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.
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.
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/
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.
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/
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.
