] Dimitrios S. Dendrinos, June 29, 2017, “On the Parthenon’s Mathematics, Astronomy and its Embedded harmony; its Skeletal Geometry, Modulus, three Key Triangles and Core Angles”; the paper can be accessed here: https://www.academia.edu/33707767/On_the_Parthenons_Mathematics_Astronomy_and_its_Embedded_Harmony_its_Skeletal_Geometry_Modulus_three_Key_Triangles_and_Core_Angles

 [5] Dimitrios S. Dendrinos, April 10, 2017, “Moving Shadows and the Temples of Classical Greece”; the paper can be accessed here: https://www.academia.edu/32383784/Moving_Shadows_and_the_Temples_of_Classical_Greece

[6] Dimitrios S. Dendrinos, June 10, 2021, “On Imperfections: Ideas, Morals, Actors, Observers, Actions”; the paper can be accessed here: https://www.academia.edu/49193216/ON_IMPERFECTIONS_Ideas_Morals_Actors_Observers_Action

[7] Dimitrios S. Dendrinos, October 9, 2021, “Quanta in Neolithic Architecture, Part I: Towards a Speculative General Theory of Evolution“; the paper can be accessed here: https://www.academia.edu/56794842/QUANTA_IN_NEOLITHIC_ARCHITECTURE_PART_I_TOWARDS_A_SPECULATIVE_GENERAL_MODEL_OF_EVOLUTION

[8] Dimitrios S. Dendrinos, October 9, 2021, “Quanta in Neolithic Architecture, Part II: Evolution from Monoliths to Corbelling and Pyramids“; the paper can be accessed here: https://www.academia.edu/56798993/QUANTA_IN_NEOLITHIC_ARCHITECTURE_PART_II_EVOLUTION_FROM_MONOLITHS_TO_CORBELLING_AND_PYRAMIDS

[9] Dimitrios S. Dendrinos, April 15, 2019, “Towards a New Epistemology“; the paper can be accessed here: https://www.academia.edu/38814232/TOWARDS_A_NEW_EPISTEMOLOGY

[10] Thomas Little Heath (translation), Dana Densmore (ed.), 2002, Euclid’s Elements, Green Lion Press, Santa Fe, NM; this is second edition, the first being a 1956 unabridged reproduction of the original three-volume text written by T. L. Heath, and published by Cambridge University Press in 1903.

[11] Thomas Little Heath, (ed.), 2002, The Works of Archimedes, Dover Publications, Inc., Mineola, NY; this is an un unabridged reproduction of the original text published by Cambridge University Press in 1897, and of a second edition by Cambridge University Press, in 1912, where the newly discovered then Methods manuscript by Archimedes was incorporated.

[12] Thomas Little Heath, 2003, A Manual of Greek Mathematics, (second edition, first published in 1963), Dover Publishers, Mineola, NY; this is the unabridged edition of Oxford University Press, 1931 edition.

[13] Johan Ludvig Heiberg, Heinrich Menge, eds., 1883-1916, Euclidis Opera Omnia, Eight Volumes, Teubner, Leipzig.

[14] https://plato.stanford.edu/entries/doxography-ancient/

(Article written by Jaap Mansfeld, 2020).

[15] Dimitrios S. Dendrinos, April 19, 2016, “Alexander’s Network Cities and their Dynamics”; the paper can be accessed here: https://www.academia.edu/24667299/Alexanders_Network_of_Cities_and_their_Dynamics

[16] https://mathshistory.st-andrews.ac.uk/Biographies/Theon/

(Article written by J.J. O’Connor and E.F. Robertson, 1999).

[17] https://historyofinformation.com/detail.php?entryid=2749

[18] http://www.papyrology.ox.ac.uk/POxy/lists/lists.html

[19] https://mathshistory.st-andrews.ac.uk/Biographies/Zeno_of_Sidon/

(Article written by J.J. O’Connor and E.F. Robertson, 1999).

[20] https://hfc-worldwide.org/blog/tag/koine/

[21] https://web.archive.org/web/20100202054101/http://www.dma.ens.fr/culturemath/histoire%20des%20maths/htm/Verdan/Verdan.htm

 

Copyright statement

©Dimitrios S. Dendrinos

The author, Dimitrios S. Dendrinos, retains the full copyrights to the contents of this paper, with the exception of the cover page photograph in Figure 1, which belongs to the public domain, and the image of Figure 2, to which attribution and credit has been supplied in the paper’s body. Copyrights to this paper are afforded the author by the relevant Laws of the United States government, and under all relevant Laws and their Provisions of the relevant International Authorities. No part of this paper, or the paper as a whole, can be reproduced without the explicit consent of, and written permission issued by the author, Dimitrios S. Dendrinos.

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