Abstract
In view of the progress made in recent decades in the fields of stemmatology and the analysis of geometric diagrams, the present article explores the possibility of establishing the stemma codicum of a handwritten tradition from geometric diagrams alone. This exploratory method is tested on Ibn al-Haytham’s Epistle on the Shape of the Eclipse, because this work has not yet been issued in a critical edition. Separate stemmata were constructed on the basis of the diagrams and the text, and a comparison showed no major differences. The greater reliability of a stemma codicum constructed on the basis of the diagrams rather than the text of a mathematical work is discussed, and preliminary conclusions are drawn.
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Notes
For example, the simple rephrasing of statements or the transposition of the words odientis oscula to make the verse Quam fraudulenta oscula odientis rhyme with Meliora sunt vulnera diligentis (Huygens 2001, p. 55).
This comparison builds on the difference between homology, i.e., a similarity inherited from a common ancestor, and homoplasy, i.e., a similarity shared by two individuals, whereas it is not found in the ancestor.
On the application of cladistic analysis in different domains, the reader is directed to the following key studies: on the approach in general (Glenisson 1979; van Reenen et al. 1996, 2004; Robinson 1996; Dees 1988; Huygens 2001; Woerther and Khonsari 2001; Macé et al. 2001; Macé and Baret 2006; Cipolla et al. 2012); in literary texts (Robinson and O’Hara 1996; Salemans 1996, 2000; Barbrook et al. 1998; Mooney et al. 2001; Windram et al. 2008; Maas 2010); and in scientific texts (Brey 2009; Pietquin 2010; Cardelle de Hartmann et al. 2013).
After constructing the stemma on the basis of a review of text accidents, Crozet says: “Cette filiation, dont nous venons d’étayer l’affirmation par le menu en considérant le texte seul, peut également se lire à l’aide des figures.” Al-Sijzī’s Barāhīn kitāb Uqlīdis fī al-‘uṣūl (Demonstrations of the Book of Euclid on the Elements) is known in three copies: B Dublin, Chester Beatty MS 3652, fols. 17r–28v, R Istanbul, Reshit MS 1191, fols. 84v–105v, and L London, India Office MS 1270, fols. 87r–100r. MSS R and B are linked to one another. In the third demonstration of prop. I.2, both carry the correct lettering of point Ǧ. In the fourth demonstration, both refer to point H, which does not appear in MS L. In prop. II.9 both manuscripts contain superfluous lines. Furthermore, MS L predates MS R for the latter is the only text where point Ǧ is placed on \(AH\), in keeping with the text stemma.
Fig. 1 
I.38 (codex P)
Fig. 2 
I.38 (codex B)
Heath assumes that \(\varDelta \) and \(E\) are the two points of intersection of \(A\varGamma \!\varDelta \) and \(E\!B\varDelta \), though it is not explicit in Euclid: “With centre \(A\) and distance \(AB\) let the circle \(B\varDelta E\) be described...” (Heath 1956, p. 96).
Shared true characters also include the so-called “co-exact” geometric properties. “Co-exact attributes are those conditions which are unaffected by some range of every continuous variation of a specified diagram” (Manders 2008, p. 92). Typically, these are topological relations, which are more stable than magnitudes or ratios of magnitudes.
With a view toward creating a matrix of characters, I have used straightforward codes: f: false, s: smaller, g: greater, e: equal, p: positive, n: negative, u: up, d: down, r: right, l: left, and ? for a missing character.
In order to locate a passage parallel to a given passage in any extant manuscript, we define a new measurement called “time,” which counts 0 as the beginning of the text and 1 as the end of the text in any complete version. Thereby one can precisely define a given passage from any text. As regards the transliteration of the Arabic, I have adopted DIN-31635 throughout, except for G ‘ayn, which does not interfere with Ǧ ǧīm or Ġ ġayn.
O for Oblongus.
This is the result of \((36+30+16+29+20)/5/65=0.40\) and \((32+24+6+20+19)/5/40=0.51\).
This is the result of \((36-16)/65=0.31\) and \((32-6)/40=0.65\).
A random selection of errors in diagrams will yield a certain combination of stated and unstated errors. At one extreme, it is possible that the random draw will include stated errors only, in which case the result will be the same as if we based our analysis on the errors the text. Otherwise, the random draw will include some unstated errors as well, and in this case the result will be better than if we had simply selected errors in the text, because unstated errors are more discriminating than stated errors (Property 2 in Sect. 4.2). Therefore, the resolution capability of a random selection of the errors contained in diagrams is greater than or equal to the resolution capability of an analysis based on stated errors.
The techniques are RHM, PAUP Parsimony, Parsimony Bootstrap, Neighbour Joining, Neighbour Joining Bootstrap, Least Squares, Least Squares Bootstrap; n-Gram Clustering; SplitsTree4 NeighborNet, SplitDecomp, ParsimonySplits, CompLearn, Hierarchical Clustering, and seven manual methods. The similarity between the stemmata produced is estimated by the average sign distance ASD, which provides a value ranging from 0 (the two stemmata are different) to 1 (the two stemmata are identical).
A branch of zero length means that there is no difference between the manuscript (terminal node) and the progenitor (intermediary node). The closest fit is MS Petersburg, which is very similar to the out-group.
This conclusion does not hold for later periods. With the spread of copperplate engraving, diagrams were frequently grouped together on pages distinct from the text in printed editions (Rider 1993; Barrow-Green 2006). Prof. Gregg de Young informs me that later manuscripts could at times reproduce this arrangement, such as Cairo, Dār al-kutub, handasa turkiyya 42, ca. 1300 H./1834 or Ṭal‘at Riyāda, handasa turkiyya 3, ca. 1250 H./1882. Quite possibly, these two manuscripts were copied from the Usūl-i hendese, a Turkish translation by Hüseyin Rıfkı Tamānī of Bonnycastle’s Elements of Geometry that was printed by the Maṭba‘at Būlāq in 1241 H./1825 (Young 2012, p. 51).
Another way to proceed might have been to search for the consensus tree between the diagram and text stemmata. We did not proceed any further in this direction, because the consensus tree algorithm is based on bootstrapping, which has certain limitations (Wiesemiller and Rothe 2006, pp. 161–5; Kitching 1998, pp. 129–31).
The diagrams in MS O contain 48 errors within a total area of \(38.26\,\hbox {cm}^2\) and the error density is therefore \(\delta _D=1.25\) per \(\hbox {cm}^2\), while the text contains 138 errors within a written area of \(847.20\,\hbox {cm}^2\): \(\delta _T=0.16\) per \(\hbox {cm}^2\). The diagrams in MS B contain 57 errors within a total area of 134.29 \(\hbox {cm}^2\): \(\delta _D=0.42\) per \(\hbox {cm}^2\), while the text has 262 errors within an area of \(4{,}034.84\,\hbox {cm}^2\): \(\delta _T=0.06\) per \(\hbox {cm}^2\). Therefore, the number of errors is \(7.0 \leqslant \delta _D / \delta _T \leqslant 7.8\) times greater in a diagram than in a text of the same area.
That is, when the diagrams are embedded in the text and show no obvious signs of emendation or alteration.
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Acknowledgments
I gratefully acknowledge Ken Saito (Osaka Prefecture University), Gregg de Young (American University in Cairo), A. Mark Smith (University of Missouri), Len Berggren (Simon Fraser University) and anonymous referees for valuable comments on a first draft of this paper, while reserving for myself full responsibility for any remaining errors.
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Communicated by: Len Berggren.
Appendices
Appendix A
Appendix B
See Figs. 10, 11, 12, 13, 14, and 15.
MS F Fātiḥ 3439
MS B Bodleian Arch. Seld. A32
MS P St. Petersburg B1030
MS O India Office 1270
MS L India Office 461
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Raynaud, D. Building the stemma codicum from geometric diagrams. Arch. Hist. Exact Sci. 68, 207–239 (2014). https://doi.org/10.1007/s00407-013-0134-0
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DOI: https://doi.org/10.1007/s00407-013-0134-0