Abstract
The analysis of a historical mathematical textbook constitutes a particular challenge when the original had been composed before the invention of the printing press and an even greater one when the very original might even not have been fixed as a manuscript. The analysis of such cases affords a focus on the relation between orality and written text. A particular clue for this relation is given by the diagrams accompanying the text. Diagrams in mediaeval manuscripts have several particularities which have been eliminated in modern editions as inappropriate for mathematical arguments. However, they may have not been so strange in the context of the direct oral teaching in which the teacher drew the diagrams in front of the pupils. The text and the structure of the Elements also contain other particular features preserved also in modern editions of which the following ones are examined: (1) no use of proposition numbers, (2) label assignment in alphabetical order which may result in different assignments in similar propositions, (3) oscillation of the appellation of the objects expressed by two or more letters such as line AB and line BA, (4) long and sometimes incomprehensible protasis (general enunciation at the beginning of each proposition). All these particularities can be interpreted as traces of oral teaching and communication of mathematics in ancient Greece.
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Notes
This article is based on a plenary lecture delivered on May 8, 2017, at ICMT2 (II International Conference on Mathematics Textbook Research and Development), in Rio de Janeiro. The author would like express his sincere gratitude to Henry Mendell for linguistic corrections and precious suggestions concerning interpretations of documents.
(Heiberg 1883–1916). The first four volumes published between 1883 and 1885 are dedicated to the 13 books of the Elements. The spurious books XIV, XV and the scholia were published in 1888 as volume 5. These five volumes have been reprinted (suppressing Latin translation) as “Heiberg–Stamatis 1969–1977”. The rest of “Heiberg 1883–1916” contains Euclid’s other works.
In this article I use the following three manuscripts. (All accessed 21 March, 2018.) P: cod. Vatican. Gr. 190. Ninth century. Heiberg’s dating to the tenth century (Heiberg 1883–1916, Vol. 1, p. VIII) has been corrected to 9th (See Mogenet and Tihon 1985, p. 23). http://digi.vatlib.it/mss/detail/178321; http://digi.vatlib.it/mss/detail/Vat.gr.190.pt.2. B: cod. Bodleian. Dorvillian. 301. Ninth century (precisely, copied in 888). http://digital.bodleian.ox.ac.uk/inquire/p/06cfa3b7-2aad-465e-88ac-0ebe3f2b5d13. F: cod. Florentin. Laurentian. XXVIII, 3. Tenth century. http://teca.bmlonline.it/ImageViewer/servlet/ImageViewer?idr=TECA0000321839&keyworks=Plutei#page/1/mode/1up.
By the expression.
$$r({{\text{A}\Delta}},{{\Delta \Gamma}})\,=\,sq({{\text{B}\Delta}}).$$(1)I mean “the rectangle contained by ΑΔand ΔΓ is equal to the square on ΒΔ”. This relation is often expressed in algebraic manner as follows:
$$\text{A}{{\varvec{\Delta}}} \times {{\varvec{\Delta}\varvec{\Gamma}}}={{\varvec{\Delta}}}{{\text{B}}^2}.$$(2)I avoid this, for Euclid never speaks of the product of two lines, nor of the second power of lines.
For further arguments, see (Saito and Sidoli 2012). The diagrams of Book I of the Elements are reproduced by Saito (2006). For those of Books II–IV, VI, XI–XIII, visit the author’s site: http://www.greekmath.org/diagrams/diagrams_index.html.
Ernst Ferdinando August (1795–1870) was a Prussian Gymnasium teacher.
This is exactly what Plato lamented about written words: “when ill-treated or unjustly reviled it (written word) always needs its father to help it; for it has no power to protect or help itself.” (Phaedr. 275e) For the text of Plato, I use Harold N. Fowler’s translation on the Perseus web site.
In Book VII, Gregory represents numbers by dotted lines (the number of dots represents the exemplar value of the truth of the proposition), and from Book VIII, he adopts the representation without lines, which August uses from Book VII.
For further information and reproduced diagrams, see the publication by Saito (2018, forthcoming).
On the other hand, a papyrus fragment (Oxy. i 29), dated to the turn of first and second centuries CE, has number 5 beside the diagram of proposition II.5 (Fowler 1999, p. 210–2 and plate 2 between p. 6 and p. 7), and Alexander of Aphrodisias (fl. ca. 200 CE) uses proposition numbers to refer to a proposition.
The detail of label assignment in these propositions is as follows: first, the circle is named by three points on it, Α, Β, \({\Gamma }\). Then, the point Δ is taken outside the circle. The line cutting the circle is named ΑΒΔ, and the tangent line (III.36), or the line satisfying the condition of equality (III.37) is named ΔΒ. Hitherto the assignment of labels is the same in both propositions. By the way, Α, Β and Γ are any points on the circumference when they appear for the first time, then Α and Γ become the point of intersection of the cutting line and the circle, and Β becomes the point where the other line (in III.36 tangent) falls on the circle. Netz (1999) describes such a phenomenon in this way: “they [the points] may first appear as underspecified, and later get complete specification” (p. 21). Now, in III.36 (more precisely, in the second case of III.36), the center is taken, and it is named Ε, in the alphabet the letter after Δ, while in III.37 the proof first draws tangent ΔΕ from Δ (ΔΒ is not assumed to be a tangent in this proposition!), then the center of the circle is taken, which cannot be Ε, for the label Ε is already used, and so it is named Ζ, the next letter after Ε.
In proposition VI.21, labels are assigned to rectilinear figures, not to the points on them.
The division of a proposition into six parts and the Greek names for each part comes from Proclus. (See Mueller 1981, 11 ff.) The six parts are as follows: protasis (general enunciation), ekthesis (setting out, i.e., restatement of protasis with lettered diagram), diorismos (specification, the assertion of what should be proved; it usually begins with “I say”), kataskeue (construction, absent in several simple propositions), apodeixis (demonstration, the demonstrative argument in narrow sense), sumperasma (conclusion, where the protasis is repeated).
In manuscripts, the diagram is usually placed at the end of each proposition (if diagrams are drawn in margin, near the final part of the text of the proposition), so it also served as separation mark between propositions.
For the English translation of the text of the Elements, I use the book by Heath (1925), replacing the labels by the original Greek letters. Heath assigns Latin letters to labels, so for example, Α, Β, Γ, Δ, Ε, Ζ, Η, Θ in the Greek text become A, B, C, D, E, F, G, H in Heath’s translation.
This is not true for all the Books of the Elements. For example, in Book X, which contains more than one hundred propositions, a proposition is usually concluded by the phrase “therefore, etc.”, which means that the sumperasma is the same as the protasis.
The labels can be compared to local variables in computer programming languages. Local variables have definite “scope” in which they are valid, if a variable of the same name appears outside the scope, it refers to something else.
Netz (1999, 74 ff.) develops illuminating arguments about such oscillation or switch of names.
I give the labels to this diagram, for it is impossible to indicate points and lines in this written article without using labels. However, with the diagram in your memory, you can surely understand the protasis without assigning labels to it.
This reminds one of Plato’s affirmation that written words served for those who knew it: “(one would be) ignorant of the prophecy of Ammon, if he thinks written words are of any use except to remind him who knows the matter about which they are written (Phaedr. 275c–d)”.
There are two more occurrences in XI.26 (included in the “Appendix”), whose text has been judged to be spurious and excluded by Heiberg. An Italian translation of the Elements (Frajese 1970) has, at the end of each proposition, a list of propositions applied in that proposition and the propositions which apply that proposition; the latter list for I.4 contains, other than those I have included in the “Appendix”, the following five propositions: I.25, VI.5, XII.16, XIII.11 and XIII.18 lemma. I have excluded them, for the application of I.4 is implicit, if applied at all, in these propositions.
I have slightly changed Heath’s translation so that the logical structure of the statement is clear: (0), (1), (2) are conditions introduced by ‘since’ (gk. epei), followed by conclusions (3), (4), (5). (See Henry Mendell’s translation at: http://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Euclid/Euclid%20I/Euclid%201.5/Euclid.1.5.html. Accessed 24 February 2018.)
This can be interpreted as a remedy for possible ambiguity of reference in the absence of proposition numbers.
This is understandable. Even if one has a whole poem in memory, it is not always easy to remember any verse in it without reciting it from the beginning. The process looks like invoking magic by reciting the spell one has learned. The spell must be pronounced strictly in the same order to invoke the magic and to get the effect one wishes. In the application of I.4, side-angle-side theorem, we tend to think that one might be able to state and use only the conclusion one needs out of the three (3), (4), (5), but the text of the Elements shows that this was not the case.
Cairncross and Henry (2015).
Cairncross and Henry (2015, p. 24).
The Greek word horoi (sg. horos) basically means limit or boundary, thus ‘boundary stone’, and mortgage horoi is its derivation. Though this same word means ‘definition’ in a mathematical context, this coincidence has nothing to do with our argument here.
Probably Archimedes was a conspicuous exception, for his major geometrical works were sent to Alexandria with prefatory letter (Quadrature of Parabola, Sphere and Cylinder Book I and Book II, Spiral Lines, Conoids and Spheroids, and Method). His other works without prefatory letter (at least as we have them now) also very probably had more distant readers than direct audience in Syracuse. However Archimedes’ works are written in the style of Euclid’s Elements—not very convenient to be read in the absence of the author. Probably, the style of mathematical writing had already been established by his time and he had no other choice in order to expose his results.
This is my interpretation of Apollonius’ preface. Another plausible interpretation is that Apollonius intended the written work and its revision for public diffusion, where at least its initial readers, Eudemus and Attalus, were in the same position as Archimedes’ Alexandrian readers, so that, at least for them, any oral features of the work would be at most residual. This is Henry Mendell’s suggestion in his personal email.
References
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August, E. F. (1826–1829). Euclidis Elementa ex optimis libris in usum tinorum (Vols. 1–2). Berlin.
Cairncross, A., & Henry, W. B. (2015). Euclid, Elements I.4 (diagram), 8–11, 14–25 (without proofs). The Oxyrhyunchus Papyri, LXXXII (Graeco-Roman memoires, No. 103), 23–38.
Commandino, F. (1572). Euclidis elementorum libri XV, una cum scholiis antiquis. A Federico Commandino Urbinate nuper in latinum conversi, commentariisque quibusdam illustrati. Pisauri (Pisauri is locative of Pisaurum (Pesaro)).
Fowler, D. (1999). The mathematics of Plato’s Academy: A new reconstruction (2nd edn.). Oxford: Oxford University Press.
Frajese, A. (1970). Gli Elementi di Euclide. A cura di A. Frajese e L. Maccioni. Torino: UTET.
Gregory, D. (1703). Euclidis quae supersunt omnia ex recensione Davidis Gregorii. Oxoniae.
Heath, T. L. (1925). The thirteen books of the Elements. 3 Vols. (2nd edn.) New York: Dover Publications (Cambridge. Reprint, 1956).
Heiberg, J. L. (1883–1916). Euclides. Euclidis opera omnia. 8 vols. and a supplement. Leipzig: Teubner.
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Acknowledgements
The research on diagrams was supported by Japan Society for the Promotion of Science (Grant nos. JP17300287, JP21300325). The author would like to express his gratitude to Henry Mendell for linguistic corrections and precious suggestions concerning interpretations of documents.
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Appendix: applications of proposition I.4 within the Elements
Appendix: applications of proposition I.4 within the Elements
1.1 Legenda
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1.
Prop.: proposition in which I.4 is applied. If I.4 is applied more than once, each application is distinguished by numbers in parentheses, e.g., I.26(1).
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2.
H-S: volume, page, line number of the beginning of the application in the publication of Heiberg and Stamatis(1969–1977). e.g., 1:12, 19 (volume 1, page 12, line 19).
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3.
Heath: volume, page, line number at the beginning of the application in Heath (1925). When it is convenient, the line number within the text of the proposition is shown (or both), e.g., 1:251, prop. 5, 15 (volume 1, page 251, line 15 of proposition 5).
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4.
(0)–(6): correspond to the arguments in the application of I.4 (see Sect. 5). (0)–(2) are premises, and (3)–(6) are conclusions.
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(0): equality of sides.
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(1): equality of sides stated in the form “two **, ** are equal to two **, **”.
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(2): equality of angles.
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(3): equality of bases (the remaining sides).
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(4): equality of (the areas of) triangles.
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(5): equality of remaining angles stated in general (protasis) form.
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(6): equality of remaining angles stated with labels.
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+: means that this argument (premise or conclusion) is present.
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⊕: means that this argument (conclusion) is present, and the conclusion is necessary in that proposition.
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−: means that this argument (premise or conclusion) is not resent.
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Prop. | H–S | Heath | (0) | (1) | (2) | (3) | (4) | (5) | (6) |
|---|---|---|---|---|---|---|---|---|---|
I.5(1) | 1:12, 19 | 1:251, prop. 5,15 | + | + | + | ⊕ | + | + | ⊕ |
I.5(2) | 1:13, 4 | 1:251, prop. 5, 28 | + | + | + | +a | + | + | ⊕ |
I.6 | 1:14, 9 | 1:255, prop. 6, 13 | + | + | + | + | ⊕ | − | − |
I.10 | 1:18, 10 | 1:267, prop. 10, 10 | + | + | + | ⊕ | − | − | − |
I.16 | 1:25, 11 | 1:279, prop. 16, 14 | + | + | + | + | + | + | ⊕ |
I.24 | 1:34, 5 | 1:297, 1 | + | + | + | ⊕ | − | − | − |
I.26(1) | 1:37, 1 | 1:302, 14 (prop. 26, 20) | + | + | + | + | + | + | ⊕ |
I.26(2) | 1:37, 10 | 1:302, 24 (prop. 26, 30) | + | + | + | ⊕ | − | − | ⊕ |
I.26(3) | 1:37, 23 | 1:303, 10 (prop. 26, 50) | + | + | + | + | + | + | ⊕ |
I.26(4) | 1:38, 8 | I:303, 22 (prop.26, 62) | + | + | + | ⊕ | + | − | ⊕ |
I.33 | 1:46, 5 | 1:323, 7 (Prop. 33, 13) | + | + | + | ⊕ | + | + | ⊕ |
I.34 | 1:48, 1 | 1:324, 30 (prop. 34, 34) | + | + | + | + | ⊕ | − | − |
I.35 | 1:48, 20 | 1:327, 5 (prop 35, 12) | + | + | + | + | ⊕ | − | − |
I.47 | 1:64, 1 | 1:350, 1 (prop. 47, 32) | + | + | + | + | ⊕ | − | − |
III.7 | 1:102, 13 | 2:15,7 (prop. 7, 44) | + | + | + | ⊕ | − | − | − |
III.8 | 1:106, 9 | 2:19, 21 | + | + | + | ⊕ | − | − | − |
III.17 | 1:120, 6 | 2:43, 24 (prop. 17, 17) | + | + | + | + | + | + | ⊕ |
III.25 | 1:128, 4 | 2:55, 12 | + | + | + | ⊕ | − | − | − |
III.26 | 1:129, 22 | 2:57,7 | + | + | + | ⊕ | − | − | − |
III.29 | 1:134, 5 | 2:60, 27 (prop. 29, 12) | + | + | + | ⊕ | − | − | − |
III.30 | 1:135, 4 | 2:61, 13 (prop. 30, 8) | + | + | + | ⊕ | − | − | − |
III.33 | 1:141, 7 | 2:68, 3 | + | + | + | ⊕ | − | − | − |
IV.5(1) | 1:157, 17 | 2:89, 3 | + | − | + | ⊕ | − | − | − |
IV.5(2) | 1:158, 12 | 2:89, 23 | + | − | + | ⊕ | − | − | − |
IV.6 | 1:160, 8 | 2:91, 9 (prop. 6, 8) | + | − | + | ⊕ | − | − | − |
IV.13 | 1:172, 7 | 2:105, 3 | + | + | + | + | + | + | ⊕ |
VI.6 | 2:51, 12 | 2:205, 7 | + | + | + | + | + | + | ⊕ |
XI.4(1) | 4:7, 10 | 3:278, 15 | − | + | + | + | + | − | ⊕ |
XI.4(2) | 4:7, 21 | 3:278, 28 | + | − | + | ⊕ | − | − | − |
XI.4(3) | 4:8, 3 | 3:279, 1 | + | + | + | ⊕ | − | − | − |
XI.6 | 4:10, 17 | 3:283, 25 (prop. 6, 21) | + | + | + | ⊕ | − | − | − |
XI.8 | 4:13, 10 | 3:288, 6 | + | + | + | ⊕ | − | − | − |
XI.20 | 4:29, 3 | 3:308, 7 | + | + | + | ⊕ | − | − | − |
XI.22 | 4:32, 10 | 3:313, 25 | − | + | + | ⊕ | − | − | − |
XI.23 | 4:36, 10 | 3:317, 15 | + | + | + | ⊕ | − | − | − |
XI.24 | 4:39, 18 | 3:324, 17 | − | + | + | + | ⊕ | − | − |
XI.26(1) | 4:43, 15 | 3:328, 16 | − | + | + | ⊕ | − | − | − |
XI.26(2) | 4:43, 17 | 3:328, 19 | + | − | + | ⊕b | − | − | − |
XI.26(3) | 4:44, 2 | 3:328,23 | − | + | + | ⊕ | − | − | − |
XI.26(4) | 4:44, 13 | (excluded by Heiberg) | − | + | + | ⊕ | − | − | − |
XI.26(5) | 4:44, 14 | (excluded by Heiberg) | + | + | + | ⊕ | − | − | − |
XI.35(1) | 4:68, 11 | 3:353, 24 | + | + | + | + | + | + | ⊕ |
XI.35(2) | 4:69, 1 | 3:354, 7 | + | + | + | ⊕ | − | − | − |
XI.38 | 4:74, 6 | 3:361, 9 | + | − | + | + | + | + | ⊕ |
XII.3 | 4:85, 11 | 3:379, 6 | + | + | + | + | ⊕c | − | − |
XIII.7 | 4:146, 12 | 3:451, 39 (prop. 7, 9) | − | + | + | + | + | + | ⊕ |
XIII.8 | 4:148, 18 | 3:454, 1 | − | + | + | + | + | + | ⊕ |
XIII.10 | 4:153, 20 | 3:459, 19 | + | − | + | ⊕ | − | − | − |
XIII.13 | 4:160, 21 | 3:468, 22 | + | − | + | ⊕ | − | − | − |
XIII.14 | 4:165, 18 | 3:475, 27 | + | − | + | ⊕ | − | − | − |
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Saito, K. Diagrams and traces of oral teaching in Euclid’s Elements: labels and references. ZDM Mathematics Education 50, 921–936 (2018). https://doi.org/10.1007/s11858-018-0929-1
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DOI: https://doi.org/10.1007/s11858-018-0929-1