# The Idea of Στοιχείον in Grammar and Cosmology: From Antique Roots to Medieval Systems

Juan Acevedo

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Juan Acevedo. The Idea of Στοιχείον in Grammar and Cosmology: From Antique Roots to Medieval Systems. History, Philosophy and Sociology of Sciences. Warburg Institute, School of Advanced Study, University of London, 2018. English. NNT: tel-03965257

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# The Idea of Στοιχεῖον in Grammar and Cosmology: From Antique Roots to Medieval Systems

Angel Eduardo Juan Acevedo

A dissertation submitted in
fulfilment of the requirements
for the degree of
Doctor of Philosophy
in Combined Historical Studies

The Warburg Institute

University of London

2018

date of examination: 30 Nov. 2018

examiners: Godefroid de Callatay (UCL Louvain), Toby Mayer (IIS, London)
I declare that the work presented in this dissertation is my own.

Signed: ____________________

Date: ____________________
Abstract

This thesis defines and follows the development of the concept expressed by the Greek
στοιχεῖον and the Latin *elementum*. From approximately the sixth century BC to the
twelfth century AD, these words had three simultaneous meanings: letter, number
and element, corresponding respectively to the disciplines of grammar, arithmetic and
cosmology. The first part of the thesis, in two chapters, draws primarily on Greek
philosophical, grammatical and arithmetical sources to delineate this polysemy, with
particular attention to Pythagorean number cosmology and the foundational and
lasting role of Plato's *Timaeus*. Once the triple concept is established, the second
part, in four chapters, tracks it through late Antiquity in Hellenistic religious texts
and in Abrahamic scriptural sources and exegetical literature, identifying semantic
analogues in Hebrew and Arabic. The third part of the thesis studies particular cases of
alphanumeric cosmology in doctrinal systems of major Jewish, Christian and Islamic
authors of the High Middle Ages, namely in the *Sefer Yetsirah*, in Aquinas and Ibn
ʻArabī. In the conclusion I gather the comparative evidence to situate the concept of
the alphanumeric element in its relations to the broader metaphysical, theological and
cosmological heritage of the International Mediterranean Middle Ages.
# Contents

Abstract 3

Acknowledgements 9

Notes to the Reader 10

Transliterations 10

Hebrew Transliteration 10

Arabic Transliteration 11

Bibliographical Style 11

Abbreviations 12

Introduction 14

The Concept 15

Thesis Structure 16

Survey of Scholarship 17

Methodology Matters 19

I Defining the Concept 22

1 Notes on Early Alphabetic Cosmology 23

Starting from Aristotle's 'Lexicon' 23

First Glimmerings 28

Etymology 28

The Concept 30

Atoms and Letters 30

Platonic Corpus 31

The Constellations 32

Latin Correspondences 33

Greek Grammarians 33
<table><tr><td>The Opening Anacoluthon</td><td>35</td></tr><tr><td>Simplicity and Pedagogy</td><td>36</td></tr><tr><td>The Semitic Origin</td><td>36</td></tr><tr><td>Nature-given or Created</td><td>38</td></tr><tr><td>More Etymologies</td><td>38</td></tr><tr><td>Number and Order</td><td>38</td></tr><tr><td>Walls and Ramparts</td><td>40</td></tr><tr><td>And what about ‘γράμματα’?</td><td>40</td></tr><tr><td>Attributes of the στοιχείον</td><td>41</td></tr><tr><td>The Four Cosmic Elements</td><td>42</td></tr><tr><td>Why Twenty-Four? More Cosmic Correspondences</td><td>43</td></tr><tr><td>The Moon and the Stars</td><td>44</td></tr><tr><td>The Planets</td><td>45</td></tr><tr><td>Astrological Correspondences</td><td>46</td></tr><tr><td>Microcosmic Correspondences</td><td>47</td></tr><tr><td>Graphical Layout</td><td>48</td></tr><tr><td>Sanchuniathon and the Snakes</td><td>49</td></tr><tr><td>Latin Grammarians</td><td>50</td></tr><tr><td>Conclusion</td><td>52</td></tr></table>

<table><tr><td>2 On the Tracks of Arithmetical Cosmology</td><td>54</td></tr><tr><td>Number and Letter</td><td>54</td></tr><tr><td>Isopsephism, Onomatomancy and Gematria</td><td>58</td></tr><tr><td>Choeroboscus on the Names of Numbers</td><td>61</td></tr><tr><td>Reckoning and Science of Numbers</td><td>62</td></tr><tr><td>Pythagoras and Pythagoreanism</td><td>65</td></tr><tr><td>Lore &amp; Science and Further Developments</td><td>67</td></tr><tr><td>Imitation or Participation</td><td>69</td></tr><tr><td>On Creation and Order</td><td>70</td></tr><tr><td>The Pythagorean Logos</td><td>70</td></tr><tr><td>The Fundamental Decad</td><td>71</td></tr><tr><td>Philolaos</td><td>72</td></tr><tr><td>Fragments of Philolaus</td><td>73</td></tr><tr><td>‘Pythagorean’ Milestones</td><td>79</td></tr><tr><td>Nicomachus and his Introduction</td><td>83</td></tr><tr><td>Nicomachus’ Introduction Tables</td><td>84</td></tr><tr><td>Conclusion</td><td>101</td></tr></table>
## II Alphanumeric Cosmology in the Abrahamic Scriptures 105

<table><tr><td>Introduction · Change and Continuity</td><td>106</td></tr></table>

### 3 Jewish Sources and Exegesis 108

<table><tr><td><em>Tanakh</em> and Translations</td><td>108</td></tr><tr><td>Genesis—Creation by Speech</td><td>108</td></tr><tr><td>Psalm 33:6</td><td>111</td></tr><tr><td>Exodus—Bezalel</td><td>111</td></tr><tr><td>Counting the Glory</td><td>113</td></tr><tr><td>The Potter of Light</td><td>114</td></tr><tr><td>Acrostics</td><td>115</td></tr><tr><td>Summary</td><td>118</td></tr><tr><td>Philo</td><td>118</td></tr><tr><td>On the Making of the World</td><td>119</td></tr><tr><td>Archetype and Noetic Creation</td><td>121</td></tr><tr><td>The Model is Λόγος</td><td>123</td></tr><tr><td>Numbers: the Tetrad and the Decad</td><td>124</td></tr><tr><td>Chapter Summary: Philo in Perspective</td><td>125</td></tr></table>

### 4 Christian Sources and Exegesis 127

<table><tr><td>Letters-Numerals of the Christic Word-Number</td><td>127</td></tr><tr><td><em>Mysteria litterarum</em>: Textual History</td><td>128</td></tr><tr><td><em>Mysteria litterarum</em>—the text</td><td>130</td></tr><tr><td>22 or 24 Letters</td><td>132</td></tr><tr><td>Apocryphal Gospels</td><td>132</td></tr><tr><td>A Spiritual ABC in Monastic Literature</td><td>133</td></tr></table>

### 5 Hermetica, Magic and Gnostic Sources 135

<table><tr><td>Who is Hermes?</td><td>136</td></tr><tr><td>Thoth and the Letters in <em>Philebus</em></td><td>138</td></tr><tr><td>The Creation Elements in the <em>Hermetica</em></td><td>140</td></tr><tr><td>Magic Phonemes</td><td>142</td></tr><tr><td>Magical Papyri</td><td>143</td></tr><tr><td>The Twenty-Four Alphabetic Names of Gods</td><td>145</td></tr><tr><td>Gnostic Writings</td><td>148</td></tr><tr><td><em>Evangelium Veritatis</em></td><td>149</td></tr><tr><td><em>Marsanes</em></td><td>151</td></tr><tr><td>Marcus the Valentinian</td><td>152</td></tr><tr><td>Conclusion</td><td>157</td></tr></table>
6 Alphanumeric Cosmology in Islamic Scripture and Exegesis 159

<table><tr><td>Introduction</td><td>159</td></tr><tr><td>The Linguistic-Logical Aspect</td><td>161</td></tr><tr><td>Starting Point and Influences</td><td>162</td></tr><tr><td>The Arabic Alphabet Order</td><td>164</td></tr><tr><td>The Arabic Letters as Numerals</td><td>166</td></tr><tr><td>The Development of the Science of Letters</td><td>169</td></tr><tr><td>ʿAlī: The Common Source</td><td>169</td></tr><tr><td>Letters in the Ḥadīth Literature</td><td>171</td></tr><tr><td>The Two Branches</td><td>173</td></tr><tr><td>Some Key Texts</td><td>176</td></tr><tr><td>Back to Basics: Philosophy & Physics</td><td>179</td></tr><tr><td>The Metaphysics</td><td>179</td></tr><tr><td>Jābir and ‘The Element’</td><td>183</td></tr><tr><td>The ‘Element’ of the Foundation</td><td>183</td></tr><tr><td>The Balance of the Letters</td><td>185</td></tr></table>

III World Construction: Systems of Cosmic Design 188

<table><tr><td>Introduction: System Building</td><td>189</td></tr></table>

7 Sefer Yetsirah: The Bright Jewel of Alphanumeric Cosmopoiesis 190

<table><tr><td>Selections from the Text</td><td>193</td></tr><tr><td>The Thirty-Two Wondrous Paths of Wisdom</td><td>194</td></tr><tr><td>The Three Modes of Inscribing</td><td>196</td></tr><tr><td>The Absolute Numerations—Sefirot</td><td>198</td></tr><tr><td>The Twenty-Two Elemental Letters</td><td>200</td></tr><tr><td>The Title: ‘Creation’ or ‘Formation’?</td><td>203</td></tr><tr><td>Authorship and Conclusion</td><td>205</td></tr></table>

8 Alphabetic Consecration & Metaphysical Foundation 206

<table><tr><td>The Abecedarium in Church Dedication</td><td>206</td></tr><tr><td>The Roman Gromatici</td><td>208</td></tr><tr><td>Christening the Land: mensuratio/γεωδαισία</td><td>210</td></tr><tr><td>Rabanus Maurus</td><td>211</td></tr><tr><td>The Celtic Connections</td><td>212</td></tr><tr><td>Eriugena</td><td>213</td></tr><tr><td>Ramon Llull</td><td>213</td></tr><tr><td>Psellos: Letters and Matter</td><td>216</td></tr></table>
<table><tbody><tr><td>Aquinas and the Second ‘Signified’ Matter</td><td>217</td></tr><tr><td>Dante and Hildegard</td><td>219</td></tr><tr><td><strong>9 Islamic Systematisation and the Sublime Letters That We Were</strong></td><td><strong>222</strong></td></tr><tr><td>Abū Ḥātim’s Letter Cosmogony</td><td>223</td></tr><tr><td>A Pure Brotherhood</td><td>225</td></tr><tr><td>How Pythagorean are the Ikhwān?</td><td>226</td></tr><tr><td>Magic Squares in the Epistles</td><td>227</td></tr><tr><td>Music: Prosody: Grammar</td><td>228</td></tr><tr><td>‘Letters’ in the Rasā’il</td><td>229</td></tr><tr><td>Risāla al-Jāmiʿa</td><td>231</td></tr><tr><td>A ‘Sufi’ Interim</td><td>233</td></tr><tr><td>Al-Būnī and the Sun of Divine Sciences</td><td>235</td></tr><tr><td>The Letter-Elements in al-Būnī</td><td>236</td></tr><tr><td>The First Paragraph</td><td>236</td></tr><tr><td>The Basics of Letter Cosmology</td><td>236</td></tr><tr><td>Letter-Number Relation</td><td>237</td></tr><tr><td>Joining the Letters</td><td>239</td></tr><tr><td>‘Magic’ Squares in the Light of Alphanumeric Cosmology</td><td>239</td></tr><tr><td>Ibn ʿArabī</td><td>240</td></tr><tr><td>The Place of the Letters</td><td>242</td></tr><tr><td>Letters and Numbers</td><td>243</td></tr><tr><td>The Three Modes</td><td>244</td></tr><tr><td>The Living Letters</td><td>244</td></tr><tr><td>Conclusion to Part III</td><td>246</td></tr><tr><td><strong>Conclusion</strong></td><td><strong>247</strong></td></tr><tr><td>Conceptual Conclusion: Echoes and Ramifications</td><td>247</td></tr><tr><td>Art, Artistry and the Arts</td><td>247</td></tr><tr><td>Physics and Modern Physics</td><td>248</td></tr><tr><td>Linguistics, Psychology and Cosmology</td><td>249</td></tr><tr><td>Metaphysics and Theology</td><td>250</td></tr><tr><td>Historical Conclusion: The Alphanumeric End</td><td>251</td></tr><tr><td><strong>Appendix A: Latin Grammarians</strong></td><td><strong>256</strong></td></tr><tr><td>Mundi elementa</td><td>258</td></tr><tr><td><strong>Bibliography</strong></td><td><strong>260</strong></td></tr><tr><td>Primary Sources</td><td>260</td></tr><tr><td>General Sources</td><td>267</td></tr></tbody></table>
Acknowledgements

A work of this extension and scope renews the awareness that gratitude tends by nature
towards infinity—so many unexpected favourable circumstances, and sources of help
and information and good will... Here is an attempt at doing justice to some of them.

The Warburg Library was described almost a century ago as 'the one open gate to the cosmopolis of knowledge,' and this holds true not only as regards the layout of the city of knowledge, its avenues and parks which we visit in our minds, but also very palpably in the citizens themselves—and what joy it is to benefit from their advice and convivial conversation. Among these cosmopolitans of knowledge I would like to thank in particular Guido Giglioni, Christopher Johnson, Jill Kraye, Berthold Kress, Jonathan Rolls, Alessandro Scafi and Claudia Wedepohl. Warm thanks to Peter Pormann, who put up with an early draft of the thesis and made valuable suggestions.

Among colleagues and friends with whom I have shared texts and dictionaries for
these past three years, I would like to mention Helena Avelar, David Bennett, Husaini
Binjamil, Yang Gao, Dorian Greenbaum, Michael Noble, Maureen O'Rourke, Adrian
Pirtea, Gregor Schwarb and Tassos Tyflopoulos. Special thanks to Bink Hallum, Liana
Saif and Sébastien Moureau for sharing their enthusiasm, knowledge and fascinating
manuscripts.

This thesis would have been unimaginable and, as I now realise, impracticable in
many ways without the inspiration, encouragement and expert guidance of Charles
Burnett, who has supervised my research and improved on every detail of my work
with the lightest touch and self-effacing generosity.

My colleagues and friends at The Matheson Trust have made possible the completion
of this work in many ways. To Tamar Drukker I owe much gratitude for her help and
patience with my Hebrew. Closer to home, I want to thank Rida, Adnan and Summer,
and in Mérida, Enrique Obediente for his friendship and example. Edgar Guzmán, my
first companion reading the *Sefer Yetsirah*, has been very present all along, as have been
my mathematicians Olga, Ramón, Cristóbal and Juan Manuel. My parents inspired in
me the love of letters and music, and somehow shared with me *il gran disio*, so whatever
is good here is as much theirs as mine.

I cannot really *thank* my family: my children teach me and fix me every day with their pure intent and their frankness; and to that lovely lady who sits at the other end of my desk and makes my life beautiful in so many ways, my accomplice and best friend, I tell you, now I will have time to ask you out.

I dedicate this thesis to the memory of José Manuel Briceño Guerrero (זצ"ל), who
taught me the Greek hexameter on a blessed Venezuelan afternoon that I never forget.
# Notes to the Reader

Translations are mine throughout, except where otherwise stated, in order to preserve consistency as far as possible, in literalness and terminology. In view of the semantic complexity of the study, I have often taken the liberty to force an overly etymological translation to bring out a particular aspect of a word.

## Transliterations

Single Greek words are indistinctly used in transliteration or Greek script depending on the context. Running Greek text is never transliterated.

Hebrew and Arabic running text is left in the original script, while single words or short phrases are transliterated according to the tables below. Vocalisation is only used in specific contexts according to need, as when a given passage is studied in depth.

## Hebrew Transliteration

My transliteration combines the two current varieties (פּשּׁוֹט ‘simple’ and מִדְיֶק ‘precise’) of the Academy of Hebrew Language, 2007, and it is almost identical to the ‘General-Purpose Style’ in the SBL Handbook of Style (2nd edn, 2014).

א

a

ז

z

מ

m

ק

q

ב

b/v

ח

h

נ

n

ר

r

ג

g

ט

ṯ

ס

s

ש

sh

ד

d

י

y

ע

ʿ

ת

t

ה

h

כ

k/kh

פ

p/f

ו

w

ל

l

צ

ts
## Arabic Transliteration

The transliteration follows the current usage of Brill's *Encyclopedia Islamica* and the
Oxford *Journal of Islamic Studies*.

أ
ب
ج
د
ه
و
ز
a
b
ج
ط
ت
س
ش
س
ت
ث
t
th
f
ح
خ
kh
q
ض
d
r
ر
dh
sh
غ
gh

## Bibliographical Style

With very minor modifications to follow Warburg Institute usage, I have done my best to follow the style specifications of the 16th edition of the *Chicago Manual of Style*, as implemented by the *biblatex-chicago package* of the TeX typesetting system (version 1.0rc5, 16.01.2018). More specifically, I have used the 'notes & bibliography' style of its 'author-date' system. This means that details of every title are given in full in a footnote upon first reference, and thereafter abbreviated. Full bibliographic details, including URLs and DOIs when available, are given in the Bibliography. Needless to say, all the inconsistencies and shortcomings in the usage of these superb tools are mine.
Abbreviations

BNP
Cancik, H., H. Schneider, C. F. Salazar, and D. E. Orton, eds, *Brill's New Pauly: Encyclopaedia of the Ancient World*.

DELG
Chantraine, P., *Dictionnaire étymologique de la langue grecque: histoire des mots*.

DGWE
Hanegraaff, W. J., *Dictionary of Gnosis & Western Esotericism*.

DK
Diels, H., and W. Kranz, *Fragmente der Vorsokratiker*.

DNP
Cancik, H., H. Schneider, and A. Pauly, eds, *Der neue Pauly: Enzyklopädie der Antike*.

DSS
Buck, C. D., *A Dictionary of Selected Synonyms in the Principal Indo-European Languages*.

Du Cange
Du Cange, C. D. F., ed., *Glossarium mediæ et infimæ latinitatis*.

Du Cange-Grec
—, ed., *Glossarium mediæ et infimæ graecitatis*.

EALL
Versteegh, K., et al., eds, *Encyclopedia of Arabic Language and Linguistics*.

EH
*Mawsūʿa al-ḥadīth al-sharīf* (*Encyclopaedia of Hadith*).

Ernout-Meillet
Ernout, A., and A. Meillet, *Dictionnaire étymologique de la langue latine*.

GALex
Endress, G., and D. Gutas, *A Greek and Arabic lexicon: Materials for a Dictionary of the Mediaeval Translations from Greek into Arabic*.

GLK
Keil, H., ed., *Grammatici latini*.

Klatzkin
Klatzkin, J., *Thesaurus philosophicus linguae hebraicae* (*Otsar ha-munahim ha-filosofiyyim*).

LBG
Trapp, E., ed., ‘Lexikon zur byzantinischen Gräzität.’

Lisān
Ibn Manẓūr, *Lisān al-ʿarab*.

LS
Lewis, C. T., and C. Short, eds, *A Latin Dictionary*.

LSJ
Liddell, H., and R. Scott, eds, *A Greek-English Lexikon*.

Munjid
al-Munjid fī al-lugha.
<table><tr><td>Nock-Festugière</td><td>Nock, A. D., and A.-J. Festugière, eds, Corpus Hermeticum.</td></tr><tr><td>OLD</td><td>Oxford Latin Dictionary.</td></tr><tr><td>PGM</td><td>Preisendanz, K., and A. Henrichs, eds, Papyri Graecae Magicae. Die griechischen Zauberpapyri.</td></tr><tr><td>RE</td><td>Pauly, A., and G. Wissowa, eds, Paulys Realencyclopädie der classischen Altertumswissenschaft.</td></tr><tr><td>RIS-Beirut</td><td>Ikhwān al-Ṣafā', Rasā'il Ikhwān al-Ṣafā' wa Khullān al-Wafā'.</td></tr><tr><td>RV</td><td>Mansfeld, J., ed., Die Vorsokratiker.</td></tr><tr><td>Schol. in Dion. Thrax</td><td>Scholia in Dionysii Thracis Artem grammaticam.</td></tr><tr><td>SQ</td><td>Nasr, S. H., C. K. Dagli, M. M. Dakake, J. E. Lumbard, and M. Rustom, eds, The Study Quran: A New Translation and Commentary.</td></tr><tr><td>Tāj</td><td>al-Zabīdī, M., Tāj al-ʻarūs min jawāhir al-qāmūs.</td></tr><tr><td>TDNT</td><td>Kittel, G., G. Friedrich, and G. W. Bromiley, eds, Theological Dictionary of the New Testament.</td></tr></table>
# Introduction

At the beginning of this research, as is so often the case in philosophy, there is a feeling of wonder, and there is Plato.

Before entering into the details of the creation of the elements, in the dialogue that bears his name, Timaeus mocks earlier philosophers who referred to the elements as στοιχεῖα (stoicheia), ‘letters’, observing that those στοιχεῖα ‘are not even syllables’ of reality,<fnref n="1" /> meaning by this that these ‘so-called elements’ (τὰ καλούμενα στοιχεῖα) are already compounds and far from elemental. The pun is involved, and in its complexity it establishes a grammatical model of the cosmos, and naturally, in this model, the letters of the alphabet are the smallest parts of reality. I had barely been exposed to the puzzling and fascinating cosmological doctrines of the Timaeus when I first read these lines, but I had already had some acquaintance with the Hebrew *Sefer Yetsirah*, the *Book of Formation*, where the divine Artist creates the universe by means of letters and ‘numbers’, and the family air between these two works, so distant from each other chronologically and culturally, made a strong impression on me. As I now try to find a concise way to introduce this research work, it occurs to me that it could be safely said to be ‘an attempt at uncovering the links or tracing the pathways between Timaeus and Sefer Yetsirah.’ Perhaps by unfolding what is implicit in such formulation, the introduction will be made clearer.

Coming from Plato means not only that this letter cosmology generated much speculation among his Greek successors, but also that through direct translation and indirect influence, it would generate a wealth of treatises and commentaries in Latin, where the word *elementum* is attested with the same combined meaning since Lucretius and Cicero,<fnref n="2" /> and then through Syriac translations and directly in Arabic and Hebrew—this is part of the story I mean to tell in the following pages. So it can be said that this is a work on the Platonic tradition, and it is a work about the transference of knowledge from Greek antiquity to the Mediterranean High Middle Ages, and thus from Greek into Arabic, Hebrew and back to Latin.

‘From Timaeus to Sefer Yetsirah’ means also from a philosophical metaphysical

<footnote n="1">1. *Timaeus*, 48b8.</footnote>

<footnote n="2">2. See below, p. 33.</footnote>
discourse to a discourse pertaining to mystic doctrines of an Abrahamic religion, and so
this can also be considered a work on comparative philosophy and comparative religion.
But this is all too general, and the reality is that my research in these pages follows a
very narrow thread through a long and winding road, 'struggling with every care,' as
Aby Warburg intended, 'to cast light on one single obscurity, thereby illuminating the
great general development pathways in all their interconnections.'<fnref n="3" /> In fact, it would
be fair too to inscribe this enquiry within the Warburgian understanding of cultural
history, *Kulturgeschichte* and *Kulturwissenschaft*, as a very broad field with room for
art, the sciences and religion to be studied in their interplay and interactions.

The Concept

Any dictionary of Ancient Greek will give two main meanings for the word στοιχεῖον,
that of ‘letter’ and that of ‘element’; κδ΄ στοιχεῖα means ‘the 24 letters’, but δ΄ στοιχεῖα
means ‘the four elements’. In addition to this grammato-physical duality, letters were
used from the sixth century BC and down to the High Middle Ages to represent
numbers: Greek, Hebrew and Arabic alphabets were used in very similar ways for all
sorts of arithmetical purposes, from everyday calculations to advanced mathematics.
The joint usage of the same notation by language and numbers allowed naturally for
certain practices halfway between linguistics and mathematics which are quite alien
to our contemporary experience of ‘number’ and which I think can be accurately
called alphanumeric. These practices were rooted in a subtly different perception of the
boundaries between letters and numbers, and this is why this work is also an attempt
at ‘a wider semiotics of writing’<fnref n="4" /> in which the alphabet is considered not just a graphic
device, but a very tight-knit integration of phonetic, graphic and numerical values<fnref n="5" />
which when combined determine the extent of its applications in other fields. This is
also why this work is concerned with grammar as much as with arithmetic, and with
phonetics and prosody as much as with calligraphy, in a synthesis that might be best
characterised as ‘alphanumeric cosmology’.<fnref n="6" />

<footnote n="3">3. A. Warburg, 'Italienische Kunst und internazionale Astrologie im Palazzo Schifanoja zu Ferrara,' in *Gesammelte Schriften: Die Erneuerung der Heidnischen Antike* (Leipzig/Berlin: Teubner, 1932), 479.</footnote>

<footnote n="4">4. D. K. Psychoyos, 'The Forgotten Art of Isopsephy and the Magic Number KZ,' *Semiotica* 2005, nos. 154-1/4 (2005): 209.</footnote>

<footnote n="5">5. Cf. J. Lougovaya, 'A Perfect Pangram: A Reconsideration of the Evidence,' *Greek, Roman, and Byzantine Studies* 57, no. 1 (2017): 186.</footnote>

<footnote n="6">6. Other denominations used in very closely related works include ‘letter mysticism’, ‘numerology’, ‘lettrism’, ‘Hurufism’. Even though some are lexically simpler to use, they have the disadvantage of being one sided or culturally and historically charged. Of course, new and descriptive compounds are possible, like ‘alphanumerism’, or reclaiming the rare ‘stichology’, but I would not like to be responsible for proliferating neologisms.</footnote>
## Thesis Structure

The examination of this triune concept of letter-number-element, and its elaboration in ancient and medieval scholarship will be the object of the thesis chapters. It is a work in two phases and nine chapters which follow rather loosely historical chronology: first a defining phase, specific, descriptive and idiographic (Part I), restricted mostly to the Graeco-Latin tradition, and then a comparative phase, illustrative, synthetic and cosmopolitan (Parts II and III).

**Part I** establishes the object of my research in all its dimensions: Chapter 1 is the most textual based of the thesis. It is devoted to the grammatical aspect, and it runs mostly as a series of glosses to passages from philosophers and to the commentaries on the grammar primer attributed to Dionysius Thrax. Chapter 2 deals with the arithmetical aspects, with a special emphasis on the Pythagorean tradition, and in particular on the fragments attributed to Philolaus and on the *Introduction to Arithmetic* by Nicomachus of Gerasa.

**Part II** includes four chapters, mining the scriptural traditions of late Hellenistic and early medieval periods, incorporating the views of evolving, growing and nascent Abrahamic religions. Chapter 3 studies Jewish Biblical and Rabbinic texts, and Chapter 4 does the same with early Christian sources. Chapter 5 tries to deal in unitary fashion with the very heterogeneous body of late Hellenistic Hermetic, Gnostic and magic texts, and Chapter 6 looks at the *Qur'ānic* and related Islamic exegetical literature.

**Part III**, in three chapters, explores some specific cases of Abrahamic alphanumeric cosmology in a dually understood 'theurgic' dimension: as the creative act of the world-making deity, and as the divinely oriented work of man; hence this part includes texts more closely related to cosmogony, liturgy, magic, and alchemy. Chapter 7 focuses on the basic structure and concepts of the above-mentioned *Sefer Yetsirah*; Chapter 8 looks at certain Celtic and Scholastic Christian practices and doctrines; and finally Chapter 9 follows the alphanumeric elements through major Islamic philosophical texts, including the *Epistles of the Brethren of Purity* and some texts by Muḥyī al-Dīn ibn ʿArabī.

The time span covered by the research is given, roughly, by the two ends of what I suggest we may call the ‘alphanumeric age,’ between the late sixth century BC, when numerals and letters first coalesced in the Greek Milesian system, and the twelfth century AD, when the introduction of the Indo-Arabic numerals around the Mediterranean was becoming generalised and letters and numbers ceased to have a single ‘body.’ This will be discussed in some more detail in the final Conclusion.
Survey of Scholarship

Aside from excellent specialised works on Jewish, Islamic and Hermetic alphanumeric cosmology, there is a remarkable dearth of English-language literature on this topic in general. There are two major contributions, both originally in German and never translated into English. The one closest to my research, though second in chronological order, is Franz Dornseiff's 1922 monograph, *Das Alphabet in Mystik und Magie*.<fnref n="7" /> Dornseiff himself expresses in his introduction the desirable opening towards more Eastern sources than he could include. I hope that this thesis will at least in some ways be a contribution towards that *desideratum*, as it is also an updating of sources regarding these topics of alphanumeric symbolism and alphanumeric speculation broadly speaking. The second major landmark is Hermann Diels' *Elementum*,<fnref n="8" /> a comprehensive historical lexicological work, tracing the history of the words στοιχεῖον and *elementum* in great detail, and of the many variations of the 'letter simile' (*Buchstabengleichnis*) and the 'lettercase simile' (*Schriftkastenbild*, assuming a set of moveable printing types).

Dornseiff's work became an undisputed reference work for the subject and had no direct continuators, but Diels' prompted several kinds of partial refutations and additions on different fronts. Lagercranz (1911),<fnref n="9" /> Vollgraff (1949),<fnref n="10" /> Koller (1955),<fnref n="11" /> Burkert (1959),<fnref n="12" /> and Schwabe (1980)<fnref n="13" /> were explicitly in dialogue with Diels mostly about the Greek term, while Rogge (1923),<fnref n="14" /> Sittig (1952),<fnref n="15" /> and Coogan (1974)<fnref n="16" /> focused on *elementum*.

<footnote n="7">7. F. Dornseiff, *Das Alphabet in Mystik und Magie*, Stoicheia, Studien zur Geschichte des antiken Weltbildes und der griechischen Wissenschaft 7 (Leipzig: B.G. Teubner, 1922). Notably, the original publication belonged to a series called ‘*Stoicheia*: Studies for the History of Ancient Worldview and of Greek Scholarship.’</footnote>

<footnote n="8">8. H. Diels, *Elementum: eine Vorarbeit zum griechischen und lateinischen Thesaurus* (Leipzig: B. G. Teubner Verlag, 1899).</footnote>

<footnote n="9">9. O. Lagercrantz, *Elementum: eine lexikologische Studie*, I, vol. 1 (Akademiska bokhandeln, 1911).</footnote>

<footnote n="10">10. W. Vollgraff, 'Elementum,' *Mnemosyne* 2, no. 2 (1949): 89–115.</footnote>

<footnote n="11">11. H. Koller, 'Stoicheion,' *Glotta* 3./4. No. 34 (1955): 161–174.</footnote>

<footnote n="12">12. W. Burkert, ‘ΣΤΟΙΧΕΙΟΝ: Eine semasiologische Studie,’ *Philologus: Zeitschrift für antike Literatur und ihre Rezeption* 103 (1959): 167–197.</footnote>

<footnote n="13">13. W. Schwabe, ‘Mischung’ und ‘Element’ im griechischen bis Platon: Wort- und begriffsgeschichtliche Untersuchungen, insbesondere zur Bedeutungsentwicklung von Stoicheion (Bouvier Verlag H. Grundmann, 1980).</footnote>

<footnote n="14">14. C. Rogge, ‘Nochmals lat. elementum,’ *Zeitschrift für vergleichende Sprachforschung auf dem Gebiete der Indogermanischen Sprachen* 51, no. 1 (1923): 154–158.</footnote>

<footnote n="15">15. E. Sittig, ‘Abecedarium und elementum,’ in *Satura: Früchte aus der antiken Welt*, by O. Weinreich (Baden-Baden: Verlag für Kunst und Wissenschaft, 1952), 131–138.</footnote>

<footnote n="16">16. M. D. Coogan, ‘Alphabets and Elements,’ *Bulletin of the American Schools of Oriental Research*, no. 216 (1974): 61–63.</footnote>
Lumpe (1962)<fnref n="17" /> gives a brief account summarising much of Diels from the perspective of conceptual history. Balázs (1965),<fnref n="18" /> barely cited elsewhere, gives what I consider an important insight into the metric and prosodic associations of the Greek. Druart (1968)<fnref n="19" /> has examined very carefully the use and scope of στοιχεῖον in Plato's works, complemented by the more recent work by Laspia, who gives a very useful summary of the *status questionis*.<fnref n="20" /> I should also mention here an important recent work by Weiss<fnref n="21" /> which takes Dornseiff as starting point.

Drawing variously from the above, the following have elaborated more on aspects of the concept itself and less on the philological aspect. Ryle (1960)<fnref n="22" /> deals with logic and the Platonic theory of forms; Lohmann (1980)<fnref n="23" /> with mathematical related terms; Vogt-Spira (1991)<fnref n="24" /> studies the phonetic-written duality, and Crowley (2005)<fnref n="25" /> treats specifically Aristotle's usage. Among encyclopedic articles, I have found Kittel's<fnref n="26" /> and Blossner's<fnref n="27" /> particularly orientating.

My primary intention in this new research on an old theme is to go back to the original texts and to expand the range of texts examined; in particular to study the semantic analogies found in Hebrew and Arabic, which with Greek and Latin constitute the main scholarly languages of the Mediterranean Middle Ages. This expansion of the field of vision is of course made possible by profiting from the insights of all the above scholars.

As may be surmised, given such precedents, this work pertains initially to philology or historical linguistics, and more specifically to lexicology, since it begins with the study of one word in one particular language, but the reader will quickly notice that

<footnote n="17">17. A. Lumpe, 'Der Begriff "Element" im Altertum,' *Archiv für Begriffsgeschichte* 7 (1962): 285-293.</footnote>

<footnote n="18">18. J. Balázs, 'The forerunners of structural prosodic analysis and phonemics,' *Acta Linguistica Hungarica* (Budapest) 15, nos. 1-2 (1965): 229-86.</footnote>

<footnote n="19">19. T.-A. Druart, 'La Notion de « stoicheïon » dans le « Théétète » de Platon,' *Revue Philosophique de Louvain* 66, no. 91 (1968): 420-434.</footnote>

<footnote n="20">20. P. Laspia, 'L'excursus fonologico del Teeteto e la testualità platonica. A cosa pensiamo quando parliamo di 'elementi' e 'sillabe'?, in *Platone e la teoria del sogno nel Teeteto. Atti del Convegno internazionale Palermo*, ed. G. Mazzarra and V. Napoli (Sankt Augustin: Academia Verlag, 2008), 188.</footnote>

<footnote n="21">21. T. Weiss, * Letters by which Heaven and Earth Were Created* (Jerusalem: Bialik Press, 2014).</footnote>

<footnote n="22">22. G. Ryle, 'Letters and syllables in Plato,' *The Philosophical Review*, no. 69 (1960): 431-451.</footnote>

<footnote n="23">23. J. Lohmann, 'Mathematik und Grammatik,' *Beiträge zur Einheit von Bildung und Sprache im geistigen Sein. Festschrift zum 80 (1980): 301-313.*</footnote>

<footnote n="24">24. G. Vogt-Spira, 'Vox und Littera: Der Buchstabe zwischen Mündlichkeit und Schriftlichkeit in der grammatischen Tradition,' *Poetica* 23, nos. 3/4 (1991): 295-327.</footnote>

<footnote n="25">25. T. J. Crowley, 'On the Use of Stoicheion in the Sense of "Element",' *Oxford Studies in Ancient Philosophy*, no. XXIX (Winter 2005): 367-394.</footnote>

<footnote n="26">26. *Theological Dictionary of the New Testament*, ed. G. Kittel, G. Friedrich, and G. W. Bromiley, 7 vols (Grand Rapids, Mich.: W.B. Eerdmans, 1985), s.v. στοιχεῖον (hereafter cited as TDNT).</footnote>

<footnote n="27">27. N. Blössner, 'Stoicheion,' *Historisches Wörterbuch der Philosophie* (Basel), 1998.</footnote>
στοιχείον is not the object of my study, but merely one of the names of my object of study, and it is valuable only because of its synthetic semantic power, and because of its place in the history of Greek philosophy. Because this is in fact the study of a polysemy, the words themselves, στοιχείον or *elementum* or *sefirah* or *harf*, are only important as facets of the ‘jewel’ (*jawhar*, Ar. for jewel, essence, Gr. *ousia*), or as gateways into the fullness of the concept. By studying the words, we see more clearly the aspects of the concept, which in turn allows us to identify other terms used for one or other aspect of the same root concept, in what is already part of a semantic enquiry, or the history of an idea.

Methodology Matters

Methodologically speaking, harking back to Ernst Curtius, my starting point is then from ‘the scientific technique which is the foundation of all historical investigation: philology,’ not however with philology as an end in itself,<fnref n="28" /> but more specifically, in the line of one of my basic references,<fnref n="29" /> assembling a diachronic semasiological study—exploring the etymologies from the first historical occurrences of the concept, and following from then onwards the shifts in meaning.

This alternating approach between word and concept is perhaps what Gadamer means when explaining, ‘what conceptual history can do is to travel the way from word to concept and back again, keeping the path clear,’ for ultimately, ‘just as music is inconceivable without overtones, the conceptual language of philosophy is only entitled to opinions by the concord of its overtones, which restore the vague, abstracted field of a given concept to the natural power at its origin.’<fnref n="30" /> It is precisely this travelling to and fro (a μέθοδος proper) between concept and terms that I intend to pursue, keeping the questioning open and paying attention to the conceptual overtones.

Thinking of ‘keeping the questioning open,’ I also agree heartily with Gadamer’s observation that ‘the purpose of enquiry in conceptual history can hardly be that of achieving a clearcut historical elucidation [...] but it should rather be the case that through the enquiry are ascertained the limitations of such elucidation.’<fnref n="31" /> This in fact brings to mind the following words addressed by Ernst Cassirer to Warburg in 1926 referring to the Kulturbibliothek:

<footnote n="28">28. E. R. Curtius, *European Literature and the Latin Middle Ages*, Bollingen Series 36 (Princeton University Press, 2013), 42.</footnote>

<footnote n="29">29. Burkert, Burkert, ‘ΣΤΟΙΧΕΙΟΝ.’</footnote>

<footnote n="30">30. H.-G. Gadamer, *Die Begriffsgeschichte und die Sprache der Philosophie*, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen: Geisteswissenschaften 170 (Wiesbaden: Springer Fachmedien, 1971), 18.</footnote>

<footnote n="31">31. *Ibid*., 13.</footnote>
May the organon of intellectual-historical studies which you have created continue to ask us questions for a long time.<fnref n="32" />

Regarding the cultural-historical aspects that at once frame and are determined by the concepts studied, I have tried to focus on the continuity and comparability of the philosophical tradition, striving for the ‘intuitive perception of an essence’ instead of trying to ascertain ‘genetic causes, currents, influences.’<fnref n="33" /> Inasmuch as the concept of philosophy that I am obliged to take as my departing point here is the broader sense of ancient and medieval philosophy, this is also a work on comparative religion, or at least on comparative theology and mysticism, since alphanumeric cosmology falls neatly within their ken.

In view of the philological starting point, that aims at drawing as much knowledge as possible from a single polysemy, and in view of the nature of the deeply entwined and interdependent medieval civilisations,<fnref n="34" /> my approach to the comparative method is a carefully balanced exercise in *untranslatability*. I shall try to ‘observe the flexibility of a religious pattern’ as it is ‘adopted and transformed across a wide range of chronological, linguistic, and religious boundaries.’<fnref n="35" /> The basic idea was expressed by Humboldt: ‘different languages are not so many designations of a thing: they are different perspectives on that same thing.’<fnref n="36" /> This means that in each language, most words do not simply denote a reality, but rather express a synthesis of related meanings.<fnref n="37" /> A fitting illustration is that of a jigsaw puzzle piece. Every language would be a different cut of the underlying picture of reality, and so, even though the final image is always the same, the pieces/concepts of each puzzle/language tessellate in different ways. I shall be comparing the many words from Greek, Latin, Hebrew and Arabic sources with the sole aim of revealing the underlying unitary concept that manifests in so many varied ways.

Ultimately, it is my intention to expose, by unravelling the linguistic complexity, what Nietzsche called the *Geistergespräch*, the conversation of minds, as it unfurls through the centuries and cultural worlds from antiquity to the High Middle Ages, ‘a

<footnote n="32">32. E. Cassirer, *The Individual and the Cosmos in Renaissance Philosophy*, trans. M. Domandi (University of Chicago Press, 2010), xiii.</footnote>

<footnote n="33">33. H. Corbin, *The Concept of Comparative Philosophy* (Ipswich: Golgonooza Press, 1981), 2.</footnote>

<footnote n="34">34. See R. Wisnovsky et al., eds, *Vehicles of Transmission, Translation, and Transformation in Medieval Textual Culture*, vol. 4, Cursor mundi (Turnhout: Brepols, 2011), 1–2.</footnote>

<footnote n="35">35. *Ibid*., 22.</footnote>

<footnote n="36">36. Cited in B. Cassin et al., *Dictionary of Untranslatables: A Philosophical Lexicon*, Translation/Transnation (Princeton University Press, 2014), xix.</footnote>

<footnote n="37">37. A remarkable early precedent for this awareness is found in the Jain logical principle and religious doctrine of ‘non-one-sidedness’, *anekāntavāda*, or multiplicity of viewpoints (*Concise Oxford Dictionary of the World Religions*, s.v.).</footnote>
conversation about fundamental human questions going on between authors ancient
and modern.'<fnref n="38" />

<footnote n="38">38. R. Goulbourne, 'Conversations with the Dead in Early Modern France,' *The Modern Language Review* 108, no. 1 (January 2013): 95.</footnote>
# Part I

## Defining the Concept
<figure><img src="image_2.png" /></figure>

# Notes on Early Alphabetic Cosmology

## Starting from Aristotle’s ‘Lexicon’

The *locus classicus* for the speculation on στοιχεῖον, and effectively the first historical account of the use of the term, is the definition given in book Δ or the ‘Lexicon’ of Aristotle’s *Metaphysics*,<fnref n="1" /> a book of central importance for ancient and medieval commentators of Aristotle.<fnref n="2" /> I will now go in detail through the various parts of this definition, following the precedent of Diels, whose comprehensive history of the word in Greek and Latin is still the basic indispensable reference.<fnref n="3" />

As I go through each section of what is really a collection of five complementing definitions, I will start taking note of the properties of the στοιχεῖον as they crop up. A list of such properties will be of the greatest usefulness for the comparative phase of this research in order to recognise the same concept in different contexts and languages. As a visual aid, I shall henceforth use this sign P on the margin to indicate the occurrences of the properties, which will be then listed together at the end. Without willing to jump ahead, I would like to note that most of the meanings that may seem to be strained out of these lines have been actually brought out over the centuries in the translations of στοιχεῖον into Latin and the Semitic languages.

1. στοιχεῖον λέγεται ἐξ οὖ σύγκειται ‘Element’ means a primary **immanent** component, formally indivisible into another form. The πρώτου ἐνυπάρχοντος ἀδιαιρέτου elements of an utterance, for example, are the τῷ εἴδει [εἰς ἔτερον εἶδος], οἶον φωνῆς στοιχεῖα ἐξ ὧν σύγκειται ἡ component parts of that utterance into which it φωνὴ καὶ εἰς ἄ διαιρεῖται ἔσχατα, is ultimately divisible, and which are not further ἐκεῖνα δὲ μηκέτ’ εἰς ἄλλας φωνὰς divisible into other phonemes formally different

<footnote n="1">1. 1014a26.</footnote>

<footnote n="2">2. Y. Halper, ‘Averroes on Metaphysical Terminology: An Analysis and Critical Edition of the Long Commentary on Aristotle’s Metaphysics Δ’ (PhD diss., Dept of Philosophy, Bar-Ilan University, 2010), 9.</footnote>

<footnote n="3">3. Diels, *Elementum*.</footnote>
ἐτέρας τῷ εἶδει αὐτῶν, ἀλλὰ κxivν from themselves. If an element is divided, the διαιρῆται, τὰ μόρια ὁμοειδῆ, οἰον parts are formally the same as the whole: e.g., a ὕδατος τὸ μόριον ὕδωρ, ἀλλ' οὐ τῆς part of water is water; but it is not so for the syl- συλλαβῆς. lable.<fnref n="4" />

In this first definition, several adjectives are predicated of στοιχεῖον, but the basic initial description is ἐξ οὖ σύγκειται, ‘from which there is composition,’ so the first property we have is that στοιχεῖον is a **component**, a constituent like a piece of a puzzle. Elements P are systemic; they do not exist in isolation.

Next, πρώτον is used: elements are ‘first things’, they are **primordial**. There is in P this some overlap with the word ἀρχή, which implies **origin** and **precedence**, that is a **cosmogonic** or generative, and a logical **causal** meaning. The elements are origins and P causes, and this is related to the next property.

A fine point made in the definition is the use of ἐνυπάρχον to characterise στοιχεῖον as a distinctly ‘immanent’ principle. In apparent contrast to ἀρχή or αἰτία, στοιχεῖον P is ‘consubstantial’ to entities, not only underlying them, but also—to elaborate on the etymology of ἐνυπάρχω—*in-under*-lying and determining them from within themselves. It should be noted, however, that the distinction between στοιχεῖον and ἀρχή, far from being clear, has a long history and is rather undecided in the Aristotelian corpus,<fnref n="5" /> as shown for instance in De gen. et corr., 329a5:

“Οτι μὲν οὖν τὰ πρῶτα ἀρχὰς καὶ στοιχεῖα καλῶς ἔχει λέγειν, ἔστω συνομολογούμενον.

Let it now be agreed that it is right to call the primary beings ‘principles’ and ‘elements’.

The next attribute of στοιχεῖον found in the first definition is ἀδιαίρετος τῷ εἶδει, ‘indivisible in form’.<fnref n="6" /> This means that the elements are **simple**, uncompounded, and P Alexander of Aphrodisias clarifies in his commentary (354.26ff): οὐ γὰρ κατὰ τὸ ποσὸν, *not as regards quantity*. The addition of τῷ εἶδει, ‘regarding the form’ or ‘essentially’, reinforces the ambiguous initial πρώτον, like τὰ πρῶτα in the above quotation from De gen. et corr., in that it has the effect of leaving the ‘materiality’ of the elements **undecided**. Τὰ πρῶτα can refer to bodies just as well as to some indeterminate manner P

<footnote n="4">4. Emphasis mine. Cf. the translation and comments in Crowley, *On the Use of Stoicheion*.</footnote>

<footnote n="5">5. About the complications arising from Aristotle's presumed attempt at a unified elemental theory, see M. Kurdzialek, 'Elementum – Die Deutung der aristotelischen Definition des στοιχείου durch David von Dinant,' in *Sprache und Erkenntnis im Mittelalter. Akten des VI. Internationalen Kongresses für mittelalterliche Philosophie*, Miscellanea Mediaevalia, 13/2 (Berlin, New York, 1981), 580–584.</footnote>

<footnote n="6">6. The appendage εἰς ἔτερον εἶδος has been convincingly shown to be 'a misguided later addition' in M. E. Kotwick, *Alexander of Aphrodisias and the Text of Aristotle's Metaphysics* (California Classical Studies, 2016), 72–4.</footnote>
of being. This subtlety did not escape Alexander, who contrasts it with the following
paragraph of the definition, as I shall do now.

2 ὁμοίως δὲ καὶ τὰ τῶν σωμάτων στοι-
χεῖα λέγουσιν οἱ λέγοντες εἰς ἄ διαι-
ρεῖται τὰ σώματα ἔσχατα, ἐκεῖνα δὲ
μηκέτ' εἰς ἄλλα εἴδει διαφέροντα: καὶ
εἴτε ἓν εἴτε πλείω τὰ τοιαῦτα, ταῦτα
στοιχεῖα λέγουσιν.

Similarly, those who speak of the 'elements'
of bodies do so referring to the parts into
which bodies are ultimately divisible, and
which are not further divisible into other
parts different in form. And whether they
speak of one or more than one such, they call
them 'elements'.

In this second definition the στοιχεῖα are unequivocally corporeal principles, and Alexander notes that this refers to the atomists or to Empedocles 'who had everything else generated by the four' (Ἐμπεδοκλέους τῶν τεσσάρων τᾶλλα γεννῶντος). Thus, according to this, the elements are corporeal and their implicit generative power is P made explicit by the verb γεννῶ of the commentary.

3 παραπλησίως δὲ καὶ τὰ τῶν διαγραμ-
μάτων στοιχεῖα λέγεται, καὶ ὅλως τὰ
τῶν ἀποδείξεων: αἱ γὰρ πρῶται ἀπο-
δείξεις καὶ ἐν πλείοσιν ἀποδείξεσιν
ἐνυπάρχουσαι, [1014b] αὖται στοιχεῖα
τῶν ἀποδείξεων λέγονται: εἰσὶ δὲ τοι-
οῦτοι συλλογισμοὶ οἱ πρῶτοι ἐκ τῶν
τριῶν δι' ἑνὸς μέσου.

The term is applied with a very similar
meaning to the 'elements' of geometrical
propositions, and generally those of demon-
strations; for the first demonstrations which
underlie the many other ensuing demon-
strations [1014b] are called 'elements' of
demonstrations. Such are the primary syl-
logisms consisting of three terms with one
middle term.

This meaning of στοιχεῖα as the principles or axioms of geometrical demonstrations (διαγράμματα), is precisely the meaning of the word in the title of Euclid's famous work. According to this definition, the elements are **axiomatic** and **fundamental**, just as the P geometric axioms which underlie and buttress Euclid's edifice. Burkert<fnref n="7" /> observes that for Menaechmus, στοιχεῖα here refers to the postulates, and this is for him the meaning of Euclid's title.

It may be observed that these meanings are already contained in *nuce* in the verb ἐνυπάρχω we had encountered, and that the various definitions would seem to develop and amplify a few basic notions. This is precisely Crowley's point in rejecting 'the common assumption that the use of the term *stoicheion* in physical, metaphysical, or more generally cosmological contexts, in the general sense of principle of body, is a

<footnote n="7">7. Burkert, ‘ΣΤΟΙΧΕΙΟΝ,’ 191–92.</footnote>
metaphorical derivation from some other use of stoicheion.'<fnref n="8" />

It is also made clear, especially by the mention of the primary syllogisms, that the elements are **organic** or interdependent, and **integral** in the sense of necessary to make a whole complete. Referring to this definition, Burkert<fnref n="9" /> explains they are 'mathematical formulations which complement each other in order to perfect a system and which are logically inherent in each other.' In the *Poetica*, Aristotle illustrates this clearly: στοιχεῖον μὲν ἐστιν φωνὴ ἀδιαίρετος, οὐ πᾶσα δὲ ἀλλ' ἐξ ἧς πέφυκε συνθετὴ γίγνεσθαι φωνή, 'στοιχεῖον is an indivisible utterance; not just any, though, but the one upon whose combination arises a composite utterance.'<fnref n="10" /> Though never so explicit in Greek grammatical tradition, this is the quality eventually called *articulatio* in Latin grammar, upon the translation of συνθετὴ φωνή as *vox articulata*.<fnref n="11" />

4 καὶ μεταφέροντες δὲ στοιχεῖον καλοῦσιν ἐντεῦθεν ὃ ἂν ἔν ὃν καὶ μικρὸν ἐπὶ πολλὰ ἰ̣ ἧ χρήσιμον, διὸ καὶ τὸ μικρὸν καὶ ἁπλοῦν καὶ ἀδιαίρετον στοιχεῖον λέγεται. ὅθεν ἐλήλυθε τὰ μάλιστα καθόλου στοιχεῖα εἶναι, ὅτι ἔκαστον αὐτῶν ἔν ὃν καὶ ἁπλοῦν ἐν πολλοῖς ὑπάρχει ἤ πᾶσιν ἤ ὅτι πλείστοις, καὶ τὸ ἔν καὶ τὴν στιγμὴν ἀρχὰς <καὶ στοιχεῖά> τισι δοκεῖν εἶναι.

Following from the above, the term ‘element’ is also applied metaphorically to any small unity which is amply serviceable; and so that which is small and simple and indivisible is called an ‘element.’ Hence it comes about that the most universal things are elements; because each of them, being a unity and uncompounded, underlies many things—everything rather, or very many things. And so it is that also the unity and the geometrical point are seen by some as first principles <and elements>.

From this definition we retain two complementary attributes: the elements are small, or rather **minuscule** (Alexander: ἐλάχιστα πάντῃ) like the geometric point, and they are P universal, most-encompassing or **pervasive**. Like the geometric point, in spite of being P dimension-less, they are boundlessly present. They are also very interestingly called ἐπὶ πολλὰ χρήσιμον, i.e. useful and used in many ways, and reliable; one could say **operative and helpful**. They are like reliable tools, which brings to mind how, according P to Hebrew lore, ‘The Holy One, praised be He, said, “I need workers.” And the Torah said to Him, “Let me provide you with twenty-two workers, the twenty-two letters which are in the Torah.”’<fnref n="12" />

<footnote n="8">8. Crowley, *On the Use of Stoicheion*, 369.</footnote>

<footnote n="9">9. Burkert, *ΣΤΟΙΧΕΙΟΝ*, 192.</footnote>

<footnote n="10">10. 1456b22, cited in Vogt-Spira, *Vox und Littera*, 305.</footnote>

<footnote n="11">11. Cf. the reference to Boethius in U. Eco et al., ‘On Animal Language in the Medieval Classification of Signs,’ in *On the Medieval Theory of Signs*, Foundations of Semiotics 21 (Amsterdam: John Benjamins B.V., 1989), 28–9, 32.</footnote>

<footnote n="12">12. Quoted in C. Bandt, *Traktat ‘Vom Mysterium der Buchstaben’: kritischer Text mit Einführung*,</footnote>
Something else to note in this definition is how the last line equates ἀρχαί and στοιχεῖα. In fact, most manuscripts only have ἀρχὰς here, but Alexander has the addition καὶ στοιχεῖα which I have reproduced. As we have seen above, the difference between the two words is rather blurred in Aristotle. Only a few paragraphs before these lines, in 1013b20, the definition of ἀρχή makes of it a genre including φύσις, στοιχεῖον, διάνοια, προαίρεσις, οὐσία and τὸ οὖ ἔνεκα !

5 ἐπεὶ οὖν τὰ καλούμενα γένη καθόλου καὶ ἀδιαίρετα (οὐ γὰρ ἔστι λόγος αὐτῶν), στοιχεῖα τὰ γένη λέγουσί τινες, καὶ μᾶλλον ἢ τὴν διαφορὰν ὅτι καθόλου μᾶλλον τὸ γένος: ὦ μὲν γὰρ ἡ διαφορὰ ὑπάρχει, καὶ τὸ γένος ἀκολουθεῖ, ὦ δὲ τὸ γένος, οὐ παντὶ ἡ διαφορά. ἀπάντων δὲ κοινὸν τὸ εἶναι στοιχεῖον ἑκάστου τὸ πρῶτον ἐνυπάρχον ἑκάστω.

Now, since what are called genera are universal and indivisible (there being no account of them), some people call the genera 'elements', and these rather than the *differentiae*, because the genus is more universal. For wherever the *differentia* is underlying, the genus also follows; but the *differentia* is not always where the genus is. What is then common to all cases is that the 'element' of every thing is that which is primary and inherent in it.

With this final genus simile the elements are characterized first as ἄλογα, 'unaccountable', **'indefinable'**, 'unexplainable', simply 'countless', or perhaps 'irrational' in the mathematical sense, which is a synonym of 'incommensurable';<fnref n="13" /> then secondly, in their function of genera, they are characterized as categorial or generific, as if each 'element' were a **genarch**, something made very explicit in acrostic compositions, where every στοιχεῖον begins and determines a verse or a section of the text. It is in this combined sense that Kahn speaks of 'categorial genera, the final answer to the *What-is-it?* question for an item within each category'.<fnref n="14" />

I have dissected at length the above lines not only because they summarise for the first time in history the various meanings of our concept, but also because they contain much that will eventually inform every discussion on the topic across cultures and centuries. And because of this, they will be of use to us as a touchstone when moving forward through the more general and specialised literature.

*Übersetzung und Anmerkungen*, Texte und Untersuchungen 162 (Berlin: Walter de Gruyter, 2007), 73, from the Midrash *Tanḥuma Yelammedenu*.

<footnote n="13">13. Famously in a letter simile in *Theaetetus* 202b3, τὰ μὲν στοιχεῖα ἄλογα καὶ ἄγνωστα εἶναι, αἰσθητὰ δέ, ‘the στοιχεῖα are not rational or knowable, but they are perceptible.’</footnote>

<footnote n="14">14. C. H. Kahn, ‘Questions and Categories,’ in *Questions*, ed. H. Hiż (Dordrecht: Springer, 1978), 251–2.</footnote>
First Glimmerings

If we step back from Aristotle now, to examine the origins of the passage just analyzed, two strands are immediately discernible, 1) the word στοιχεῖον itself, and 2) the concept which eventually would be *par excellence* expressed by this term. These two strands of enquiry have to do with the etymology and the semantics.

Etymology

The first occurrence of a related root in Greek literature comes from *Iliad* 23, where
in verses 358 and 757 the chariot racers are said to stand μεταστοιχί, explained by the
scholiast as ἐπὶ στοῖχον, ἐπὶ τάξιν, ‘in a row’, ‘in order’.<fnref n="15" /> One of the main aims of
Burkert’s article was precisely to ‘establish the root στοῖχος conclusively, through the
widest possible sampling, as the semantic origin of στοιχεῖον.’<fnref n="16" /> This characteristic of
the στοιχεῖα is often mentioned in the grammatical literature, but it is not made quite
explicit in the *Metaphysics* definition: they are **sequential**, gradual in the sense of Lat. ῆ
**gradus**, a step, related to στείχω, to walk, and to στίχος, a line of verse, or a line in
general.

The first ever use of the word itself seems to come from a third-hand report by
Plutarch of the cosmology of the obscure Petron of Himera,<fnref n="17" /> who posited the existence
of 183 worlds (κόσμοι) arranged in the form of a triangle, with one at each vertex
and sixty along each side, which are said to be next to each other and to κατὰ
στοιχεῖον ἄπτεσθαι—which we could gloss as ‘to cohere in orderly fashion’, ‘like an
ABC.’ A French translation gives ‘ils se touchent les uns les autres par leurs éléments
fondamentaux.’<fnref n="18" />

The reference to this extraordinary cosmology appears twice in Plutarch's dialogue.<fnref n="19" />
Initially one of the characters presents it and adds: ἄπτεσθαι δὲ τοὺς ἐφεξῆς ἀλλήλων
ἀτρέμα περιυόντας ὥσπερ ἐν χορείᾳ, ‘they hang on to each other in a row, going
round in circles, gently, as in a dance,’ but a few lines down, the narrator, presumably
Plutarch, comments that he himself does not know what to make of the κατὰ στοιχεῖον
ἄπτεσθαι.<fnref n="20" /> Now, János Balázs, who makes a strong case for the musical-rhythmical

<footnote n="15">15. *Scholia in Iliadem* (scholia recentiora Theodori Meliteniotis, *e cod. Genevensi gr. 44*), Repr. 1966, vol. 2, Les scolies genevoises de l'Iliade, ed. by J. Nicole (Geneva: Georg, 1891), 23.757.</footnote>

<footnote n="16">16. Burkert, *ΣΤΟΙΧΕΙΟΝ*, 169.</footnote>

<footnote n="17">17. See Diels, *Elementum*, 62–3; Vollgraff, ‘Elementum,’ 91ff. The uncertainty is directly related to the dating of Petron. Diels himself, presenting this as the earliest ever textual evidence, writes that ‘he *seems* to appear in the sixth century BC’ [my italics].</footnote>

<footnote n="18">18. Plutarch, ‘Pourquoi les oracles ont cessé,’ in *Oeuvres morales de Plutarque*, trans. D. Ricard, vol. 2 (Paris: Lefévre, 1844), 328.</footnote>

<footnote n="19">19. At 422b3 and later 422d7.</footnote>

<footnote n="20">20. See H. Diels and W. Kranz, *Fragmente der Vorsokratiker*, 6th ed., Repr. 1966 (Berlin: Weidmann,</footnote>
origin of στοιχεῖον, points out how Petron’s 183 worlds are ‘contiguous in a choric
dance with one another.’<fnref n="21" /> Balázs also points out that ‘the verb στείχω meant not going
in general, but procession in *well-ordered ranks*,’ and he establishes associations with
the Greek appreciation of dance as the all-encompassing artistic form. Perhaps we
should further specify and instead of simply considering the elements to be ‘gradual’,
we might say they are choreutic, or even *dancey*. A similar usage in a cosmological *P*
context is found in *Sch. to De divinibus nom.* 256,22, «ἀλληλουχίαι δὲ τῶν ὁμοστοίχων»
εἰσὶν αἱ τοῦ κόσμου *κατατάξεις* ἐκ μιᾶς οὐσίας τῆς ὕλης εἰς σώματα πληθυνθεῖσαι
στερεωμάτων καὶ οἰονεὶ σειρὰ ἀλλήλων ἐχόμεναι, the ‘interconnections of those who
have the same element’ are the *orders* of the universe, which from one essence are
multiplied into solid bodies of the matter, and which hold to each other ‘like links
of a chain.’<fnref n="22" /> Even further on the musical aspect, in later Greek alchemical literature,
Stephanus of Byzantium would describe how ‘Orpheus made melody with rhythmical
sounds so that the symphony should re-echo the co-ordinated movement of the
elements,’ where the elements are called ὁμοταγεῖς οὐσίαι, the ‘co-ordinated essences.’<fnref n="23" />

In any case, eschewing Petron as a dubiously dated testimony, if we are looking
for a completely certain earliest testimony to the use of the word, we must also
dismiss Anaximander<fnref n="24" /> and other pre-Socratic authors whose *ipsissima verba* are
not preserved,<fnref n="25" /> and acknowledge that στοιχεῖον first occurs in Aristophanes’s
*Ecclesiazusae* (651), ὅταν ἦ δεκάπουν τὸ στοιχεῖον, ‘when the shadow of the gnomon
is ten feet long.’ And thus we can see how the earliest certain recorded meaning of
στοιχεῖον has to do with gnomonics, referring to the shadow cast on the sundial, and
by extension to the module of its advance.<fnref n="26" /> This powerful image combines different
meanings contained in the etymology: it is a walking shadow-line, and it walks in
circles, determining events as it progresses.

<footnote n="1951">1951), 28 (hereafter cited as DK).</footnote>

<footnote n="21">21. Balázs, *The forerunners*, 233. This article seems to have escaped the notice of Burkert and others. See also, along similar lines, the more recent Laspia, *L'excursus fonologico del Teeteto*, 204.</footnote>

<footnote n="22">22. *Ioannis Scythopolitani prologus et scholia in Dionysii Areopagitae librum 'De divinis nominibus' cum additamentis interpretum aliorum*, vol. 4,1, Patristische Texte und Studien 62, ed. by B.R. Suchla (De Gruyter, 2011), 236; in spite of some common 'sociological' readings, the medieval commentaries agree in giving to ὁμόστοιχος here a physical sense.</footnote>

<footnote n="23">23. F. Sherwood Taylor, 'The Alchemical Works of Stephanos of Alexandria, part 2,' *Ambix*, June 1938: 126-27.</footnote>

<footnote n="24">24. See DK, 11ff.</footnote>

<footnote n="25">25. Cf. Druart, *La Notion de « stoicheïon »*, 422, 'si les textes doxographiques utilisent le mot stoicheïon, les citations probables des Présocratiques, elles, l'ignorent.'</footnote>

<footnote n="26">26. Cf. Vollgraff, *Elementum*, 102.</footnote>
## The Concept

As regards the concept itself, it is well known<sup>27</sup> that it was Empedocles who, following the Pythagorean predilection for the holy Tetractys, introduced the quaternary of the elements, calling them the 'roots of all things,' ῥιζώματα τῶν πάντων. He thus invented what would become the prevailing concept of 'element', without employing the name which would eventually be used for it, στοιχεῖα. He characterizes them as 'primordial and jointly originating,' πρῶθ' ἥλικά τ' ἀρχήν,<sup>28</sup> hence also the primal elements [Urelemente], from which are compounded the four visible elements of the current world order. Πρῶθ' ἥλικά τ' ἀρχήν could be alternatively rendered 'primordial and contemporary in origin,' to reveal an interesting new property of the elements: they are **coeval**, all coming to be at the same 'time', without question of precedence among them.

<figure><img src="image_3.png" /></figure>

Another important basic quality of the elements made explicit above, recalling also the shadow of the gnomon, is their 'halfway' ontological status, being **intermediary** P between the transcendent and the material.

Not even the atomists had decided on a term for their corpuscles, 'calling them alternatively εἴδεα, σχήματα, ἰδέαι, φύσεις, ναστά, ἄτομα, among others'.<sup>29</sup> Perhaps the most important gain from reading Diels's work is to realize that the word στοιχεῖον was originally not limited to the meaning of physical constituents of matter, but that there was, in the words of Burnet,<sup>30</sup> 'the ambiguity of the term “element.”' The examples abound, and this is for instance why texts would appear like the Pseudo-Plutarch Tίνι διαφέρει ἀρχὴ καὶ στοιχεῖα, *What is the Difference between ἀρχὴ and στοιχεῖα?*<sup>31</sup>

## Atoms and Letters

The comparison to letters, if not a straightforward identification, seems to be a very early feature of the concept of στοιχεῖα in their 'atomic' aspect, as *naturale simplex*

<footnote n="27">27. See Diels, *Elementum*, 15.</footnote>

<footnote n="28">28. Cf. DK, 21.B.38.</footnote>

<footnote n="29">29. Diels, *Elementum*, 16.</footnote>

<footnote n="30">30. J. Burnet, *Early Greek Philosophy*, 3rd ed. (London: A & C Black, 1920), 18.</footnote>

<footnote n="31">31. *Placita philosophorum* 875C, 4. Note the plural of στοιχεῖα in contrast to the singular of ἀρχὴ.</footnote>
*corpus*. Recent analysis of some of Heraclitus’ fragments,<fnref n="32" /> and recent approaches to some of the aspects of Democritean atomism,<fnref n="33" /> make very possible a hitherto unexplored continuity of this image at the very onset of the Greek philosophical tradition, including a very conscious usage by Plato of an imagery which was already part of the common heritage.<fnref n="34" />

## Platonic Corpus

It was already affirmed in ancient times that ‘no one before Plato had used the term στοιχεῖον to refer to the physical principles’.<fnref n="35" /> The classical reference is attributed to Eudemus<fnref n="36" /> and the present context warrants a good reading of it:

Πλάτων τά τε τῶν Πυθαγορείων καὶ τῶν Ἐλεατικῶν ἐπὶ τὸ σαφέστερον προαγαγὼν τά τε ὑπὲρ τὴν φύσιν ἐξύμνησεν ἀξίως κἀν τοῖς φυσικοῖς καὶ γενητοῖς τὰς στοιχειώδεις ἀρχὰς τῶν ἄλλων διέκρινε καὶ στοιχεῖα πρῶτος αὐτὸς ὠνόμασε τὰς τοιαύτας ἀρχάς.

Clarifying greatly the doctrines of Pythagoreans and Eleatics, Plato composed worthy ‘hymns’ about the preternatural. He discerned the **elementary principles** in what is manifested and natural, and he was the first to call those principles ‘elements’.

The expression στοιχειώδεις ἀρχὰς is noteworthy: ‘those principles which are like στοιχεῖα.’ According to this, clearly enough, the στοιχεῖα would be one species within the genus ‘principles.’ Unfortunately, the Platonic corpus does not really warrant such a clear distinction. In fact, when comparing the rather motley use of στοιχεῖα across the Platonic corpus against the wealth of pre-Socratic metaphysical speculation, it seems more appropriate not to insist too much on any sort of a programmatic intention on the part of Plato, but rather to assent to the laborious conclusions of Crowley, ‘The core sense of *stoicheion*, then, is that of a basic part of a whole,’<fnref n="37" /> and in particular to the work by Thérèse-Anne Druart, who has devoted two meticulous articles to the ‘Platonic *stoicheiology*’ and the matter of the letter-simile, Diels’ *Buchstabengleichnis*.<fnref n="38" />

<footnote n="32">32. A. Lebedev, ‘The Metaphor of *Liber Naturae* and the Alphabet Analogy in Heraclitus’ *Logos Fragments*,’ paper delivered in October 2013 at the Conference «Heraklit von Ephesos und seine Zeit», Selçuk, Turkey, 2013, (Now in print in the volume *Heraklit im Kontext*), esp. 3–16.</footnote>

<footnote n="33">33. See D. W. Graham, *The Texts of Early Greek Philosophy: the Complete Fragments and Selected Testimonies of the Major Presocratics*, 2 vols (Cambridge University Press, 2010), 617.</footnote>

<footnote n="34">34. Lebedev, ‘The Metaphor of *Liber Naturae*,’ 26.</footnote>

<footnote n="35">35. Diels, *Elementum*, 17.</footnote>

<footnote n="36">36. Eudemus of Rhodes, *Fragmenta*, 2nd ed., ed. F. Wehrli, Die Schule des Aristoteles 8 (Basel: Schwabe, 1969), fr. 31. Cf. also the valuable observations in Crowley, ‘*On the Use of Stoicheion*,’ 369–70.</footnote>

<footnote n="37">37. *Ibid*., 392.</footnote>

<footnote n="38">38. T.-A. Druart, ‘La Stoicheïologie de Platon,’ *Revue Philosophique de Louvain* 73, no. 18 (1975): 392, and Druart, ‘*La Notion de « stoicheïon »*.’</footnote>
Her conclusions run along similar lines: 'la valeur paradigmatique de la « comparaison des lettres » a été source, elle aussi, d'une extension de la problématique par un dépassement du cadre proprement linguistique.'<sup>39</sup> And in her earlier article devoted to *Theaetetus*, 'Le passage incessant de composant matériel à constituant explicatif, nous révélerait alors que, par les stoicheïa, Platon entend, tend plus exactement à élaborer une notion de constituant, de principe explicatif interne.'<sup>40</sup>

Whatever the case, Plato's impact, which is reflected not only in Aristotle's definition above, but also in his own flexibility regarding the term,<fnref n="41" /> opened the field for very broad subsequent developments. One case in point is Euclid, who came to be dubbed ò στοιχειωτής<fnref n="42" /> and whose influence is prominent in later Neo-Platonic literature, both in mathematical and arithmological speculations.

## The Constellations

Another major semantic development of the term, perhaps partly influenced by early Christian literature, and mostly by the lasting impact of the New Testament,<fnref n="43" /> is the astronomical meaning of constellation, whence zodiacal sign and eventually the astrological concept of ruler of a sign.<fnref n="44" /> Variously conceived as 'elemental spirits,' or 'cosmic spirits' worshipped 'by Jews and Gentiles,' they had a bad connotation which according to Diels is ultimately at the origin of the late medieval and modern Greek στοιχείο for a demon, genie, ghost or bugbear.<fnref n="45" />

All this rich context makes it all the more impressive that the term became eventually so important in grammatical treatises, as I shall discuss in the next section. All the pre-Socratic doctrines and all the Platonic syntheses and the Aristotelian treatments of στοιχεῖα were bound to inform, even if indirectly, all succeeding scholarly appropriations of the term. Let us then step back somehow, or aside into the grammatical tradition, but not before a brief excursus to mention the Latin lexical and philosophical correspondences which I have hitherto left unmentioned.

<footnote n="39">39. Druart, 'La Stoicheiologie,' 261.</footnote>

<footnote n="40">40. Druart, 'La Notion de « stoicheïon »,' 433.</footnote>

<footnote n="41">41. See Crowley, 'On the Use of Stoicheion,' 370ff.</footnote>

<footnote n="42">42. Diels, *Elementum*, 26.</footnote>

<footnote n="43">43. See *ibid*., 50ff.</footnote>

<footnote n="44">44. See Manetho, *Apotelesmatica*, 4, 624, οὐρανίων ἄστρων στοιχεῖα, D. L. 6, 102, τὰ δώδεκα στοιχεῖα, and cf. also 'Lexikon zur byzantinischen Gräzität,' s.v. στοιχειωματικός, 'belonging to the horoscope.'</footnote>

<footnote n="45">45. See *Glossarium mediæ et infimæ graecitatis*, ed. C. D. F. Du Cange (Lyon: Jean Anisson, 1688), (reprint Collège de France, 1943), 1.1453 (hereafter cited as Du Cange-Grec).</footnote>
Latin Correspondences

The word *elementum* appears in Lucretius<fnref n="46" /> and then in Cicero<fnref n="47" /> with the two meanings of letter and element that στοιχεῖον had already developed in the Greek world by that time. It is a matter of debate how ontological or simply rhetorical Lucretius’ usage of the letter image was,<fnref n="48" /> but Cicero was one of the translator of Timaeus, and *elementum* appeared in other later authors partaking of the contemporary debates in Greek philosophy, which included Stoic atomistic and Middle Academy developments.<fnref n="49" /> Interestingly, these first attestations show ready-made the identification between στοιχεῖον and ἄτομος, the ‘physical’ meaning which was never articulated as such in pre-Socratic literature in spite of being conceptually there. Examples from Horace, Ovid, Quintilian and other later grammarians abound. For Boethius (*De institutione arithmetica*, 2,48), the interval of fourth, being the basis of harmony, can be considered virtually an *elementum*.

Regarding the etymology of *elementum*, in spite of some elaborate, and at times very curious, hypotheses by Diels,<fnref n="50" /> Sittig,<fnref n="51" /> and others,<fnref n="52" /> Burkert appears to be right in considering it a clear case of undecidability.<fnref n="53" />

Greek Grammarians

It might be said without exaggerating that the Greek grammatical tradition is all in one way or another related to Dionysius Thrac's Τέχνη γραμματική or *Ars grammatica*.<fnref n="54" /> According to Uhlig, the editor of the still standard Teubner edition, 'no other book in the entire world of profane literature has been preserved of such a great impact, not only because this *techne* is the ancestor of all the grammars authored in Europe, but also because there is hardly any of them in which the traces of the original are totally

<footnote n="46">46. 'Five times in the first two books of the *De rerum natura* the arrangement of atoms in an object is compared to the arrangement of letters in a word'; see A. Dalzell, 'Language and atomic theory in Lucretius,' *Hermathena*, no. 143 (1987): 19.</footnote>

<footnote n="47">47. *Academica*, I, 7, 26. See Diels, *Elementum*, , 69.</footnote>

<footnote n="48">48. See especially the reference to Friedländer in Dalzell, 'Language and atomic theory in Lucretius,' 28n.</footnote>

<footnote n="49">49. See Diels, *Elementum*, 68ff.</footnote>

<footnote n="50">50. *Ibid*., 81ff.</footnote>

<footnote n="51">51. Sittig, 'Abecedarium.'</footnote>

<footnote n="52">52. S. B. Platner, 'Notes on Elementum,' *The Classical Review* 8, no. 8 (1894): 344–345; Rogge, 'Nochmals lat. elementum.'</footnote>

<footnote n="53">53. Burkert, 'ΣΤΟΙΧΕΙΟΝ,' 167.</footnote>

<footnote n="54">54. Dionysius Thracius, *Ars grammatica*, ed. G. Uhlig and A. Merx, Grammatici Graeci 1 (Leipzig: B. G. Teubner Verlag, 1883).</footnote>
obliterated.’<fnref n="55" /> It must be noted that this appraisal is written very nearly two thousand years after the appearance of the treatise, and that it does not mention the definitive and lasting impact on Byzantine scholarship.

The Τέχνη is traditionally attributed to Dionysius (fl. 100 BC), a pupil of Aristarchus of Samothrace. If the attribution is to be trusted, this little vademecum would be the only Hellenistic grammatical treatise to survive to our times, but Dionysius’ authorship has been doubted and discussed since antiquity; some scholars maintain that the entire treatise is a compilation of the third or fourth century AD, others defend its complete authenticity, dating it to the end of the second century BC, and a third group of scholars argue for a middle position, allowing for some portions of the book to go back to Dionysius.<fnref n="56" />

Dionysius’ Τέχνη, ‘in essence a descriptive taxonomy of classical Greek phonology and morphology,’<fnref n="57" /> is divided into twenty sections. Section 6, Περὶ στοιχείου, comprises forty lines of text, of which only the first five lines concern this enquiry. After them, the text explains the phonetic classification of the letters and the role some letters play on the nominal inflection. My objective in the following pages will be to survey the literature dedicated to these first five lines of section 6, which read:

Περὶ στοιχείου· γράμματά ἐστιν εἰ-κοσιτέσσαρα ἀπο τοῦ α μέχρι τοῦ ω. γράμματα δὲ λέγεται διὰ τὸ γραμμαῖς καὶ ξυσμαῖς τυποῦσθαι· γράψαι γὰρ τὸ ξῦσαι παρὰ τοῖς παλαιοῖς, ὡς καὶ παρ’ Ὀμήρωι· «Νῦν δέ μ’ ἐπιγράψας ταρσὸν ποδὸς εὖχεαι αὖτως». τὰ δὲ αὐτὰ καὶ στοιχεῖα καλεῖται διὰ τὸ ἔχειν στοῖχόν τινα καὶ τάξιν.

On the element—The letters [γράμματα] are twenty-four, from α to ω. They are called γράμματα because they are drawn with lines [γραμμαί] and scratches, since for the ancients drawing was like making scratches, as in the *Iliad*, ‘Now, having but scratched my ankle with your arrow, you boast vainly.’ These same letters are called ‘elements’ because they have a sort of gradation and order.<fnref n="58" />

<footnote n="55">55. Dionysius Thrax, *Dion. Thrax*, vi. Regarding its influence on Syriac and other Semitic grammatical traditions, see D. King, ‘Elements of the Syriac Grammatical Tradition as these Relate to the origins of Arabic grammar,’ in *The Foundations of Arabic Linguistics: Sībawayhi and Early Arabic Grammatical Theory*, ed. A. Marogy (Leiden, Boston: Brill, 2012).</footnote>

<footnote n="56">56. For dating and attribution questions, see L. Pagani, ‘La “Techne grammatike” attribuita a Dionisio Trace e la nascita della grammatica nell’antichità greca,’ *Rivista di filologia e di istruzione classica* 138, no. 3 (2010): 390–409; E. Dickey, *Ancient Greek Scholarship: A Guide to Finding, Reading, and Understanding Scholia, Commentaries, Lexica, and Grammatical Treatises, from Their Beginnings to the Byzantine Period*, Classical Resources 7 (New York: Oxford University Press, 2007), 77; F. Ildefonse, *La Naissance de la grammaire dans l’Antiquité grecque*, Histoire des doctrines de l’Antiquité classique 20 (Paris: J. Vrin, 1997).</footnote>

<footnote n="57">57. R. H. Robins, *The Byzantine Grammarians: Their Place in History*, vol. 70, Trends in Linguistics: Studies and Monographs (Berlin/New York: Mouton de Gruyter, 1993), 41.</footnote>

<footnote n="58">58. Dionysius Thrax, *Dion. Thrax*, 6</footnote>
This is the basic text. I will now present the relevant scholia and commentaries by topic, drawing mainly from Hilgard's rich collection, which includes mostly undated material spanning at least ten centuries of exegesis,<fnref n="59" /> and making use also of the works by Lallot<fnref n="60" /> and Pecorella.<fnref n="61" />

## The Opening Anacoluthon

A recurring topic in the scholia is 'the puzzling anacoluthon' between the heading Περὶ στοιχείου and the initial words γράμματά ἐστιν...

Ἀπορίαν τινὰ εὐθέως καὶ ζήτησιν ἔχει ἡ ἐπιγραφή, τίνος ἔνεκεν ἐπιγράψας «περὶ στοιχείου» οὐκ ἐπήγαγε «στοιχεῖον ἐστιν» [...] Ταύτην οὖν τὴν ζήτησιν σὺν θεῷ εἰσόμεθα.

The heading presents us immediately with an impasse and demands an inquiry: why, after writing «περὶ στοιχείου», did he not follow with «στοιχεῖον ἐστιν»? [...] This is a matter we will only know with God's help.<fnref n="62" />

It goes without saying that this nomenclature problem is not just a quirk of the author, but an indication of the polysemy of the underlying concept. In any case, a little further down, a solution is offered:

Καὶ ἔστι μὲν εἰπεῖν, ὡς καὶ αὐτὸς μετ' ὀλίγον ἐρεῖ, ἐπειδὴ ταὐτόν ἐστι στοιχεῖον καὶ γράμμα· φησὶ γὰρ ὑποκατιὼν ὁ τεχνικός «τὰ δὲ αὐτὰ καὶ στοιχεῖα καλεῖται»· τὸ δὲ ἀληθές, ὅτι στοιχεῖον μέν ἐστιν ἡ ἐκφώνησις, γράμματα δὲ αἱ εἰκόνες καὶ οἱ χαρακτῆρες. Καὶ χαρακτῆρες μέν εἰσιν κδ', ἐκφωνήσεις δὲ πολλῷ πλείους·

It can be said, as he himself does a little later, that it is because the element and the letter (γράμμα) are the same. Indeed, a few lines below the author says 'These same are also called elements.' But the truth is that the element is the phoneme, whereas the letters are the images (εἰκόνες) and characters. And the characters are twenty-four, but the phonemes ['allophones' in linguistic jargon] are many more.<fnref n="63" />

After this, the scholiast follows with a total count of sixty-six alphabetic sounds (στοιχεῖα), including forty-five vocalic and twenty-one consonantal, which correspond to the twenty-four letters (γράμματα). With this passage, we come across another

<footnote n="59">59. *Scholia in Dionysii Thracis Artem grammaticam*, Grammatici Graeci 1.3, ed. by A. Hilgard (Leipzig: B. G. Teubner Verlag, 1901) (hereafter cited as Schol. in Dion. Thrax)—when no details are given, page numbers refer to Hilgard throughout this chapter.</footnote>

<footnote n="60">60. *Dionysius Thrax, La grammaire de Denys le Thrace, 2e éd. rev. et augm.*, ed. J. Lallot, *Sciences du langage (Paris: CNRS Éditions, 1998)*.</footnote>

<footnote n="61">61. *Dionysius Thrax, Techne grammatike*, ed. G. Pecorella (Bologna: Cappelli, 1962).</footnote>

<footnote n="62">62. pp. 30–31.</footnote>

<footnote n="63">63. p. 32.</footnote>
fundamental duality of the concept: the στοιχεῖα are not only ‘letter’ and elemental atom, but they are also a sound and a written sign. The fluid alternation between στοιχεῖα and γράμματα came from earlier times and would be persistent; when Aristotle wanted to be unequivocal in speaking about the written signs, he used τὰ γραφόμενα.<fnref n="64" />

## Simplicity and Pedagogy

Ἐστέον ὅτι τὰ ὀνόματα τῶν στοι-
χείων ἄκλιτά εἰσι. Καὶ διὰ τί εἰσιν
ἄκλιτα; Ἐπειδὴ ἀρχαί εἰσι, καὶ αἱ ἀρ-
χαὶ ἀπλαῖ καὶ ἀποίκιλοι ὀφείλουσιν
εἶναι· ἤ ὅτι καὶ οἱ θεμέλιοι τὸ ἀμε-
τάθετον ὀφείλουσιν ἔχειν· ἄλλως
δὲ διὰ τὸ ἀρτιμαθές τῶν παίδων οὐ
κλίνονται, ἵνα μὴ πολλὰ γινόμενα
δυσκατάληπτα αὐτοῖς γένωνται.

One should know that the names of the let-
ters are indeclinable. Why are they indeclin-
able? Because they are principles (ἀρχαί), and
principles must be simple (ἀπλαῖ) and homo-
geneous (ἀποίκιλοι). Or also because like the
foundations, they must have to do with what
is unalterable. Or maybe they are not declined
because then there would be too many hard to
grasp new concepts for the children who are just
starting to learn.

The fact that they are ‘indeclinable’, reveals another aspect of their being ἀδιαίρετος, but emphasizing their being **unchangeable**, invariable. Here for the first time in the P grammatical literature we encounter the very important ‘constructive’ image found in the Aristotelian definition, the fact of being a ‘component’, ἐξ ὧν σύγκειται. We also find for the first time the association with children that seems to pertain to the στοιχεῖα through the ages and cultures.<fnref n="65" /> I shall have later occasion to elaborate on this particular facet which is of course not unrelated to the primordiality of the elements.

## The Semitic Origin

The mythical Phoenician origin of the alphabet gives rise to a number of etymological speculations and reveals new aspects of the letters.

Καλοῦνται δὲ τὰ στοιχεῖα φοινίκεια,
ἐπεὶ ὁ Κάδμος Φοῖνιξ ὤν εἰς Ἕλλη-
νας ταῦτα μετήνεγκεν· ἤ ὡς φωνίκειά
τινα ὄντα, ἦτοι φωνῆς ἐγγραμμάτου
δυνάμεις καὶ εἰκόνες, τοῦ <ω> μετα-

The letters are called Phoenician because it
was Cadmus, a Phoenician, who brought them
over to the Greek. Or maybe, if an ω replaces
the diphthong οι, they are called Phoenician
because they are phonic (φωνίκεια), being ut-

<footnote n="64">64. In *De interpretatione*, with Ammonius’ comment, cited in Vogt-Spira, *Vox und Littera*, 304.</footnote>

<footnote n="65">65. Cf. Dornseiff, *Alphabet*, 17–20.</footnote>
βληθέντος εἰς τὴν <οι> δίφθογγον...
otti
Phoenician

Phoenician

βληθέντος

εἰς

τὴν

<οι>

δίφθογγον...

Η

φοινίκεια,

ἐπεὶ

διὰ

μίλτου,

ἄρ

ἐστὶ

χρῶμα

φοινικοῦν,

πρότερον

ἐγράφετο

ἐν

πετά-

λοις

φοινικείοις

ἐγράφετο·

ἐν

πυρρώδει

κηρῷ

ἐτυποῦντο·

ἔπερ

ἐστὶ

κρεῖττον,

ἔτι

φοινίσσεται

ὑπ'

αὐτῶν

ο

νοῦς

ῆγουν

λαμπρύνεται.

terances and counterparts of written speech...

Or they are called Phoenician because it was

with ochre, which is a **reddish** colour, that

people first wrote. Or because the earliest writ-
ing was made on red clay tablets; or because

they impressed them on red-looking wax. Or,

what is more likely, the mind **turns golden red**

through them, as if it were polished.<fnref n="66" />

The variety of the explanations gives an idea of the uneven texture of the genre, but it
is not difficult to discern some valuable features. First, there is a notable original Semitic
relation. If we assume that there is a Semitic consonantal substrate in the alphabet, and
given the nature of the abjads, we may conceive of an original feature of the στοιχεῖα,
prior to the vocalic evolution of the Greek alphabet: they are **unvocalised**, in what
might be expressed as a sort of quiescence.

P

P

I would also like to note and retain the curious ‘gilding’ or ‘polishing’ of the mind,
which shows the letters as enlightening, and which goes together with the reference to
the golden-red phoenix and its well-known solar connotations.<fnref n="67" /> In another scholium
(p. 312) is found a common saying directly related to this: φῶς νοῦ ἐτυμολογεῖται ἡ
φωνή, ‘the word φωνή comes from φῶς νοῦ, the “light of the mind”’.<fnref n="68" />

P

P

Further down (p. 32), there is a little remarkable variation on the *Phoenician* label,
according to which they were ‘sent down’ by Hermes, thus making the elements
*Hermetic* in a certain way.

P

P

Τινὲς δέ φασι τοὺς χαρακτῆρας τῶν
στοιχείων τοὺς παρ' ἡμῖν ὑπὸ Ἐρμοῦ
ἐν φοίνικος φύλλω γεγραμμένους κα-
ταπεμφθῆναι τοῖς ἀνθρώποις, διὸ καὶ
φοινίκεια λέγεται τὰ γράμματα·

Some say that the characters of the elements
as we have them were **sent down** to men by
Hermes written on a date [φοίνιξ] leaf, and
this is why the letters are called φοινίκεια.

It will noted that here the writer equates the letters (γράμματα) with χαρακτῆρες τῶν στοιχείων, the graphical traces of the elements/phonemes. The unpredictable identification and contrast with γράμμα, pointing to the phonetic nature of στοιχεῖα, reinforces the impression of their intermediary nature: like air or breath, they are

<footnote n="66">66. p.191-92.</footnote>

<footnote n="67">67. Cf. Herodotus, *Historiae*, 3rd ed., ed. K. Hude, *Scriptorum Classicorum Bibliotheca Oxoniensis, Repr.* 1990 (Oxford: Oxford University Press, 1927), II. 73, ὄρνις ἱρός, τῷ οὖνομα φοῖνιξ [...] τὰ μὲν αὐτοῦ χρυσόκομα τῶν πτερῶν, τὰ δὲ ἐρυθρά·, ‘a sacred bird by the name of Phoenix [...] some of its feathers are golden and some red.’</footnote>

<footnote n="68">68. Found also in G. Choeroboscus, *Περὶ τρόπων ποιητικῶν*, ed. L. Spengel, vol. 3, Rhetores Graeci (Leipzig: Teubner, 1856), (reprint 1966), 249.19, and other contemporary authors.</footnote>
τοὺς ἀριθμοὺς τυποῦσθαι· στοῖ-
χος γὰρ παρὰ τοῖς παλαιοῖς ὁ
ἀριθμός· τοιγαροῦν οἱ Σικυώνιοι
κατὰ φυλὰς ἑαυτοὺς τάξαντες καὶ
ἀριθμήσαντες Διὸς Στοιχαδέως ἱε-
ρὸν ἰδρύσαντο. Ἀλλοι παρὰ τὸ
στιχηδὸν γράφεσθαι στιχεῖον, καὶ
πλεονασμῷ τοῦ «ο» στοιχεῖον·

στοιχεῖα] because numbers are written with
them, **for among the ancient, a graduated line
was the number.** This is why the Sicyonians,
when they ordered and numbered themselves in
*philai*, built a temple to Zeus Stoichadeus. Yet
others say the word is really στιχεῖον [line], due
to their being written in lines [στιχηδὸν], and
that the added <ο> makes it στοιχεῖον.

Here is made explicit for the first time the all-important connection between στοιχεῖον and number. The formulation, based on the step-by-step linear order of a στοῖχος, looks from our modern understanding of arithmetic like a clear reference to the number line of the positive integers, hence the natural reference to a rank of soldiers. There is also in this image a neat analogy with the sundial etymology we had encountered before, and it also makes it easy to develop similar comparisons with astronomical movements in their regular progress through the graduated spheres.<fnref n="70" /> It should also be related to the association with Pythagoras through the geometrical shaping of the characters,<fnref n="71" /> and to discussions about the number of the elements which we shall soon find in the scholia. Also, when related to the phonetic nature of the στοιχεῖα, we have here the idea of **ordered** sound which becomes utterance, 'for order is στοῖχος', as explained in P the following fragment:

Οἱ δὲ παρὰ τὸ στοῖχος στοιχεῖον,
τὸ ὡς ἐκ τάξεως φωνὴν ἤγουν
ἐκφώνησιν ἀποτελοῦν· στοῖχος
γὰρ ἡ τάξις.

Some say στοιχεῖον is modelled on στοῖχος, be-
cause they believe that the voice (φωνή) becomes
an utterance (ἐκφώνησις) thanks to its order, for
order is στοῖχος.

The image here, which is of capital importance and has a long philosophical history, is that of the division of the acoustic continuum into phonetic units, and in view of the above, it is conceived as a sort of ‘numbering’ of the raw phonetic material of language, in order to create meaning.

<footnote n="70">70. This is precisely the image found in R. Caballero, ‘El Comentario anónimo al *Tetrabiblos* de Tolomeo: Edición crítica y traducción castellana de los escolios metodológicos del libro I (in Ptol. Tetr. 1.1.1–1.3.1)’, *MHNH* (Universidad de Málaga), no. 13 (2013): 225, «Τάξει καὶ δυνάμει.» Τουτέστιν ἀριθμῷ, “regarding order and efficacy”, that is, number,” with a reference to the calculation of the movements of the heavenly bodies.</footnote>

<footnote n="71">71. Lagercrantz, *Elementum*, 5.</footnote>
## Walls and Ramparts

Bringing into focus new facets through more etymologies, some passages in the
collection explain in the following way the association of the elements with walls:

Οἱ δὲ παρὰ τὸ τοῖχος τοιχεῖον, καὶ πλεονασμῷ τοῦ <ς> στοιχεῖον· ἐν γὰρ
τοίχοις ἔγραφον τὸ πρότερον...

[36, 9] ὥσπερ γὰρ τὸ τεῖχος φυλακή ἐστι καὶ ἀσφάλεια τῶν ἐν τῇ πόλει,
οὕτω καὶ τὰ στοιχεῖα φυλακή ἐστι τῶν παλαιῶν πραγμάτων· δι’ αὐτῶν
γὰρ ἡμῖν τοῖς μεταγενεστέροις ἐφυλάχθη καὶ ἐν γνώσει ἐγένετο.

Others say the original word is *τοῖχος*, *τοιχεῖον* (wall), with an added σ, because at first people wrote on the walls...

Just as a wall is a protection and a reassurance for what is within the city,
the letters are a protection for the matters of the ancients, since it is through
them that such things have been kept for us the later born, and brought to
our knowledge.

This makes the elements have also a protective, defensive, **safeguarding** aspect, P
which has to do with the above mention of Zeus *Stoichadeus*, god of the *phylai* or of
battle ranks, inasmuch as they are considered groups of numbered individuals. Perhaps
later on, in the comparative phase of this work, it would be relevant to explore the
semantic parallel with the Hebrew YHWH *Tzeba'ot*, 'Lord of Hosts'.<sup>72</sup> In the Qur'an,
besides the recurring ascription of the 'armies' (*junūd*) of heaven and earth to God, we
find mention of the angels ordered in ranks, *saffāt* (37: 1),<sup>73</sup> cognate to *saffāf*, a typesetter
or composer.<sup>74</sup>

## And what about ‘γράμματα’?

The etymology of γράμματα also comes up in another passage which tries to clarify the
difference between them and στοιχεῖα:

οἱ δέ φασιν, ὅτε μὲν γράφονται, λέγονται
γράμματα, ὅτε δὲ ἀναγινώσκονται,
λέγονται στοιχεῖα. Καὶ στοιχεῖα εἴρηται
παρὰ τὸ ἔχειν στοῖχόν τινα καὶ τάξιν,
γράμματα δὲ διὰ τὸ γραμμαῖς καὶ ξυσμαῖς

Some say that when written they are
called γράμματα, and when read aloud
στοιχεῖα. They are called στοιχεῖα be-
cause they have some sort of rank and
order, and γράμματα because they are

<footnote n="72">72. Cf. Ps. 46: 7; and see below pp. 111, 116, and 193.</footnote>

<footnote n="73">73. Cf. the gloss in F. Hamza, trans., *Tafsīr al-Jalālayn* (Amman: Royal Aal al-Bayt Institute for Islamic Thought, 2016), 37: 1, ‘the angels who range their souls in worship or their wings in the air awaiting their orders.’</footnote>

<footnote n="74">74. Cf. E. W. Lane, *Arabic-English Lexicon*, 2 vols (Cambridge: Islamic Texts Society, 1984), s.v. ṣ-f.</footnote>
τυποῦσθαι· ἤ παρὰ τὸ γλάπτω, τὸ κοιλαίνω, γλάμμα, καὶ κατὰ μετάθεσιν τοῦ «λ» εἰς τὸ «ρ» γράμμα· ἤ παρ' ὅσον ἐκ τῶν παρὰ τοῖς γεωμέτραις λεγομένων γραμμῶν ἐσχημάτισται· καὶ γάρ, ὥς φασιν, ὁ Πυθαγόρας τοῦ κάλλους αὐτῶν ἐπεμελήθη, ἐκ τῆς κατὰ γεωμετρίαν γραμμῆς ῥυθμίσας αὐτὰ γωνίαις καὶ εὐθείαις καὶ περιφερείαις.

The mention of the verb γλάπτω, a form of γλάφω, meaning carving or engraving, introduces the considerations about the crafts related to the production of the letters. As I shall have the occasion to show later, this demiurgic dimension has an important bearing on the cosmogonic role of the alphabet.

As in Aristotle's definition, there is here again the mention of the geometric diagrams, but this time with Pythagoras involved in the development of the letters' shapes. Pythagoras is related to the origin and the nature of the στοιχεῖα in many more ways that we will explore in the next chapter.

## Attributes of the στοιχεῖον

A recurring topic is that of the attributes or qualities of the individual στοιχεῖα which allow them to be grouped in categories. In the following commentary, four are enumerated, but they can be five or six according to other authors.

Τούτω οὖν τῷ στοιχείῳ, τουτέστι τῇ ἐκφωνήσει, παρέπεται τέσσαρα· χαρακτὴρ μέν, ὡς τρίγωνον σχῆμα ἤ ἡμικύκλιον ἤ στρογγύλον καὶ τὰ ἕξῆς ὄνομα δέ, ὡς τὸ ἄλφα ἤ βῆτα καὶ τὰ λοιπά· δύναμις δέ, ὡς μακρόν, βραχύ, δασύ, ψιλόν, φωνῆεν, σύμφωνον καὶ τὰ λοιπά· τάξις δέ, ὡς τὰ μὲν προτακτικὰ τῶν φωνηέντων καὶ συμφώνων, τὰ δὲ ὑποτακτικά. Καὶ τὴν μὲν οὖν ἐκφώνησιν τῶν στοιχείων ἡ φύσις τοῖς ἀνθρώποις ἐξ ἀρχῆς ἐδωρήσατο, ταῦτα δὲ τὰ τέσσαρα, ἄ προειρήκαμεν παρεπόμενα τῷ στοιχείῳ, ἀνθρώπων ἐστὶν ἐφεύρεσις· διὸ καὶ παρ' ἡμῖν μὲν οἱ χαρακτῆρες τῶν στοιχείων τοιοῖδέ τινές εἰσιν, οὖς ἐδιδάχθημεν, παρὰ δὲ

drawn with lines [γραμμαῖς] and scratches. Or, because with a metathesis of λ to ρ, carving becomes γλάπτω, whence γλάμμα and then γράμμα. Or also, following the use of geometricians, because letters are fashioned with lines [γραμμῶν]; they say accordingly that Pythagoras cared about their beauty, harmonising their lines with angles, straight edges and circumferences.

This element, that is the phoneme [ἐκφώνησις], has four attributes. First, the **character**, that is, a shape like a triangle or a semicircle or a curve and so on. Then, the **name**, like *alpha* or *beta*, etc. Third, the **pronunciation**, such as long, short, aspirated, unaspirated, vocal, consonant, etc. And finally, the **order**, by virtue of which some vowels and consonants come first while others follow. Now, the **phonetic individuality** itself of the elements was a gift of nature to mankind from the beginning, but the four attributes are human inventions, and this is why the characters are such as we have learnt them,
Πέρσαις ἔτεροι, παρὰ Σύροις δὲ ἄλλοι, καὶ παρ' ἄλλοις ἔθνεσιν ἄλλοι.<fnref n="75" />

but they are different by the Persians, and other by the Syrians, and yet others among other peoples.

Here the στοιχεῖον is identified to ἐκφώνησις, which the context makes a kind of Nature-given 'phonetic individuality', and to it are added the humanly invented, or rather conventional, attributes, one graphic, χαρακτήρ, one acoustic, δύναμις, one numerical, τάξις, and one conceptual, ὅνομα—all relative, depending on cultural variations. According to this distinction, the στοιχεῖον is clearly and essentially a **phonemic unit**: a discrete acoustic linguistical unit. The final lines also make clear that while there is an ethnic, cultural level of the letters, there is also a deeper level in which they are **cosmopolitan**, perceived as native wherever they are found.

P

Such attributes as the above allow for the elements to be grouped among themselves, just as sounds can be vowels or consonants, and as letters can be round or square, etc. This means that the letter-elements too are **classifiable**, something strikingly apparent in the comparison between the periodical table of chemical elements and the consonant table of the International Phonetic Alphabet. To exemplify the analogy, it can be said, for instance, that a voiced alveolar fricative is the same anywhere in the world, regardless of it being called *zed*, *zayin* or *za*, or being written *z*, *r* or *j*.<fnref n="76" />

P

THE INTERNATIONAL PHONETIC ALPHABET (revised to 2015)

CONSONANTS (PULMONIC)

© 2015 IPA

<table><thead><tr><th></th><th>Bilabial</th><th>Labiodental</th><th>Dental</th><th>Alveolar</th><th>Postalveolar</th><th>Retroflex</th><th>Palatal</th><th>Velar</th><th>Uvular</th><th>Pharyngeal</th><th>Glottal</th></tr></thead><tbody><tr><td>Plosive</td><td>p b</td><td></td><td></td><td>t d</td><td></td><td>ṭ ḍ</td><td>c j</td><td>k g</td><td>q G</td><td></td><td>ʔ</td></tr><tr><td>Nasal</td><td>m</td><td>ŋ</td><td></td><td>n</td><td></td><td>ŋ</td><td>ɲ</td><td>ŋ</td><td></td><td>N</td><td></td></tr><tr><td>Trill</td><td>B</td><td></td><td></td><td>r</td><td></td><td></td><td></td><td></td><td></td><td>R</td><td></td></tr><tr><td>Tap or Flap</td><td></td><td>>V</td><td></td><td>ʃ</td><td></td><td>ṭ</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Fricative</td><td>φ β</td><td>f v</td><td>θ ð</td><td>s z</td><td>ʃ z</td><td>ṣ z</td><td>ç j</td><td>x γ</td><td>χ ψ</td><td>ħ ʕ</td><td>h fi</td></tr><tr><td>Lateral fricative</td><td></td><td></td><td></td><td>l ʃ</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Approximant</td><td></td><td>ʊ</td><td></td><td>ʃ</td><td></td><td>l</td><td>j</td><td>ψ</td><td></td><td></td><td></td></tr><tr><td>Lateral approximant</td><td></td><td></td><td></td><td>l</td><td></td><td>l</td><td>ʌ</td><td>L</td><td></td><td></td><td></td></tr></tbody></table>

Symbols to the right in a cell are voiced, to the left are voiceless. Shaded areas denote articulations judged impossible.

## The Four Cosmic Elements

The issue of the correspondence between the letters and the physical elements, or put otherwise, of the physical reality of the letter-elements, can be approached from different angles, which are all represented in the grammatical commentaries. The first

<footnote n="75">75. p. 31.</footnote>

<footnote n="76">76. An admirable treatment of this kind of 'collection and division' in the Platonic dialogues is given in M. L. Gill, 'The Divine Method in Plato's Philebus,' in *Plato's Philebus: Selected Papers from the Eighth Symposium Platonicum*, ed. J. Dillon and L. Brisson (Sankt Augustin: Academia Verlag, 2010), 42–46.</footnote>
image is that of the quaternary of the cosmic elements:

Εἶποι δ' ἄν τις αὐτὰ στοιχεῖα εἰρῆσθαι
καὶ κατὰ μίμησιν τῶν τεσσάρων στοι-
χείων, πυρός, ὕδατος, ἀέρος, γῆς· ὥσπερ
γὰρ ἐκ τούτων τῶν τεσσάρων στοιχείων
σύγκειται πάντα καὶ ἐν ὑποστάσει ἐστὶ
τὰ ἐν τῷ κόσμῳ, οὕτω καὶ ἐκ τῶν κδ' στοι-
χείων σύγκειται καὶ ἐν ὑποστάσει ἐστὶ
πᾶς λόγος καὶ πᾶν σύγγραμμα καὶ πᾶν
βιβλίον.

Someone might say that they are called
στοιχεῖα because of their **imitation** of the
four elements, fire, water, air, earth. Just
as all things in the universe are composed
of these four elements and **are sustained**
by them, every word and every piece of
writing and every book is composed by
the twenty-four letters and is sustained by
them.

I will reserve for next chapter<sup>77</sup> a more detailed reference to the topic of the μίμησις,
‘existential imitation’ or ‘representation’, which appears here for the first time and is
central to this thesis, but it is important to clarify my reading of that relation here.
Examining the structure of the sentence and the two verbs used, it is the second one
that draws my attention: the things of the world, like books and words, ἐν ὑποστάσει
ἐστὶ τῶν στοιχείων, that is, literally, ‘they are/exist in the hypostasis of the στοιχεῖα.’
I interpret this as ‘on the hypostasis,’ meaning that the element-letters are viewed as a
supporting underlying ‘material’ on which the world, be it physical or linguistic, rests.
The στοιχεῖα are here quite literally the *infrastructure*, whereas reality would be the
*superstructure*.

Why Twenty-Four? More Cosmic Correspondences

The correspondence just mentioned is found above all in astronomical observations and
knowledge, and in the *Scholia Marciana* (p. 196), we find for the first time the recurring
motif of the correspondence with the hours of day and night, which is another echo of
the early gnomonic meaning of στοιχεῖα:

Κδ' λέγουσιν αὐτὰ εἶναι κατὰ μίμησιν τῶν κδ' ὡρῶν τοῦ νυχθημέρου.

They are said to be twenty-four because they represent the twenty-four
hours of day and night.

And more specifically: τὰ φωνήεντα τὴν ἡμέραν μιμοῦνται, τὰ δὲ σύμφωνα τὴν νύκτα,
'the vowels represent the day, and the consonants the night.'<sup>78</sup>

The letters are limited in number, which is related to their being integral. Their
number can vary greatly depending on the historic stage of the alphabet, whether
we count final letters and those used only for arithmetic purposes, like the *sampi*, or

<footnote n="77">77. See below p. 69.</footnote>

<footnote n="78">78. *Schol. in Dion. Thrax*, 323.</footnote>
whether we count phonemes or only the written signs... but whatever the choices, they
are numerable, forming a **countable set**. This reminds of an interesting observation ὁ
made by Burkert:<fnref n="79" /> 'one letter alone is nothing without its relation to the other letters, to
the alphabetic system.' And he notes how what is found mostly in relevant discussions is
the plural στοιχεῖα, which would be the equivalent of using an expression for the whole
set, viz. ἀλφάβητος, *abecedarium*, or our usual ABC. Indeed, as will become clearer in
the following pages, the subject of this thesis is above all that plural concept in its
integrality, and only secondarily its inner properties, classification, and architecture.

The Moon and the Stars

Following on the astronomical correspondences, the *Scholia Londinensia* (p. 491)
present an interesting association of the twenty-four γράμματα with the moon in
particular, which reminds us of the Arabic association of the twenty-eight letters with
the lunar mansions. The passage is lengthy and at times puzzling too, but well worth
quoting.

Οὐκ ἀλόγως δὲ τούτῳ τῷ ἀριθμῷ οἱ πα-
λαιοὶ ἐχρήσαντο· ἀλλ' ἀπὸ τῶν νυκτὸς
καὶ ἡμέρας ὡρῶν μετέθεσαν αὐτὰ ἐπὶ
τὴν τῶν στοιχείων τάξιν, διὰ τὸ τὴν δύ-
ναμιν τῶν στοιχείων πάνυ τῷ σεληνι-
ακῷ ἐοικέναι δρόμῳ, φωτίζουσάν τε τὰ
πράγματα καὶ φωτιζομένην ὑπ' αὐτῶν,
καὶ οὖσαν σχῆμα τοῦ δρόμου τῆς σε-
λήνης, ὁτὲ μὲν αὐξανομένης ὁτὲ δὲ μει-
ουμένης διὰ τῆς οἰκείας δυνάμεως· καὶ
πανσέληνον μὲν μιμεῖται διὰ τῆς τῶν
φωνηέντων φύσεως, διχότομον δὲ διὰ
τῶν ἡμιφώνων, ἀμφίκυρτον δὲ διὰ τῆς
τῶν ἀφώνων τοῦ φθόγγου μειώσεως.

It is not without reason that the ancients
used this number, but they did so by trans-
posing them [γράμματα] from the hours of
the night and day, and following the order of
the letters [στοιχεῖα], because the power of
the phonemes resembles greatly the course
of the moon, shedding light on our matters
and being illuminated by them, and having
the disposition of the path of the moon, as it
waxes and wanes by virtue of its own power.
Indeed, the full moon is represented by the
nature of the vowels, the half-moon by the
semi-vowels, and the gibbous moon by the
weakness in the utterance of the mute ones.

Thus, as their lunar correspondence is revealed, their number is shown to be P
determined by it. And then this association with the moon takes an even more
unexpected and practical turn:

Εἶτα ὥσπερ αἱ Θεσσαλαί, φασί, γοητεύ-
ουσαι διὰ τῶν μαγγανειῶν βούλονται
κατάγειν τὴν σελήνην, οὕτω καὶ ἡ γραμ-

It is like what they say of the Thessalian
witches, who cast spells to bring down the
moon. In the same way, **written language**

<footnote n="79">79. Burkert, ‘ΣΤΟΙΧΕΙΟΝ,’ 169.</footnote>
ματικὴ μαγεία τὴν φωνὴν τοῦ ἀέρος χορηγοῦσαν, εἰσιοῦσάν τε εἰς τὸ νοερὸν ἡμῶν καὶ αὖθις ἐξιοῦσαν, ἐπισκοπουμένη καὶ εἰς τοὺς ἑαυτῆς τύπους κατακλεί- ουσα διὰ τῆς ἀνακυκλήσεως τοῦ κδ' ἀριθμοῦ περὶ τὴν τῶν γραμμάτων παγι- δεύεται διαφοράν.

leads the speech from the air into our
mind through magic, and then brings it out
again; watching over and holding down
her own types [τύποι], she is ensnared
by the revolution of the number twenty-
four along the variation of the letters
[γράμματα].

The letters are perceived as magical or perhaps rather alchemical in the psycho-
linguistic operation they perform, the ‘alcemy of meaning’ of which they are raw
materials: the transformation of simple sounds in semantic aggregates. It is no wonder
that some kind of magic comes up, since we touch here on the aporetic matter of the
origin of language qua mental process.<fnref n="80" /> But I shall refrain from entering now into these
deep linguistic waters, to return instead to the heavens where wider associations are at
play.

The Planets

“Οσπερ δὲ ἐν τοῖς μετεώροις οἱ ἑπτὰ πλανῆται τὸ κῦρος τῆς διοικήσεως τῶν φαινομένων ἔχοντες καὶ τῶν ζωδιακῶν οὐκ ἐκκλίνοντες, ἀλλ' ἐπὶ τούτων μένοντες καὶ στρεφόμενοι διοικοῦσι τὸ φαινόμενον, οὐτωσὶ καὶ τὰ φωνήεντα τὸ κῦρος τῆς ἐγγραμμάτου φωνῆς κεκτημένα διὰ τοῦ σχηματίζεσθαι καὶ συμπλέκεσθαι τοῖς συμφώνοις οὐχ ὑπερβαίνει τοὺς τύπους τῶν κδ' γραμμάτων, ἀλλ' ἐν τούτοις ἀεὶ καὶ διὰ τούτων τὴν αὐτὴν ἀνακυκλοῦντα πᾶσαν ἀποτελεῖ φωνὴν ἐγγράμματον, μηδενὸς εἰς τοῦτο προφερομένου ἐτέρου μήτε φθόγγων μήτε γραμμάτων, ὥσπερ οὐδὲ νὐξ οὐδὲ ἡμέρα παρὰ τὰς κδ' ὥρας εἰς ἀνακύκλησιν προσδεῖταί τινος, ἀλλὰ διὰ τούτων τὸν κύκλον καὶ τοὺς ἐλιγμοὺς

Just as the seven planets in heaven have
the sovereignty in the administration of
the observable motions, never leaving the
zodiac signs, but remaining always above
them and revolving as they go through the
visible heaven, thus do the vowels have
the sovereignty of the written speech, as
they are given shape and are combined with
the consonants without ever transgressing
the characters of the twenty-four letters,
but rather by always being among them
and through them, they bring into being
the ever recurring wholeness of the written
speech, nor do the voice or the letters move
ahead of one another in order to accomplish
this. Just as neither night nor day need
anything aside from the twenty-four hours
for their revolution, but rather complete

<footnote n="80">80. Whorf's observation is most apt here, and refreshing too, coming as it does from someone who had reflected so deeply about language: 'thinking is most mysterious, and by far the greatest light upon it that we have is thrown by the study of language,' B. L. Whorf, 'Language, Mind, and Reality,' in *Language, Thought, and Reality* (MIT Press, 1956), 252.</footnote>
ποιεῖται τῆς οἰκείας περιφορᾶς. Καὶ τὰ μὲν γράμματα διὰ τοῦτο καὶ οὕτω πως περάσματος καὶ ἀριθμοῦ ὡρισμένου τετύχηκεν.

through them the cycle and rotations of their own course, the letters also, through this and in this way, obtain finiteness and a defined number.

If the letters as a whole had been compared to the moon in their general function, now they are, in their specifics, compared to the planets and the zodiac as they interact with each other. This remarkable simile, where the travelling vowels go through the twenty-four divisions of the sphere, immediately reminds one of the Arabic term for vocalisation marks, ḥarakāt, meaning ‘movements’. It also brings up the question of the status of vowels in their relation to consonants: in the Greek language, for the first time, the vowels acquired the status of letters which they never had in the Semitic abjads. And yet, in this very Hellenistic exegesis, the vowels are clearly considered of a different class altogether. This question is that of two different conceptions of what an alphabet is: either a homogeneous or a hybrid series.

## Astrological Correspondences

To complete the previous passage on the planetary relations, here are the correspondences of the seven vowels:

Καὶ τί δήποτε ὁ τεχνικὸς τῶν φωνηέντων τὸν ἀριθμὸν μέχρι τοῦ ἑπτὰ ὁρίζεται; Καί φαμεν ὅτι ἴσα αὐτὰ τῷ ἀριθμῷ τῶν χορδῶν τῆς λύρας τοῦ Ἀπόλλωνος ἐποίησεν, ἥ κατὰ μίμησιν τῶν ἑπτὰ ἀστέρων τῶν πλανήτων τοῦ οὐρανοῦ· πλάνητες γάρ εἰσιν ἑπτά, Κρόνος, Ζεύς, Ἀρης, Ἠλιος, Ἀφροδίτη, Ἐρμῆς καὶ Σελήνη. Ταῦτα γὰρ τὰ φωνήεντα τοῖς πλάνησιν ἀνάκεινται· καὶ τὸ μὲν «α» φασὶ τῇ Σελήνῃ ἀνακεῖσθαι, τὸ δὲ «ε» τῷ Ἐρμῷ, τὸ δὲ «η» τῇ Ἀφροδίτῃ, τὸ δὲ «ι» τῷ Ἠλίῳ, τὸ δὲ «ο» τῷ Ἄρει, τὸ δὲ «υ» τῷ Διί, τὸ δὲ «ω» τῷ Κρόνῳ.

And why would the Craftsman determine the number of the vowels as seven? We say it is because he made them equal in number to the seven strings of Apollo's lyre, or as a representation of the seven wandering stars of the sky. For the planets are seven, Saturn, Jupiter, Mars, Sun, Venus, Mercury and the Moon. Now, the vowels are dedicated to the planets: α to the Moon, ε to Mercury, η to Venus, ι to the Sun, ο to Mars, υ to Jupiter, ω to Saturn.<fnref n="81" />

As found elsewhere, here we have another demiurgic name, ὁ τεχνικός, the Craftsman, who assigns cosmical correspondences, and once again speaks of a heavenly or principal nature of the letters or some kind of ‘arch letters’.<fnref n="82" /> The image of Apollo,

<footnote n="81">81. p. 198.</footnote>

<footnote n="82">82. See below, p. 49.</footnote>
and of the lyre devised by Hermes and given to his brother as an endearing gift,<fnref n="83" />
is anything but casual and emphasizes and clarifies the connection between music,
language and heavenly order. Harking back to the previous paragraphs, it is important
to keep in mind that we are still here, simultaneously, on both levels, the psychological
and the astronomical, in 'a realm of patterned relations inconceivably manifold and
yet bearing a recognizable affinity to the rich and systematic organization of language,
including *au fond* mathematics and music.'<fnref n="84" />

Microcosmic Correspondences

A natural consequence of the macrocosmic correspondences of the letters is that there
will also have to be a direct correspondence with the parts of the human compound:

τὰ μὲν φωνήεντα τῇ ψυχῇ ἐοίκασι, τὰ δὲ
σύμφωνα τῷ σώματι· καὶ ὥσπερ ἡ ψυχή, εἰ καὶ
χωρὶς τοῦ σώματος δύναται εἶναι, ἀλλὰ δεῖται
τοῦ σώματος εἰς τὸ ἀποτελέσαι τὴν σύστασιν
τοῦ ζώου, τὸν αὐτὸν τρόπον καὶ τὰ φωνήεντα,
εἰ καὶ καθ' ἑαυτὰ δύνανται παραλαμβάνεσθαι
καὶ ἀφ' ἑαυτῶν ἐκφωνεῖσθαι, ἀλλὰ δέονται τῆς
τῶν συμφώνων συντάξεως εἰς τὸ ἀποτελέσαι
τὴν ἐγγράμματον φωνήν.<fnref n="85" />

The vowels resemble the soul, and
the consonants the body. Just as the
soul, even though it can exist outside
the body, needs the body to produce
the compound of life, just so the
vowels, even though they can be used
and uttered on their own, need the
addition of the consonants in order to
produce the written speech.

And more fundamentally in the *Collectio Marciana* (p. 317):

Ποσαχῶς τὸ στοιχεῖον; Τριχῶς·
στοιχεῖον ἡ ἐκφώνησις, στοιχεῖον ὁ
χαρακτήρ, στοιχεῖον καὶ τὸ ὄνομα.
Καὶ διὰ τί λέγεται τριχῶς; Διότι καὶ
τῷ ἀνθρώπῳ τρία παρέπεται, τὰ
τῆς ψυχῆς, τὰ τοῦ σώματος καὶ τὰ
τῶν ἐκτός·

In how many ways do we use ‘element’? —
In three: the pronunciation is an ‘element’, the
character is an ‘element’, and the name [of each
letter?] is also an ‘element’. And why do we
have these three uses? — Because man also has
threefold attributes: those of the soul, those of
the body and those of the external.

This is the basis of a kind of letter *melothesia* which evolved over the centuries into countless applications in what was effectively a common ground of astrology, medicine and magic.<fnref n="86" />

<footnote n="83">83. *Homeric Hymn to Hermes*, 46ff.</footnote>

<footnote n="84">84. Whorf, 'Language, Mind, and Reality,' 247–8.</footnote>

<footnote n="85">85. p.198.</footnote>

<footnote n="86">86. See Dornseiff, *Alphabet*, 81n.</footnote>
Graphical Layout

The layout of the lines and the direction of writing are yet another aspect of the letters discussed in the scholia:

Τῶν ἀρχαίων οἱ μὲν βουστροφηδὸν ἔγραφον, οἱ δὲ κιονηδόν, οἱ δὲ πλινθηδόν, οἱ δὲ σπειρηδόν·

<figure><img src="image_4.png" /></figure>

Some of the primordial people wrote as the oxen go, some vertically as in columns, others as if forming a rectangle and yet others in zigzag lines.

There had been mention before of the shapes and the proportions of the letters, but this catalogue of possibilities makes me wonder about an architecture among the elements, a geometry established by their inter-relations. And we may surmise that the elements, having to do by definition with ranks and order, are bound to certain kinds of order and disposition. Could we say they are geometric, or perhaps even somehow stereometric, having three-dimensional volume? It will be seen that this is, with a slightly different emphasis, an aspect of the same organicity already listed above (p. 26) as one of the properties of the elements. In fact, these layout variations should be considered in the light of the prevailing layout of the classical inscriptions, the stoichedon grid, which would influence the development of Hebrew square script and that, as shown by its name, has a direct relation to the kind of order implicit in the word στοιχεῖον, the battle ranks mentioned above.

<figure><img src="image_5.png" /></figure>

INSCRIPTION IN STOICHEDON STYLE, ATHENS, LATE 5TH CENTURY BC<fnref n="87" />

<footnote n="87">87. Detail of the ‘Decree of Kallias’, now at the Louvre.</footnote>
Sanchuniathon and the Snakes

One last Greek grammatical passage worthy of close attention comes from Eusebius and has to do with the mythical Phoenician figure of Sanchuniathon and the celestial origin of the letters:<fnref n="88" />

... θεὸς Τάαυτος μιμησάμενος τὸν οὖρανον τῶν θεῶν ὅψεις, Κρόνου τε καὶ Δαγῶνος καὶ τῶν λοιπῶν, διετύπωσε τοὺς ἱεροὺς στοιχείων χαρακτῆρας.

... having assimilated the sky, the images of the gods, of Kronos and Dagon and the others, the divine Taautos [identified with the Egyptian Thoth] impressed through [διετύπωσε—or shaped, cast, stamped, modelled] the sacred characters of the element-letters.

We had come across the root of τύπος, an impression or print, as one among the means of reproducing letters, but this is the first time that we find it clearly elevated to a 'divine' function, which is certainly not unrelated to the concept of the *arch-type* or *archetype*, the principal mould, which we shall discuss at a later stage. Thoth 'prints' or rather, less anachronistically, 'stamps' the letters after the celestial divine pattern. Τύπος denotes most stages of the printing procedure: from the mould or model to the pattern and the final impression itself.<fnref n="89" />

Further down there is another mention of Taautos and more details about his creation of the letters:

Ο δ' αὐτὸς πάλιν περὶ τῶν Φοινίκων στοιχείων ἐκ τῶν Σαγχουνιάθωνος μεταβαλών, θέα όποιά φησι περὶ τῶν έρπυστικῶν καὶ ἰοβόλων θηρίων, ἄ δὴ χρῆσιν μὲν ἀγαθὴν ἀνθρώποις οὐδεμίαν συντελεῖ, φθορὰν δὲ καὶ λύμην, οἷς ἂν τὸν δυσαλθῆ καὶ χαλεπὸν ἰὸν ἐγχρίμψειεν, ἀπεργάζεται.

He in turn, succeeding Sanchuniathon with regard to the Phoenician letters, brings them to completion divine, as he says they are, fashioned after venomous reptiles, such as will not be of any use or benefit to mankind, but rather for destruction and corruption, for those whom he approaches with this deadly and cruel poison.<fnref n="90" />

There is here an obvious reference to the harsh critique of writing recounted in Plato's *Phaedrus*, when Thoth brings to Egypt the dubious gift of the written word—writing is deadly as poison. But there is also an important association with the snakes,

<footnote n="88">88. Eusebius of Caesarea, *Praeparationis evangelicae libri XV*, ed. Friedrich Adolph Heinichen (Sergiana Libraria, 1842), 45.</footnote>

<footnote n="89">89. See *A Greek-English Lexikon*, ed. H. Liddell and R. Scott (Oxford: Oxford University Press, 1968), s.v. τύπος (hereafter cited as LSJ); cf. also below p. 120.</footnote>

<footnote n="90">90. Eusebius, *Praeparatio*, 48.</footnote>
in the immediate visual symbolism of their moving and gliding lines, like the lines of
writing do with certain materials, and also in the traditional mythical symbolism of
their telluric force, as a furtive dynamic power from the earth.

With these passages I shall add two new properties to the elements: they are of **divine** origin, which was already confirmed by their being 'Hermetic' and their descent from P heaven (see p. 37); and they can also be lethal, harmful, **deleterious**, or in any case P sharing in a 'slippery' or crafty nature, remarkably close again to the attributes of Hermes.

Latin Grammarians

In order to avoid needlessly burdening this first chapter, I have consigned the bulk of
my Latin citations for this chapter to Appendix A, where the interested reader can find
them with a translation in parallel columns.

During the course of this chapter I have already made a few references to the
Latin grammatical tradition and to the concept of *elementum* when appropriate. In the
following lines I will only add a few more particulars that stand out from the specialised
literature<fnref n="91" /> and may complement the information on some of the topics brought up in
the previous pages.

<figure><img src="image_6.png" /></figure>

The Romans had a rich and ancient tradition of grammatical studies,<fnref n="92" /> and it is
remarkable how, in spite of the great influence that Greek scholarship had on the
development of its Roman equivalents, and even though the discipline itself preserved
for the most part its Greek name,<fnref n="93" /> the Latin grammatical tradition never ceased to
exhibit traits of its own. It is remarkable how the Latin *artes* tend to follow a different
order of exposition, as if the foundations were conceived in a different way, but there
was in Latin grammars a customary initial section devoted to phonetics, the 'phonetic
complex', consisting of two consecutive chapters dedicated respectively to *vox* (as

<footnote n="91">91. Quotations are mostly from Keil's collection of Latin grammarians, H. Keil, ed., *Grammatici latini*, reprint of the 1855-80 Leipzig edn, 8 vols (Hildesheim: Georg Olms, 1961) (hereafter cited as *GLK*); and other Latin grammatical sources. A most valuable online repository is *Corpus Córporum: repositorium operum Latinorum apud universitatem Turicensem*, ed. P. Roelli, Developed by the Institute for Greek and Latin Philology, University of Zurich (Zurich, November 13, 2015).</footnote>

<footnote n="92">92. See M. Seppänen, 'Defining the Art of Grammar: Ancient perceptions of γραμματική and grammatica' (PhD diss., Dept of Classics, University of Turku, 2014), 104ff.</footnote>

<footnote n="93">93. It could also be called *litteratura*; cf. A. Reinikka, *Ars Pseudo-Scauri: A Critical Edition and Commentary*, academic dissertation (Helsinki: Unigrafiä, 2012), 45.</footnote>
translation of φωνή) and to *litterae*<fnref n="94" /> (as translation of both στοιχεῖα and γράμματα—note the plural),<fnref n="95" /> and these sections are comparable to the section 'On the στοιχεῖον' found in the Τέχνη of Dionysius.

The pair στοιχεῖον-γράμμα did not find an exact parallel in the Latin pair of *elementum*-littera, but littera dominated as the broader term which designated both a graphical sign and the minimal unit of articulate speech.<fnref n="96" /> As in the Greek tradition, this ever unclear definition of the στοιχεῖον/littera has a bearing on modern linguistic discussion about the concept of phoneme, and as a liminal entity existing between orality and writing, it brings into modern discourse the awareness of an inclusive perspective that works at ease with the ambiguity.<fnref n="97" />

As in the Greek grammarians, the metaphysical or logical meaning of, in this case, *elementum*, is mingled with the properly grammatical. It is common to find the grammatical teachings in the form of a dialogue:

> *Grammaticae artis initia a voce oriantur, quae elementis constat. Ele-
> mentum quid est? unius cuiusque rei initium, a quo sumitur incrementum*
> *et in quod resolvitur.*

> The beginnings of the art of grammar originate from the sound, which is
> formed of elements. **And what is an element?** The beginning of any single
> thing, from which it grows and into which it is resolved.<fnref n="98" />

The common doctrine of the *litterae* included the following assumptions: a ‘letter’ is a minimal phono-graphic unit (*elementum*) of ‘writable utterance’ (*vox litterata*); the concept of ‘letter’, as with the Greek alphabet, entails distinctions by ‘properties’. In this grammatical model, every level of discourse is ‘resolved’ or ‘broken up’ (*solvitur*) into the parts of the level immediately below it, and all discourse is ultimately resolvable into ‘letters’, the ‘elements’ or ‘atoms’ of writing.<fnref n="99" /> Or in the words of Sergius (c. 450):<fnref n="100" />

> *Omnis oratio soluatur in uerba, uerba denuo soluantur in syllabas, rursum
> syllabae soluantur in litteras, littera sola non habet quo soluatur. ideo a
> philosophis atomos dicitur.*

<footnote n="94">94. The single or double *t* of *litera/littera* had to do with the history of the word and its etymology, and ended up being a matter of personal or local preference; see L. Buzássyová, ‘The “Phonetic Complex” in Renaissance Latin Grammar Petrus Ramus’s Dichotomies and Their Reflections in Two Vernacular Grammatical Texts,’ *Graeco-Latina Brunensia* (Bratislava) 21, no. 2 (2016): 82n.</footnote>

<footnote n="95">95. *Ibid*., 82</footnote>

<footnote n="96">96. *Ibid*.</footnote>

<footnote n="97">97. See, Vogt-Spira, ‘*Vox und Littera*,’ 296.</footnote>

<footnote n="98">98. Audacis de Scauri et Palladii libris excerpta (Keil VII).</footnote>

<footnote n="99">99. M. Irvine, *The Making of Textual Culture: ‘Grammatica’ and Literary Theory 350-1100* (Cambridge: Cambridge University Press, 2006), 100.</footnote>

<footnote n="100">100. *GLK*, 4, 475.</footnote>
Every sentence is resolved in words; words resolve in syllables; syllables resolve in letters, but the letter alone has nothing to resolve into. This is why it is called *atomos* by the philosophers.

And also on this physical note Priscian<fnref n="101" /> gives the Latin equivalent of the cosmological association with the elements.

literas autem etiam elementorum vocábulo nuncupaverunt ad similitudinem mundi elementorum [...] litera igitur est nota elementi [...] abusive tamen et elementa pro literis et literae pro elementis vocantur.

They called the letters with the name of the elements because of their resemblance to the elements of the cosmos [...] The letter is the sign [nota] of the element [...] It is a misuse when they call the elements letters and the letters elements.

Even though *elementum* had been used with a physical meaning from the time of Lucretius, this awareness and its consequences do not appear in the Latin grammatical literature in the same way that it did among the Greek scholars. Even though several of the properties I have noted for the Greek στοιχεῖα are also found among Latin grammarians, like the divine origin of writing, there is one of them which is remarkably absent, and it is the relation to number and the great emphasis given to their harmonious order. This arithmetical dimension of the Latin script, or rather the lack of it, will be one of the topics treated in the next chapter, dedicated to number.

## Conclusion

By way of conclusion I would simply like to bring together all the different properties or attributes of the στοιχεῖα which have been labelled through the previous pages, in order to have a more compact vision of where we stand before continuing with the research.

According to the notes above, and although several of these are rather tentative, I would now say that the στοιχεῖα/elementa are, in turns or simultaneously, and in no particular order:

1. constitutive

2. primordial

3. cosmogonic

4. causal

5. immanent

6. simple

7. ambiguous

8. corporeal

9. generative

10. axiomatic

11. organic

12. integral

<footnote n="101">101. *Institutionum Gramaticarum* (Keil vol. II).</footnote>
13. minuscule

14. pervasive

15. operative

16. indefinable

17. genarchic

18. sequential

19. dancey

20. coeval

21. intermediary

22. unchangeable

23. unvocalised

24. enlightening

25. solar

26. Hermetic

27. ordered

28. safeguarding

29. cosmopolitan

30. classifiable

31. numerable

32. lunar

33. divine

34. deleterious

They generate and destroy, they dance and do not change, they define and are indefinable, they protect and help, they enlighten and are minuscule, the are the real atoms, requiring one another, everywhere, subtle and yet corporeal, Hermetic... With such a list of attributes at hand, it should not be too difficult to identify semantic analogues of these cosmic 'letters', and I feel equipped to embark on the next phase of this research, but first, in the next chapter, I plan to complete the definition of the object of research, and at the same time to establish a time frame based on the definition itself, by completing the picture of the στοιχεία with their numerical association.
## Chapter 2

# On the Tracks of Arithmetical Cosmology

In the previous chapter we found a definition of ‘number’, ἀριθμός, among the grammarians’ attempts at defining the στοιχεῖον,<fnref n="1" /> and the association they established is with order as στοῖχος, a ‘row’ or a ‘queue’, and as τάξις, an ‘array’. They go as far as saying that the name of the στοιχεῖα is due to the fact that numbers are written with them, as if στοιχεῖα meant something like ‘arithmeticals’, or, indeed, ‘numerals’.

In order to elucidate the relation between the elemental letters and the numbers, my first step will be to address the relation between numbers and letters in general, and then to clarify the meaning of ἀριθμός in the Greek tradition.

### Number and Letter

‘Numbers seek form in words and in signs.’<fnref n="2" /> In this statement by Menninger, numbers are set apart as entities whose meaning is not exhausted by either their names or their written expression. In the Greek language, for instance, τριάκοντα, ‘thirty’, means etymologically ‘three times ten,’<fnref n="3" /> but for the greater part of the history of the language, down to the late Middle Ages,<fnref n="4" /> it was written as λ, which does not reflect in any way

<footnote n="1">1. See above p. 39.</footnote>

<footnote n="2">2. K. Menninger, *Zahlwort und Ziffer: eine Kulturgeschichte der Zahl*, 2 vols. (Göttingen: Vandenhoeck & Ruprecht, 1957), 67; page numbers refer to the German edition; English transl. by Paul Broneer, *Number Words and Number Symbols: A Cultural History of Numbers*, MIT, 1969.</footnote>

<footnote n="3">3. As incidentally does English *thirty*, from OE ðrítig. See A. L. Sihler, *New Comparative Grammar of Greek and Latin* (Oxford University Press, 2008), 418ff. and C. D. Buck, *A Dictionary of Selected Synonyms in the Principal Indo-European Languages* (University of Chicago Press, 2008), 936 (hereafter cited as DSS).</footnote>

<footnote n="4">4. The details of this history, including the earlier acrophonic numeral system and other less common developments are found in S. Chrisomalis, *Numerical Notation: A Comparative History* (Cambridge University Press, 2010). See also Menninger.</footnote>
the composition of the number as expressed in τριάκοντα or, for example, as in the
very clear Roman XXX, or as in our ‘30’. While the basic number-words are patterned
on the structure of the human body,<fnref n="5" /> and while they tend to parallel the arithmetical
efficiency of the abacus array, numerals can exhibit a bewildering variety of expression.<fnref n="6" />
This lack of correspondence between number-words and numerals *tout court*, or ‘lexical
numeral words’ and ‘graphic numerical notation’<fnref n="7" /> (*Zahlwort* or *Zahlsprache* and *Ziffer*
or *Zahlschrift*, in the useful German terms) can be easily ascertained again and again,
until we recognise that numerals are far from being mere representations of number
words, but that they exist as a parallel, and often only partially interdependent, system
for notating numbers.

The sequence of number words in a language often predates and outlasts any given
system of numerals as they may become established and replaced. We may instinctively
think that numerals are simply ‘copies’ of number words, that there is a ‘linguistic fit’
of numerals,<fnref n="8" /> but the rules that govern their syntax and usage do not in fact correspond
to the rules of the number-word sequence, which usually follow calculation logic and
stepwise gradation.<fnref n="9" />

Now, in the context of this work, observing this relation comparatively in Greek and
Latin languages, we can appreciate a striking paradox of cultural history in that the
Romans, who possessed such a subtly and precisely ordered lexical number sequence,
used a system of numerals so cumbersome that it is hard to see how it could be a product
of the same culture.<fnref n="10" />

This paradoxical disjunction throws some light on the essential alterity of numbers,
but from the point of view of this research it brings into focus, above all, the contrast
between Greek and Roman numerals, and the fact that letters in Latin were never really
equivalent to numerals; the Latin alphabet was never fully alphanumeric as the Greek
was for so many centuries, and this is one reason why it figures only marginally in
these pages.

Alphabetical numerical notation appears among the Greeks in the sixth century BC

<footnote n="5">5. V. Blažek, *Numerals: Comparative Etymological Analyses of Numeral Systems and their Applications* (Brno: Masaryk University, 1999), 325.</footnote>

<footnote n="6">6. See S. Chrisomalis, ‘Re-evaluating Merit: Multiple Overlapping Factors Explain the Evolution of Numerical Notations,’ *Writing Systems Research* 9, no. 1 (2017): 5.</footnote>

<footnote n="7">7. Chrisomalis, ‘Re-evaluating Merit,’ 5, attributes it ultimately to the ‘difference between visual-graphic and oral-aural modalities.’</footnote>

<footnote n="8">8. *Ibid.*, 4–6.</footnote>

<footnote n="9">9. Some linguists argue that, in fact, what numeral systems do is to reflect, even more transparently than other parts of language, the very same principles of ordering that underpin human linguistic and mathematical capacities; see J. Gvozdanović, ‘Remarks on Numeral Systems,’ in *Indo-European Numerals* (Berlin and New York: Mouton de Gruyter, 1992), 8.</footnote>

<footnote n="10">10. Menninger, *Zahlwort und Ziffer*, 64–65.</footnote>
with what is called the Milesian numeral system. This system, based on the Phoenician
alphabet's substrate,<fnref n="11" /> was adapted from the Egyptian demotic numeral system, also
based on three enneads or groups of nine letters for each decimal order,<fnref n="12" /> as in
the illustration below; it was adopted into Hebrew, Syriac and other languages, and
expanded greatly between the fourth and seventh centuries AD, including its adoption
into Arabic, until around the twelfth century, when the letters started being replaced
by the Indo-Arabic positional numerals.<fnref n="13" /> Alphabetic systems are still used today only
in limited contexts such as liturgical texts, numbered lists, and divinatory magic, and in
Greek itself, the letters are still used like numbers in the way we use Roman numerals
in English.

<figure><img src="image_7.png" /><figcaption>CLASSICAL GREEK ALPHANUMERIC CORRESPONDENCES<fnref n="14" /></figcaption></figure>

The Greek Milesian system is precisely the initial object of study of this thesis and what could be called a **fully alphanumeric** writing system, that is, a system were the same set of alphabetic signs is used both for everyday language and for mathematics. This and the other replicate alphanumeric systems were remarkably and unexplainably long-lived. They were regularly used for more than a thousand years across linguistic families (Indo-European, Semitic, Kartvelian, like Georgian) and religions.

In cases such as the Greek system, this might be partially explained by the political
importance of the system's users, but this is hardly the case of smaller communities,
like Armenian and Ethiopian; that they continued to use alphabetic numerals requires
explanation. Chrisomalis suggests that alphabetic numerical notation systems, as the
scripts they are, are conveyors of cultural identity.<fnref n="15" /> There is no question that such

<footnote n="11">11. The Ionic alphabet, the one Panhellenic script from the fourth century BC 'until the Byzantine reform that established the minuscule script almost 1,300 years later, in the ninth century AD'; Psychoyos, 'The Forgotten Art of Isopsephy,' 159.</footnote>

<footnote n="12">12. Chrisomalis, *Numerical Notation*, 58ff., 134ff. Psychoyos, 'The Forgotten Art of Isopsephy,' 181, 184.</footnote>

<footnote n="13">13. For the centuries-long process of adoption, see Chrisomalis, *Numerical Notation*, 123, 221, and Psychoyos, 'The Forgotten Art of Isopsephy,' 199ff. About the end of their generalised usage, see refs below, p. 251.</footnote>

<footnote n="14">14. The units in the first line are called the πύθμενες or bases of the counting system. Cf. Egyptian demotic and Greek tables in Chrisomalis, *Numerical Notation*, 55, 139.</footnote>

<footnote n="15">15. *Ibid*., 185-7.</footnote>
persistence requires explanation and that it cannot be simply attributed to political
causes. Perhaps, though, it is easier to find an answer by comparing it to the following
statement by Menninger: 'In essence, letters are the embodiment of words, not of
numbers.'<fnref n="16" /> In this statement is embedded a conceptual limitation that is clearly
revealed when contrasted with a truly alphanumeric understanding of the system.

Let us recall our previous chapter on the many dimensions of the στοιχεία from
a 'grammatical' point of view, which did include basic qualities like the order of the
alphabet, shared cosmological correspondences, references to the shapes of letters, and
the following kind of exegesis: προτέτακται δὲ τὸ «α» ὅτι σημεῖόν ἐστι τῆς μονάδος,
ἀρχὴ δὲ πρώτη ἀριθμοῦ ἡ μονάς— 'the alpha goes first because it is a sign of the unit,
and the unit is the first principle of number.'<fnref n="17" />

In the light of such identifications, we could at least suspect that what Menninger’s
assertion does is point to an essential limitation of our contemporary concept of letter
(or shall I say ‘phoneme’? or ‘element’?). We are so used to having separate notations for
sounds and numbers that it does take an effort at estrangement or defamiliarization (the
French *dépaysement*), to allow for an enlargement of our concept. In any case, here we
are brought to a subtle distinction: as we paraphrase and affirm that ‘in essence, Greek
letters are the embodiment of sounds,’ we have to wonder, ‘how like the other sounds,
the other words of a given language, are numbers? How special are numbers within
language? Zahlwort and Zahlsprache—do they differ much from the ordinary Wort and
Sprache?’ We must remember that letters too have in Greek, as in Semitic languages,
special names which are not just onomatopoeic: *alpha*, *beta*, *gamma*..., and perhaps
they are comparable in their status to the names of numbers. Psychoyos is quite right
to point out that to speak of an Ionic writing system and a Milesian numeral system
as separate constructs misses the mark. These two systems were one united system for
its users, who lived naturally between the phonetic and the arithmetical values of its
signs. And indeed, grammar and linguistics may not be ‘the best guides for one to study
the emergence and evolution of the alphabet and other scripts. This should take place
in the context of a wider semiotics of writing,’<fnref n="18" /> which is precisely one of the possible
ways to describe my approach to this topic in the following pages.

I shall return later to these questions that pertain to the heart of my enquiry, but
before moving on, I need to turn the attention to a historically important phenomenon,
isopsephism or systematic alphanumeric equivalence, which is and will be necessarily
in the background throughout this chapter, if not throughout the entire research.

<footnote n="16">16. Menninger, *Zahlwort und Ziffer*, 67.</footnote>

<footnote n="17">17. Dionysius Thrax, *Dion. Thrax*, 381, 485.</footnote>

<footnote n="18">18. Psychoyos, 'The Forgotten Art of Isopsephy,' 208–209.</footnote>
Isopsephism, Onomatomancy and Gematria

A scarcely attested term, ἰσόψηφος, synonym with the equally infrequent ἰσάριθμος,<fnref n="19" /> means literally ‘pebble-equal’, thus going back to the etymology of ‘calculation’, and it is applied to words in which the values of letters added together make up the same sum, like ΝΟΜΟΣ (law) = ΑΡΙΘΜΟΣ (number) = 430, ΔΙΟΣ (of Zeus) = ΘΕΟΣ (deity) = 284; naturally anagrams will share the same value, like ΚΡΑΤΟΣ (strength) = ΑΡΚΤΟΣ (a bear) = 691.<fnref n="20" /> A famous example reported by Suetonius (Nero 39, 2)<fnref n="21" /> is an indictment of the matricidal emperor,

Νέρων ἰδίαν μητέρα ἀπέκτεινε.

NERO = he slew his own mother.

where both sides of the equality total 1005.<fnref n="22" />

What takes place fundamentally in isopsephic practice is a *ratification*. The post-classical Latin verb, *ratificare*, ‘to confirm, approve’, has its precedent in the classical *ratum facere*, which means to make something ‘fixed by calculation’,<fnref n="23" /> *i.e.* confirmed as if by calculation, made unalterable, ‘settled’ in the very same sense that accounts are settled. The numerical calculation, as it were, endows meaning with the mathematical accuracy of the movement of the stars, which are *rati*.<fnref n="24" />

Such ἰσόψηφα, found in coins, sculptures, paintings and poems<fnref n="25" /> are at the basis of ὀνοματομαντεία, a divination through names, or rather a divination of names,<fnref n="26" /> whereby concepts are explained through numerical identity. The point, though, is not ‘to find a coincidental equivalence between words. It means that there is a direct relationship between these words’,<fnref n="27" /> recalling ‘the meaning of “symbol” in the ancient world—objects that participated in and made present the person or object they symbolized.’<fnref n="28" /> Hippolytus (*Haer.* 4.13) gives a detailed explanation of the procedure

<footnote n="19">19. See LSJ.</footnote>

<footnote n="20">20. See examples in Dornseiff, *Alphabet*, 96ff.; W. Kroll, ‘Onomatomanteia,’ in *RE*, vol. 18,1 (1939), 517–520; C. Luz, *Technopaignia, Formspiele in der griechischen Dichtung*, Mnemosyne Supplements. Monographs on Greek and Roman Language and Literature 324 (Leiden: Brill, 2010), 247–326.</footnote>

<footnote n="21">21. Quoted by F. Bücheler, ‘Νεόψηφον,’ *Rheinisches Museum für Philologie* 61 (1906): 308.</footnote>

<footnote n="22">22. See extended examples, like isopsephic epigrams, in *Psychoyos*, ‘*The Forgotten Art of Isopsephy*,’ 178–9.</footnote>

<footnote n="23">23. See *A Latin Dictionary*, ed. C. T. Lewis and C. Short (Oxford: Oxford University Press, 1966), s.v. *ratus* (hereafter cited as LS). Also *Glossarium mediæ et infimæ latinitatis*, ed. C. D. F. Du Cange (Niort: L. Fabre, 1883), s.v. *ratificatio* (hereafter cited as Du Cange).</footnote>

<footnote n="24">24. Cf. Cicero (*N.D.* 2, 37, 95): *astrorum rati immutabilesque cursus*.</footnote>

<footnote n="25">25. For some epigraphic examples and others in the *Anthologia Palatina*, see LSJ, *l.c*.</footnote>

<footnote n="26">26. See Kroll, *‘Onomatomanteia.’*</footnote>

<footnote n="27">27. G. Locks, *The Spice of Torah: Gematria / גומטריא תורה ו Fond מטריא* (New York: Judaica Press, 1985), xv.</footnote>

<footnote n="28">28. See J. Kalvesmaki, ‘Reading the Early Christian Theology of Arithmetic: Methods of Research and the Search for a Method,’ lecture presented 19/11/2002 at the Catholic University of America.</footnote>
used, and he speaks of the practice as a 'Pythagorean reckoning' (Πυθαγορείος ψήφος) heeded by 'those who invent a serious philosophy' (δι' ἀριθμῶν καὶ στοιχείων, 'through numbers and letters', who believe they can 'prophesy by means of reckonings and numbers, letters and names' (διὰ ψήφων τε καὶ ἀριθμῶν, στοιχείων τε καὶ ὀνομάτων μαντεύεσθαι—4.14), and also that they can 'discern life' (τὸ ζῆν διακρίνειν—4.15).<fnref n="29" /> It is remarkable that from the earliest datable mention of this practice in Greek literature (Iamblichus, VP 18, 147) there is an association with Pythagoras.<fnref n="30" />

Now, in order to see why and how ἰσοψηφία is relevant to our research, we need to steer clear of two distractions: 1) magical, oneiromantic and related occult literature, which has been for centuries the prolific and enthusiastic field of application of this technique; and 2) a facile superficial explanation of what is at work in the alphanumeric identity—or rather the ‘alphanumeric equality’. Bücheler remarks how such were found everywhere and he calls the ἰσόψηφα *Spielereien*, ‘amusements, play’, and there is certainly a ludic dimension to them, which would partly explain why gematria has been called the ‘spice of the Torah’,<fnref n="31" /> but like all traditional games, they were rooted in a cosmological worldview, in the symbolic correspondence mentioned above, and were thus considered a means of acquiring knowledge, usually knowledge transmitted upon initiation.<fnref n="32" />

This practice continued well into the Christianised Hellenic world, where entire isopsephic hymns<fnref n="33" /> were not uncommon.<fnref n="34" /> The most famous example of isopsephy in general is undoubtedly the ‘Number of the Beast’ mentioned in Rev. 13:18, which was ‘calculated’ in various ways over the centuries.<fnref n="35" />

In any case, from our contemporary perspective, this technique of calculating the numerical values of words has inevitable esoteric and mystical connotations, partly due to the advent of a different system and the rupture of the alphanumeric connection, but mostly due to the prominence it acquired among Jewish and Muslim scholars, as a *gimטריה* gimatriyah, derived from a Greek word, and in the Islamic world as *hisāb al-jummal*, the ‘reckoning of the total’, and particularly among schools and individuals related to Kabbalists and Sufis, who would influence countless developments in the

<footnote n="29">29. Regarding the use of these techniques by astrologers (*mathematici*), and other relevant references, see A. García Molinos, 'Tipología de la adivinación en los papiros griegos mágicos' (PhD diss., Facultad de Filosofía y Letras, Universidad de Valladolid, 2014), 206.</footnote>

<footnote n="30">30. Dornseiff, *Alphabet*, 93.</footnote>

<footnote n="31">31. Locks, *The Spice of Torah*.</footnote>

<footnote n="32">32. See Luz, *Technopaignia*, 325.</footnote>

<footnote n="33">33. *Ibid*., 292.</footnote>

<footnote n="34">34. Kalvesmaki has written extensively on this Christian aspect.</footnote>

<footnote n="35">35. Cf. Menninger, *Zahlwort und Ziffer*, II, 72 and, for a later occurrence, Victorinus Petavionensis, *Commentarii in Apocalypsin* (*Corpus Cúrporum*, University of Zurich), 119, 3.</footnote>
fields of magic, alchemy and other esoteric disciplines.<fnref n="36" /> It must not be overlooked, however, that even today, within the Hebrew and Arabic-speaking communities, the practice of numerology (as generally labelled nowadays) retains an immediacy and a matter-of-fact presence that is easy to ignore from a Latin alphabetic worldview. From child-naming to toponyms, medicine and politics, the use of gematria is very alive on the streets of large parts of Africa and Asia.<fnref n="37" />

It would be all too easy to dismiss this survival as mere superstitious impulse, but returning to the open question about the longevity of alphanumeric notation, it may be possible to see a clearer cosmological reason for it. Indeed, if numbers and letters are interchangeable, arithmetic can be seen as an aspect of grammar, and any physical application of arithmetic ‘makes sense’ as a discourse does—numbers ‘say’ something. ‘Medicine, astrology, and numerology were grounded on common assumptions about the relationship between orders of reality, the relation of language to nature, and what is required if one is to have certain knowledge of anything.’<fnref n="38" /> Some pages back I mentioned Chrisomalis’ suggestion of a ‘cultural identity’ reaffirmation as a factor in preserving alphanumeric notation, but it may be more appropriate to consider the possibility of a deeper ‘cultural’ attachment, the attachment, or a natural inclination, to a worldview in which quantities ‘speak’ a human language, and literature can reach ‘mathematical accuracy’. The crucial Greek word that condenses these meanings is of course λόγος, which ‘in its mathematical sense of “relation, ratio, proportion”, has been attributed to the Pythagoreans and to Pythagoras himself [...] The sense “calculation” comes from the basic sense of the root λέγ- almost more directly than the sense “word”.’<fnref n="39" /> This inner identity within number has also been seen as an ‘inner polarisation’ of Pythagorean number which ‘connects the quantitative with the qualitative domain’.<fnref n="40" />

<figure><img src="image_8.png" /></figure>

Before moving on with my argument, and to illustrate the sometimes baffling complexity of this inner dynamic of the number-letters, I would like to present a

<footnote n="36">36. For an example of orthodox Halachic use in Judaism see L. Jacobs, *The Jewish Religion: A Companion* (Oxford: Oxford University Press, 1995), 182. Cf. also R. Patai, *The Jewish Alchemists: A History and Source Book* (Princeton University Press, 1994), 155.</footnote>

<footnote n="37">37. For examples from Israeli and Indonesian contexts, cf. M. Kravel-Tovi and D. Moore, *Taking Stock: Cultures of Enumeration in Contemporary Jewish Life*, The Modern Jewish Experience (Indiana University Press, 2016), 32, 71; C. Holt, *Culture and Politics in Indonesia* (Equinox Publishing, 2007), 81; and C. Leslie and A. Young, *Paths to Asian Medical Knowledge*, Comparative Studies of Health Systems and Medical Care (Berkeley: University of California Press, 1992), 261.</footnote>

<footnote n="38">38. *Ibid*., 260.</footnote>

<footnote n="39">39. W. Burkert, *Lore and Science in Ancient Pythagoreanism* (Cambridge, Mass.: Harvard University Press, 1972), 438, 440.</footnote>

<footnote n="40">40. See S. H. Nasr, *Science and Civilization in Islam*, 2nd ed. (Cambridge: Islamic Texts Society, 1987), 26.</footnote>
relatively unknown Byzantine grammatical fragment.

## Choeroboscus on the Names of Numbers

These lines are a good example of the semantic problems posed by the word στοιχεῖον, and they are directly related to the distinction between Zahlwort and Ziffer, the names of the numbers—or number-nouns—and their notation. Georgius Choeroboscus (fl. eighth–ninth century AD) is notable for his extensive use of earlier grammatical treatises and also for the influence of his works on later scholars.<fnref n="41" />

πᾶς ἀριθμὸς ἐντὸς τῶν στοιχείων θέλει ἔχειν στοιχεῖον ἐκ τῶν ἐντὸς τῆς δεκάδος.

Every number wants to have among its ‘letters’ a ‘letter’ from within the Decad.

οἱ εἶς ἀριθμὸς ἔχει τὸ ε, ὅπερ ἐκ τῶν ἐντὸς τῆς δεκάδος ἐστίν·

Number one (εἶς) contains ε, which is within the Decad.

πάλιν ὁ δύο ἔχει τὸ δ, ὅπερ ἐκ τῶν ἐντὸς τῆς δεκάδος ἐστίν·

Number two (δύο) contains δ, which is within the Decad.

πάλιν ὁ τέταρτος ἀριθμὸς ἔχει τὸ ε καὶ τὸ α, ἄτινα ἐκ τῶν ἐντὸς εἰσὶ τῆς δεκάδος.

The fourth number contains ε and α, which are within the Decad.

τὸ δὲ ὀκτὼ δοκεῖ ἀντικεῖσθαι, τοῦτο γὰρ οὐκ ἔχει στοιχεῖον ἐκ τῶν ἐντὸς τῆς δεκάδος.

Now, eight (οκτώ) would seem not to comply, as it does not contain any
‘letter’ from within the Decad.

ἀλλ' ἔστιν εἰπεῖν ὅτι δυνάμει ἔχει στοιχεῖα ἐκ τῶν ἐντὸς τῆς δεκάδος·

But it is possible to say that it does potentially contain ‘letters’ from within
the Decad.

λέγουσι γὰρ οἱ φιλόσοφοι ὅτι παρὰ τὸ ἄγω δύο ἐστίν·

Because the philosophers say that it comes from ‘I add two’ (ἄγω δύο).

τὸ δὲ ἄγω δύο ἔχει στοιχεῖα ἐκ τῶν ἐντὸς τῆς δεκάδος· λέγω δὴ τὸ α καὶ
τὸ γ, καὶ τὸ δ·

And ἄγω δύο does indeed comprise ‘letters’ from within the Decad, namely
α, γ and δ.

<footnote n="41">41. See Dickey, *Ancient Greek Scholarship*, 80.</footnote>
οὐκτὼ γὰρ λέγεται παρὰ τὸ ἄγειν, δύο γὰρ τετράδας ἄγει.

This is because ὀκτώ is said to be formed by addition, since it adds together two tetrads.42

One way of appreciating the subtlety of the semantic play here is to translate in a different way the first line, which is the key to the understanding of the whole passage: ‘Every number-noun wants to have among its elements an element of the Decad.’ We can see that the meaning of ἀριθμός here seems to be *Zahlwort*. As for στοιχεῖον, here it means ‘letter’ first, in ἐντὸς τῶν στοιχείων, ‘among its letters’, but then we have στοιχεῖον ἐκ τῶν ἐντὸς τῆς δεκάδος, where στοιχεῖον accords with the meaning of δεκάς, that is both ‘the first ten numbers’ and ‘the first ten letters’. To be able to make sense of the passage, we need to enter for a moment the alphanumeric mentality, and to realise that ‘the decad’ (viz. 10) means ‘from ἄλφα to ἰῶτα’ *at the same time* that it means ‘from one to ten’.

<figure><img src="image_9.png" /></figure>

Now, after such a good taste of the intricacies of alphanumeric notation, and thus brought to confront this duality of number which is either identity or inner polarisation, and still wondering at the possible reasons for the longevity of alphanumeric notation, we may embark on a wider examination of the Greek theory of number, hoping also to see more clearly how deep the alphanumeric duality goes.

## Reckoning and Science of Numbers

A basic Greek distinction was mentioned by Plato between λογιστική and ἀριθμητική, the ‘art of calculation’ and ‘the science of numbers,’ or ‘the art of reckoning’ and ‘the study of number’, both disciplines sharing the same object of study, ἀριθμός.43 A *locus classicus* is Plato’s *Respublica* 525a9, λογιστική τε καὶ ἀριθμητικὴ περὶ ἀριθμὸν πᾶσα, ‘reckoning and number theory are both solely concerned with number.’44 The first one is more properly concerned with what is ἀριθμητόν, countable or numerable, as explained in a scholium to *Charmides* 165e2,

λογιστική ἐστι θεωρία τῶν ἀριθμητῶν, οὐχὶ δὲ τῶν ἀριθμῶν μεταχειριστική, οὐ τὸν ὄντως ἀριθμὸν λαμβάνουσα, ὑποτιθεμένη τὸ μὲν ἓν ὡς μονάδα, τὸ δὲ ἀριθμητὸν ὡς ἀριθμόν, οἶον τὰ τρία τριάδα εἶναι καὶ τὰ δέκα δεκάδα.

<footnote n="42">42. G. Choeroboscus, *De orthographia* (epitome), in *Anecdota Graeca e codd. manuscriptis bibliothecarum Oxoniensium*, ed. J. Cramer, repr. Amsterdam: Hakkert, 1963 (Oxford, 1835), 267.</footnote>

<footnote n="43">43. See D. H. Fowler, 'Ratio in Early Greek Mathematics,' *Bulletin of the American Mathematical Society* 1, no. 6 (1979): 808 and J. Gow, *A Short History of Greek Mathematics* (Cambridge: Cambridge University Press, 1884), 22.</footnote>

<footnote n="44">44. See also *Pol.* 259e1ff.</footnote>
Reckoning is the study of what is countable, and it does not deal with numbers, as it does not use number *per se*, but rather uses 1 as if it were the Monad, the countable as if it were the number, taking 3 for the Triad and 10 for the Decad.<fnref n="45" />

But the perception of this duality of number is considered an achievement in the study of mathematics, and by no means a given or a starting point. Initially, as summed up by Fowler, 'within the surviving Greek classical mathematical tradition, number (*arithmos*) always denotes a positive integer.'<fnref n="46" /> At first, following the progression in *Resp*. 524d, *arithmos* never appears to mean anything other than 'a definite number of definite objects', and yet theoretical arithmetic grows out of it, from the understanding that in the process of 'counting off', or 'reckoning' any objects, we make use of a prior knowledge of 'counting-numbers' which are already in our possession.<fnref n="47" />

The conception of 'pure' numbers (the object of ἀριθμητική), as opposed to visible or tangible numbers (the working tools of λογιστική), arises out of the natural phenomenon of counting, for 'it would indeed seem strange if there should be numbers of—nothing. How will he who is involved in "natural" counting and calculating become aware of these monads which are in no way to be touched or seen?'<fnref n="48" />

Klein finds these matters first treated explicitly as fundamental problems within Pythagorean and Platonic philosophy, and the following conceptual shift described: through the continual practice of counting and calculation, and the acquired familiarity with numbers and their operations—what Plato calls ‘arithmetic and logistic art’—we are brought to ponder on those numbers we already possess before we begin counting or calculating, and which must be independent of the particular things which happen to be counted—of what are these the numbers? This question concerns the special nature of the object of arithmetic and logistic as that which alone of all things is in the strict sense knowable, being in fact always to some degree already known, an object which has a purely noetic character and which exhibits at the same time all the essential characteristics of the countable as such.<fnref n="49" />

Here we have come across another remarkable analogy between ἀριθμός and στοιχεῖον: they both seem to have a dual character which makes them readily perceptible and yet intangible, even transcendent. The στοιχεῖα are notably sounds,

<footnote n="45">45. *Scholia Platonica*, ed. by W.C. Greene (Haverford: American Philological Association, 1938).</footnote>

<footnote n="46">46. Fowler, *Ratio in Early Greek Mathematics*, 809.</footnote>

<footnote n="47">47. See J. Klein, *Greek Mathematical Thought and the Origins of Algebra*, trans. E. Brann (Cambridge (MA): MIT Press, 1968), 7.</footnote>

<footnote n="48">48. *Ibid.*, 50, 77.</footnote>

<footnote n="49">49. Cf. *ibid.*, 49–50. It is most interesting to compare here the Chomsky’s view that what underlies both the human linguistic and arithmetical faculties is the same kind of computational complexity which is equipped to deal with discrete infinities; cited in Gvozdanović, *Remarks on Numeral Systems*, 1.</footnote>
the phonemes heard and readily understood, practically independent of their graphical notation, which can also be called στοιχεῖα. Now we see that numbers can be either the simple referents of counting or something else of ‘purely noetic character’. Plato himself, through his theory of the *arithmoi eidetikoi*, ‘eidetic numbers’,<fnref n="50" /> remained true to the Pythagorean tradition; indeed, the eidetic numbers might, in their foundational function, be compared to the Pythagorean arithmetical ‘foundations’ or πυθμένες, the coefficients of the powers of ten, as seen above.<fnref n="51" />

Nussbaum confirms, ‘the most general sense of *arithmos* in ordinary Greek of the fifth century would be that of an ordered plurality of its members, a countable system or its countable parts.’<fnref n="52" /> It may be remembered that τάξις is one of the very basic qualities of the alphabetic series, and that in one of our grammarians we had encountered precisely a very explicit and related association with number, when the Sicyonian soldiers ordered and numbered themselves.<fnref n="53" /> The Pythagoreans were renowned for seeing the true grounds of worldly things in their countableness, since the condition of being a κόσμος, a ‘world of beauty’, is primarily determined by the presence of an ordered arrangement (τάξις).<fnref n="54" />

What comes to the fore repeatedly in these reflections and definitions, including the long passages of the *Metaphysics* devoted to criticise Plato’s ideal numbers,<fnref n="55" /> is the reference to the Pythagorean tradition, as when there was mention of the Pythagoreans looking after the shapes of the letters,<fnref n="56" /> and of the Pythagorean lineage of isopsephy.<fnref n="57" />

I shall now follow the direction of the references and turn my attention more specifically to Pythagorean number doctrines, particularly in their relation to physics

<footnote n="50">50. Known to us from the Aristotelian polemic against it (cf. above all *Metaphysics* M 6–8), and through later Platonist developments.</footnote>

<footnote n="51">51. 99 Klein, *Greek Mathematical Thought*, 137.</footnote>

<footnote n="52">52. M. C. Nussbaum, ‘Eleatic Conventionalism and Philolaus on the Conditions of Thought,’ *Harvard Studies in Classical Philology*, no. 83 (1979): 90, echoed by C. Huffman, *Philolaus of Croton, Pythagorean and Presocratic: A Commentary on the Fragments and Testimonia with Interpretive Essays* (Cambridge: Cambridge University Press, 1993), 173, and cited by Roochnik (‘Counting on Number,’ 546).</footnote>

<footnote n="53">53. See above p. 39.</footnote>

<footnote n="54">54. Klein, *Greek Mathematical Thought*, Cf.</footnote>

<footnote n="55">55. See J. Annas, *Aristotle’s Metaphysics: Books Mu and Nu*, Clarendon Aristotle Series (Oxford: Clarendon Press, 1976).</footnote>

<footnote n="56">56. Above, p. 39.</footnote>

<footnote n="57">57. Above, p. 59.</footnote>
and metaphysics.

Pythagoras and Pythagoreanism

It is becoming something of a truism to speak of an ‘explosion’ of Pythagorean-related scholarly literature,<fnref n="58" /> and the field has been transformed so much in the last few decades that, in the words of Huffman,<fnref n="59" /> ‘the Pythagoras of current scholarship is not your mother’s let alone your grandmother’s Pythagoras.’ He seemed to be taken for granted, ‘one of the most familiar names among the Greek philosophers, one we are told very much and we know very little about,’<fnref n="60" /> until with the 1962 publication of Burkert’s *Weisheit und Wissenschaft: Studien zu Pythagoras, Philolaos und Platon*,<fnref n="61" /> along with von Fritz’s complementary article in the *RE*,<fnref n="62" /> this field of studies received a powerful wake-up call to examine anew centuries of basic assumptions which had effectively become obstacles to a distinct vision of Pythagoras and Pythagoreanism.<fnref n="63" /> Saito speaks of the ‘blow of the epoch-making study’ and ‘dismantling the myth of origins’ of Pythagoreanism. This reexamination keeps its momentum and has been extending its scope into later developments, trying to include everything that came to be labelled as Pythagorean over the centuries.<fnref n="64" /> Also, because of the enduring influence

<footnote n="58">58. Cf. P. S. Horky, *Plato and Pythagoreanism* (Oxford, New York: Oxford University Press, 2013), x, also *BNP*, s.v. ‘Pythagoras’ and related entries.</footnote>

<footnote n="59">59. C. Huffman, ed., *A History of Pythagoreanism* (Cambridge: Cambridge University Press, 2014), 1.</footnote>

<footnote n="60">60. See G. Cornelli, *In Search of Pythagoreanism: Pythagoreanism as an Historiographical Category*, Studia Praesocratica 4 (Berlin/Boston: Walter de Gruyter, 2013), vii, for the history of modern Pythagorean scholarship. L. Zhmud, “‘All Is Number’? ‘Basic Doctrine’ of Pythagoreanism Reconsidered,” *Phronesis* 34, no. 3 (1989): 270–292 gives a succinct and engaging account.</footnote>

<footnote n="61">61. Revised and translated into English as *Lore and Science in Ancient Pythagoreanism*. Praise has been showered on this book from all quarters for decades. For one telling testimony from an earlier authority in the field, see K. von Fritz, review of *Lore and Science in Ancient Pythagoreanism*, by W. Burkert and E. L. Minar (tr.), *Isis* 65 (1974): 531–531, and his earlier K. von Fritz, review of *Weisheit und Wissenschaft. Studien zu Pythagoras, Philolaos und Platon*, by W. Burkert, *Isis* 55, no. 4 (1964): 459–461, where it is called ‘a most energetic effort to solve the problems posed by a complicated and confused ancient tradition.’ See also Cornelli, *In Search of Pythagoreanism*, 42.</footnote>

<footnote n="62">62. K. von Fritz, ‘Pythagoras,’ in *RE*, vol. XXIV (1963).</footnote>

<footnote n="63">63. For the most up-to-date relevant references see in particular Huffman, *History*; Horky, *Plato and Pythagoreanism*; K. Saito, ‘Mathematical Reconstructions Out, Textual Studies In: 30 years in the historiography of Greek mathematics,’ *Revue d’histoire des mathématiques* 4, no. 1 (1998): 131–142; D. J. O’Meara, *Pythagoras Revived: Mathematics and Philosophy in Late Antiquity* (Oxford: Oxford University Press, 1990); and the bibliography in *BNP*, s.v. ‘Pythagoras’. For a taste of a ‘hyper-critical’ position according to which ‘aucun des fragments considérés comme authentiques par Walter Burkert et Carl Huffman ne possède une légitimité propre,’ see L. Brisson, ‘Platon, Pythagore et les Pythagoriciens,’ *Eikasia. Revista de Filosofia*, no. 12 (2007): 39–66.</footnote>

<footnote n="64">64. See L. Zhmud, ‘What is Pythagorean in the pseudo-Pythagorean literature?,’ August 2015.</footnote>
of all sorts of ‘Pythagorean’ writings on Arabic literature, with ulterior ramifications
into Hebrew literature and right into the European Middle Ages, the reassessment of
Pythagoreanism is already having a wider impact on many areas of Islamic, Jewish and
Medieval studies.<fnref n="65" /> There is also a somewhat parallel line of Pythagorean scholars who
in different degrees argue for a more ‘inclusive’<fnref n="66" /> view of the ancient and medieval
material, and according to whom there is a respectable doctrinal continuity from the
obscure origins of the Pythagoras myth down to the medieval Pythagorean apocrypha
in various languages and all sorts of contexts.<fnref n="67" />

Nowadays, after all the sifting and winnowing, some scholars argue that there is
no reliable evidence that Pythagoras was a mathematician, a shaman or anything,
yet there is talk of a ‘vague image’ we still feel compelled to flesh out.<fnref n="68" /> The
words of a recent reviewer of Huffman’s anthology appear very apt: ‘we cannot
give a sufficient account or express a valuable analysis of Pythagoreanism without
dealing with the interpretations and the judgments of ancient and contemporary
interpreters. Comprehension of Pythagoreanism, in fact, necessarily passes through
the reading of late sources and through the understanding of a constantly renewed
interest—differently motivated in different eras—in the figure of Pythagoras and in his
doctrines.’<fnref n="69" /> Cornelli recalls Guthrie’s ‘bottomless pit of research on the Pythagoreans’,
steps back somehow and issues what seems to be a balanced overview of the latest
research, arguing against an ‘overly presentist view of ancient philosophy’, and
explaining why there is no need to ‘choose between an acousmatic and a mathematical
Pythagoreanism.’<fnref n="70" />

But let us take Burkert as point of departure, and attempt a summary of the current
views on ‘Pythagoras the arithmetician’ while availing ourselves of the wealth of recent
scholarship.

<footnote n="65">65. For a recent partial overview with references to relevant sources, see A. Izdebska, ‘The Attitudes of Medieval Arabic Intellectuals towards Pythagorean Philosophy: different approaches and ways of influence,’ in *Cultures in Motion: Studies in the Medieval and Early Modern Periods*, ed. A. Izdebski and D. Jasiński (Jagiellonian University Press, December 2014), 25–44.</footnote>

<footnote n="66">66. See Cornelli, *In Search of Pythagoreanism*, 192, and L. Zhmud, ‘Mathematics vs Philosophy: An alleged fragment of Aristotle in Iamblichus,’ *Hyperboreus* 13, nos. 1–2 (2007): 77.</footnote>

<footnote n="67">67. In addition to some of Zhmud's works, which remain closer to scholarly consensus, two examples are P. Kingsley, *Ancient Philosophy, Mystery, and Magic: Empedocles and Pythagorean tradition* (Oxford: Oxford University Press, 1995) and A. Uždavinys, *The Golden Chain: An Anthology of Pythagorean and Platonic Philosophy* (Bloomington: World Wisdom, Inc, 2004). Thomas Taylor, ‘the Platonist’ (1758–1835) may also be mentioned as an important precedent of this view.</footnote>

<footnote n="68">68. See Huffman, *History*, 4, 24ff.</footnote>

<footnote n="69">69. G. De Cesaris, review of *A History of Pythagoreanism*, by C. Huffman, *International Plato Journal*, December 2014: 101–103.</footnote>

<footnote n="70">70. Cornelli, *In Search of Pythagoreanism*, 192, 196.</footnote>
*Lore & Science and Further Developments*

The declared primary purpose of this work was to determine, among the variegated mass of the Pythagorean tradition, what can be considered original Pythagorean, and what is either interpretation, be it Platonic, Aristotelian, Neo-Platonic, or addition and attribution by later authors.

Burkert starts by noting how the earliest reliable evidence for Pythagorean literature, and by far the most influential, is, by one of the curious turns of the history of philosophy, found in the Corpus Aristotelicum. Historical passages like the one in *Metaphysics* A had a decisive role in defining later views, and it is known that there were additionally some writings specifically on the Pythagoreans which were lost.<fnref n="71" /> The thrust of the Aristotelian accounts accumulates until a fundamental stumbling block is identified: ultimately ‘the Pythagorean doctrine cannot be expressed in Aristotle’s terminology’ (Burkert, *L & S*, 45).

Finally, the conclusion is reached that ‘already in the earliest evidence available two conceptions of Pythagorean philosophy are in sharp contrast with each other’ (p. 79):

1. the Aristotelian one, which aside from its own occasional contradictions tends to posit a divide or a clear break between Pythagorean and Platonic doctrines, and

2. the non-Aristotelian, Old Academy one, traced back to Speusippus, which is the older of the two.

This latter one ‘equates Pythagoreanism with the doctrines of the *Timaeus* and with the Platonic number philosophy: The highest principles, immaterial, are the One and the Indefinite Dyad; from them come the pure, incorporeal numbers; the numbers produce the pure geometrical shapes, line, plane, and solid, as well as the perceptual functions of the “Living Creature Itself”; from the mathematical regularity of the regular polyhedra come the elements and therewith the multifariousness of the empirical world; and this whole process takes place, in this order, **only in thought**, which traces back changeless Being to its ultimate principles and understands it by means of them’ [my highlighting]. Since ‘this Academy tradition with its “derivation system” must be considered to be actually Old Academy’ (p. 82), when we study the Old and Middle Academy we are actually studying what for centuries has been labelled Pythagoreanism. There is accordingly no need of obscure far-fetched sources or *testimonia*, since there are no Pythagorean *ipsissima verba*. We are left reassured that the cosmology of *Timaeus* and other dialogues where ‘theoretical arithmetic’ is introduced, are very likely the continuation of a real Pythagorean doctrinal thread.

<footnote n="71">71. Regarding these, and their fragmentary transmission by later authors, see *inter alia* Horky, *Plato and Pythagoreanism*, 3.</footnote>
Now, if we still try to go back to the earliest sources in a quest for the *authentic* Pythagorean texts, the fragments of Philolaus (probably written some time before 400BC) are especially important as the most likely to convey original Pythagorean contents. They are also of especial interest to us because they treat of number theory and cosmology, while those of Archytas, which are second in likelihood of authenticity, deal more with other topics like acoustics (pp. 22 and 267). In fact, one of the very tangible effects of Burkert’s insights was a renewed interest in Philolaus and Archytas, with several monographic works dedicated to them over the last few decades.<fnref n="72" />

Before having a closer look at Philolaus, I shall comment on some of the above findings:

* A sort of ‘Pythagorean’ arithmetical genealogy or philosophical lineage appears to be delineated, starting from an undecidable legendary origin with Pythagoras, then first established in the fragments of Philolaus and Archytas, and adopted by Plato, mostly in his *Timaeus*, to be thereafter scattered to the four intellectual winds, as it were, through the later Academy and subsequent Neo-Platonic developments. In all this, Aristotle is the witness *par excellence*.

* The mention of the two highest principles, One and Dyad,<fnref n="73" /> is an opportunity to set further limits to this enquiry: while it seems inescapable that a logical reduction to last principles will yield a binary system,<fnref n="74" /> my current research has boundaries, an ‘alphabetic constraint’, meaning that we need a minimal number of elements which can function in combination as conveyors of human language. This consideration—what is the minimal working alphabet—opens onto a vast and tangential field of studies that is mostly within the ken of system theory, computer science, logic architecture and similar. For a taste of such, the question might be translated as ‘how many symbols do we need for obtaining computational completeness?’<fnref n="75" /> We should not let go unnoticed the interesting connections that suddenly crop up: computing (λογιστική), logic, architecture,

<footnote n="72">72. See Huffman, *Philolaus*; and C. Huffman, *Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King* (Cambridge: Cambridge University Press, 2010). There are also more recent chapters on the same two philosophers by D. W. Graham and M. Schofield respectively, in Huffman, *History*.</footnote>

<footnote n="73">73. Corresponding to odd/even or limited/unlimited, in other contexts; see below.</footnote>

<footnote n="74">74. As Leibniz explained (see G. W. Leibniz, ‘Explanation of Binary Arithmetic,’ transl. and edited by Lloyd Strickland, 2007) in relating it to the Chinese binary principles of the 64 hexagrams of the *I Ching*.</footnote>

<footnote n="75">75. See A. Alhazov, R. Freund, and M. Oswald, ‘Tissue P Systems with Antiport Rules and Small Numbers of Symbols and Cells,’ in *Developments in Language Theory, 9th International Conference (Springer, 2005)*, 101. Closer to us, R. Lehmann, ‘27-30-22-26 - How Many Letters Needs an Alphabet? The Case of Semitic,’ in *The Idea of Writing: Writing Across Borders*, ed. A. de Voogt and J. Quack (Leiden: Brill, 2012), 11–52, has reflected on a similar issue within the context of Semitic alphabets.</footnote>
but we should also not dwell on them. What we thought important to observe
is that we shall consciously refrain from discussing further the ultimately basic
'complete' system of elements, which would be binary, to stay on the level of
what should then be strictly considered an 'intermediary' level, where we shall
deal with perhaps ten elements (the Decad), or sixteen (the number of letters of
the primordial Greek alphabet), or twenty-two, twenty-four or twenty-eight, like
the letters of the Hebrew, later Greek and Arabic alphabets.

* The progression from incorporeal numbers to pure geometrical shapes to the elements—and all this **only in thought**—sets further the scene for the interplay of our alphabetic principles, and we shall keep it in mind for comparison with other sources.

Imitation or Participation

'In the Hippocratic writings the relation between microcosm and macrocosm becomes a matter of "imitation", but this imitation may be turned either way.' The body imitates the cosmos and parts of the cosmos imitate human organs. 'Imitation is a two-sided correspondence.' 'Nothing more is meant than the correspondence of cosmos and number, in the sense that one explains and illuminates the other.'<sup>76</sup> This is compared by Burkert to the very important passage in *Metaph.* 987b which is used as an argument in 'all attempts to prove a Pythagorean origin of the theory of ideas':

οἱ μὲν γὰρ Πυθαγόρειοι μιμήσει τὰ ὄντα φασὶν εἶναι τῶν ἀριθμῶν, Πλάτων
δὲ μεθέξει, τοῦνομα μεταβαλών. τὴν μέντοι γε μέθεξιν ἢ τὴν μίμησιν ἥτις
ἄν εἴη τῶν εἰδῶν ἀφεῖσαν ἐν κοινῷ ζητεῖν.

Pythagoreans say that things exist by 'imitation' of numbers, whereas Plato says that they exist by 'participation'—merely changing the term. As to what this 'participation' or 'imitation' of the forms may be, they apparently gave up on the question.

I may recall here those passages in the grammatical survey where the letters are
said to relate κατὰ μίμησιν to the four elements, or also to the twenty-four hours
of day and night, and more specifically, the seven vowels to the seven planets. The
question posed in those cases is the same mentioned by Aristotle: how are we to
understand this μίμησις? The enormity of the gap between the two possibilities reported
by Aristotle, namely μιμήσει and μεθέξει, has been explored many times,<sup>77</sup> and the
many interpretative possibilities of the simple term μίμησις alone have also given

<footnote n="76">76. Burkert, *L & S*, 44–45.</footnote>

<footnote n="77">77. Cf. Horky, *Plato and Pythagoreanism*, 32, 46, 255.</footnote>
rise to countless pages, including notably in the field of aesthetics, for art can be
conceived as essentially μίμησις, and the artists as copyists. The original meaning of
the word referred to the liturgical actions of the sacrificial priest, i.e. dance, music,
song.<fnref n="78" /> ‘True “imitation” is not a matter of illusory resemblance (όμοιότης) but of
proportion, true analogy, or adequacy (αὐτὸ τὸ ἴσον, i.e., κατ’ ἀναλογίαν)’;<fnref n="79" /> the
geometric, mathematical connotation is unmissable.

On Creation and Order

'Pythagoras says that the cosmos is a created thing in the realm of thought but not in that of time (κὰτ' ἐπίνοιαν οὐ κατὰ χρόνον).'<fnref n="80" /> Aristotle dismisses offhand this Pythagorean doctrine as absurd, but it appears in later literature as Pythagorean, and it agrees with the cosmology of *Timaeus*, and with the important doxographical statement that Pythagoras coined the term 'cosmos', ἐκ τῆς ἐν αὐτῷ τάξεως, 'because of the order in it.'<fnref n="81" />

The association of Pythagoras with the henceforth allied concepts of ‘universal orderly beauty’ (κόσμος) and ‘sequential order’ (τάξις), established from early on a sort of conceptual amalgam that would be perpetuated through the Middle Ages. The intemporal and conceptual character of this creation, this universe and this order is to be related to the immaterial aspect of letter and number in their intrinsic duality.

Discussing Philolaus' theories, Burkert observes: 'the Greek word ἀριθμός is not completely equivalent to the modern concept of “number”. Ἀριθμός is always a whole number, and tied up with the actual process of counting. Thus it is closely connected with things, and in fact is itself a thing, or at least an ordering of things. Ἀριθμός means a numerically arranged system, or its parts.'<fnref n="82" /> This is in direct relation to the alphabet and the letters, as I shall have occasion to review in coming pages.

The Pythagorean Logos

The term λόγος, in its mathematical sense of ‘relation, ratio, proportion’, has also
been attributed to the Pythagoreans and to Pythagoras himself. Its origin would be in

<footnote n="78">78. W. Tatarkiewicz, *A History of Six Ideas: An Essay in Aesthetics*, Melbourne International Philosophy Series 5 (The Hague/Boston/London: Martinus Nijhoff, 2012), 266.</footnote>

<footnote n="79">79. See 'Imitation, Expression and Participation' in A. Coomaraswamy, *Selected Papers 1: Traditional Art and Symbolism*, ed. R. Lipsey, Bollingen Series 89 (Princeton: Princeton University Press, 1977), 276ff. and a recent criticism, A. Chakrabarti, *The Bloomsbury Research Handbook of Indian Aesthetics and the Philosophy of Art*, Bloomsbury Research Handbooks in Asian Philosophy (Bloomsbury Publishing, 2016), 73-4.</footnote>

<footnote n="80">80. Aëtius, 2.4.1.</footnote>

<footnote n="81">81. Burkert, *L & S*, 71, 77.</footnote>

<footnote n="82">82. *Ibid*., 265—my highlighting.</footnote>
musical theory, the numerical ratio in an interval, with obvious parallels in geometry,
where the ratios determine angles and shapes. 'Thus the logos would be "the group
or bundle of numbers that lie hidden in a thing", by use of which it can be not only
described but reproduced.'<sup>83</sup> It is important to note, as mentioned above, that 'the sense
"calculation" comes from the basic sense of the root λέγ- almost more directly than the
sense "word".'<sup>84</sup> This observation about the original sense of λόγος is very important,
and a serious invitation to think twice before routinely translating λόγος as 'word'.
The consequences for the understanding of many 'doctrines of the λόγος' are wide and
profound. For one remarkable example and by way of experiment, John 1:1 might be
rendered: 'In the beginning was the proportion.'

The Fundamental Decad

‘One is νοῦς and οὐσία, two is δόξα... ten is the perfect number, which comprehends the whole nature of number and determines the structure of the cosmos, and with it ends the symbolic interpretation of the numbers.’<sup>85</sup>

This text comes with a reference to *Metaph.* 986a8, where Aristotle is speaking about
οἱ καλούμενοι Πυθαγόρειοι. I quote part of the passage because it is also a fundamental
reference for this work:

τὰ τῶν ἀριθμῶν στοιχεῖα τῶν ὄντων στοιχεῖα πάντων ὑπέλαβον εἶναι,
καὶ τὸν ὅλον οὐρανὸν ἁρμονίαν εἶναι καὶ ἀριθμόν.

they assumed that the elements of numbers are the elements of all things,
and the entire heaven to be a harmony or number.

τέλειον ἡ δεκάς εἶναι δοκεῖ καὶ πᾶσαν περιειληφέναι τὴν τῶν ἀριθμῶν
φύσιν

the decad is considered to be something perfect and to comprise the whole
nature of the numbers.

Finally, in 986a15ff.

φαίνονται δὴ καὶ οὖτοι τὸν ἀριθμὸν
νομίζοντες ἀρχὴν εἶναι καὶ ὡς ὕλην
τοῖς οὖσι καὶ ὡς πάθη τε καὶ ἔξεις,
τοῦ δὲ ἀριθμοῦ στοιχεῖα τό τε ἄρτιον
καὶ τὸ περιττόν, τούτων δὲ τὸ μὲν

These thinkers obviously consider **number to be a first principle, and like material for things** and as constituting their conditions and states. The elements of number are the Even and the Odd; of these the former is

<footnote n="83">83. Burkert, *L & S*, 438.</footnote>

<footnote n="84">84. *Ibid.*, 440; see also Lohmann, 'Mathematik und Grammatik,' 310, where the arithmetical meaning of ὅνομα is mentioned as well.</footnote>

<footnote n="85">85. Burkert, *L & S*, 467–68.</footnote>
πεπερασμένον τὸ δὲ ἄπειρον, τὸ δ' ἕν ἐξ ἀμφοτέρων εἶναι τούτων (καὶ γὰρ ἄρτιον εἶναι καὶ περιττόν), τὸν δ' ἀριθμὸν ἐκ τοῦ ἑνός, ἀριθμοὺς δέ, καθάπερ εἴρηται, τὸν ὅλον οὐρανόν. limited and the latter unlimited. The unity consists of both (since it is both odd and even); number is derived from the unity; and numbers, as we have said, are the entire heaven.

Regarding the decad, Schofield considers the Pythagorean influence on even such an Aristotelian trademark as the ten categories. In view of passages like the above, he observes: 'if the *Categories* insists that there are just ten basic categories through which we can articulate reality, the treatise surely must have a pedigree in Pythagorean teaching on number as principle of all things, and on ten as the perfection and completion of the number series.'<fnref n="86" />

Finally, Burkert sums up the Pythagoreans view: 'Number is not quantity and measurability, but **order and correspondence**, the articulation of life in rhythmical pattern, and the perspicuous depiction of the whole as the sum of its parts. To see a "consistently quantitative view of the world" in Pythagorean number theory is a mistake.'<fnref n="87" /> This concept of number is certainly closer to a language than to what we know as arithmetic and mathematics.<fnref n="88" />

## Philolaos

The extraordinary significance of Philolaus for this thesis is due to the fact, now established by Burkert and Huffman, that the Pythagorean theory described and criticized by Aristotle is in most cases that of Philolaus, and that in fact, we know of no philosophical/scientific, cosmological views of Pythagoreans before those of Philolaus. Cornelli, though adding some temperating comments, concludes that Philolaus' doctrine is 'the *solution* to the problem of attributing a doctrine of "all is number" to ancient Pythagoreanism, in epistemological, ontological and numerological dimensions,' and even more decisively, that 'Philolaus can be considered the solution to our problem: ancient Pythagoreanism, or at least fifth-century Pythagoreanism, did hold the doctrine of "all is number", whether in a mystical or epistemological sense.'<fnref n="89" /> It was through the public philosophical activities of Philolaus that the hitherto

<footnote n="86">86. M. Schofield, 'Archytas,' chap. 3 in *A History of Pythagoreanism*, ed. C. Huffman (Cambridge: Cambridge University Press, 2014), 79–80.</footnote>

<footnote n="87">87. Burkert, *L & S*, 477.</footnote>

<footnote n="88">88. For more on this topic in its relation to the abacus, and including the remarkable preservation of certain aspects of 'medieval number' in our system of double-entry bookkeeping, see A. Heeffer, 'On the curious historical coincidence of algebra and double-entry bookkeeping,' in *Foundations of the Formal Sciences VII: bringing together philosophy and sociology of science*, vol. 32 (College Publications, 2011), 125.</footnote>

<footnote n="89">89. Cornelli, *In Search of Pythagoreanism*, 184, 187.</footnote>
‘shadowy’ world of Pythagorean lore emerged into daylight and became part of the Greek philosophical conversation. In this transformation it is likely that Philolaus played the role of inventor of ‘Pythagorean’ philosophy more than of transmitter, and this may be the reason why Aristotle often referred to them in general as οἱ καλούμενοι Πυθαγόρειοι, ‘the so-called Pythagoreans’.<fnref n="90" />

In the following lines I present and comment on those fragments of Philolaus which are relevant to my enquiry into the nature of ἀριθμός. Regarding the authenticity or spuriousness of the fragments, Huffman’s *Philolaus*, cited above, can be considered the current authoritative account.<fnref n="91" /> Given that this enquiry is not so much concerned with ‘canonic’ philological Pythagoreanism as with the history and influence of one particular Pythagorean concept, I shall widen the scope to include within my selection some of the spurious or doubtful fragments; after all, they too were for many centuries received and transmitted as veritable Pythagorean teachings, contributing to the Medieval view of ancient philosophy.<fnref n="92" />

## Fragments of Philolaus

### Fragment 4 (authentic)

καὶ πάντα γα μὰν τὰ γιγνωσκόμενα ἀριθμὸν ἔχοντι· οὐ γὰρ οἶόν τε οὐδὲν οὕτε νοηθῆμεν οὕτε γνωσθῆμεν ἄνευ τούτου.

Everything we are perceiving now is with number—for it is not possible that anything at all be understood or known without it.

If, as has been suggested by the specialists, Philolaus is the ‘mathematical’ Pythagorean *par excellence*, then this fragment, along with 11, must surely be at the very basis of the ever popular perception of Pythagoreanism as the doctrine that ‘all is number’, or ‘everything is number’, as reported and immortalised by Aristotle.<fnref n="93" /> My translation is an effort to make the most of the crucial ἔχοντι (Attic ἔχουσι), in the sense brought out by Huffman himself but not reflected in his translation of the fragment, ‘the sense of being constituted by an ordered plurality in some sense.’ Our intended meaning for ‘being with number’ instead of the rather vague ‘having’, relies partly on the opposition to ἄνευ, and it is close to that of a woman ‘being with child’, thus close to the Hippocratic example mentioned by Huffman, where lame parents are said to be able

<footnote n="90">90. E.g. *Met*. A8 989b29. See D. W. Graham, ‘Philolaus,’ chap. 2 in *A History of Pythagoreanism*, ed. C. Huffman (Cambridge: Cambridge University Press, 2014), 48.</footnote>

<footnote n="91">91. Latest views in Huffman, *History*, especially chapters 2 and 8.</footnote>

<footnote n="92">92. Cf. the defining role played by the Christian apocrypha on Medieval and Renaissance arts. See J. Goethals, V. McGuire, and G. Zhang, *Power and Image in Early Modern Europe*, EBSCO ebook academic collection (Newcastle: Cambridge Scholars, 2009), 98–9.</footnote>

<footnote n="93">93. Cornelli, *In Search of Pythagoreanism*, 137, gives thorough lists of Aristotle’s mentions. Cf. also Graham, *Philolaus*, 54.</footnote>
to produce healthy offspring because ‘the lame part has the same number as the healthy part,’ with reference to the proper constitution of the body by the four humours.<fnref n="94" />

The other main aspect of the fragment is the epistemological one. We have three words related to cognition, namely γιγνωσκόμενα, νοηθῆμεν, and γνωσθῆμεν (the latter two are passive infinitives). I have brought out the present tense of γιγνωσκόμενα as a deliberate contrast to the gnomic second part of the fragment, for which I keep Huffman’s wording almost intact. Zhmud translates ‘All that is cognizable, certainly has number. For it is impossible for us to think or perceive something without this.’<fnref n="95" /> Nussbaum speaks of the ‘close association between numerability and knowability’,<fnref n="96" /> and it is along this epistemological line that the fragment is interpreted by most. There is no question of the epistemological import of the text, but it is important to keep in mind, with Viltaniotis, that ‘many examples in Greek thought show that the concept of number refers to something far more complex than an “ordered plurality,”’<fnref n="97" /> and that, ‘if it is impossible for a thing to be known without number, this is because it is impossible for a thing to *be* what it is without number,’ for ‘it is number, along with the basic powers to limit and to be limited, which makes things what they are.’<fnref n="98" /> In other words, and quite along the lines of the Latin concept of *realitas* (from *reor*, what ‘I think’), here epistemology is ontology.<fnref n="99" />

I would also take note of Viltaniotis’ assertion that ‘number has a precise metaphysical role in Philolaus’ system, and that this role is to be understood in close connection to the role of *harmony*, the third principle without which it would have been impossible for limiters and unlimiteds to be fitted together.’<fnref n="100" />

This ‘harmonic’ aspect, finally, is related directly to an intersecting area of Pythagoreanism that we shall henceforth encounter repeatedly, that of musical theory, in which the identity between perceptibility and intelligibility—or ‘number’ as substantial structure and ‘number’ as enabler of cognition—mutes the dichotomy by its being ‘heard’; as Hicks observes, the octave is ‘the sense-perceptible manifestation of the ratio 2:1’, and ‘the point is simple: who would deny that the octave sounds consonant?’<fnref n="101" /> It is not difficult to surmise, in this light, that probing the depths of

<footnote n="94">94. Huffman, *Philolaus*, 175, my italics.</footnote>

<footnote n="95">95. Zhmud, “‘All Is Number’?,” 275.</footnote>

<footnote n="96">96. Nussbaum, ‘*Eleatic Conventionalism*,’ 91.</footnote>

<footnote n="97">97. I.-F. Viltanioti, ‘Powers as the Fundamental Entities in Philolaus’ Ontology,’ *Journal of Ancient Philosophy* VI, no. 2 (2012): 21.</footnote>

<footnote n="98">98. *ibid.*, 22, my italics.</footnote>

<footnote n="99">99. Leaving aside for the moment Semitic parallels, cf. the popular etymology that relates *think–thing*, or Ger. *denken–Ding* (e.g. S. Urban [Cave, Edward], ‘English Grammar and English Grammarians,’ *The Gentleman’s Magazine* (London) 14 (1840): 477), and see Cassin, *Dictionary of Untranslatables*, 901.</footnote>

<footnote n="100">100. Viltanioti, ‘Powers,’ 22, my italics.</footnote>

<footnote n="101">101. A. Hicks, ‘Pythagoras and Pythagoreanism in Late Antiquity and the Middle Ages,’ chap. 20 in *A*</footnote>
the ‘bottomless pit’ may require in the first place a salutary dose of music.

Wer den Dichter will verstehen
Muss in Dichters Lande gehen.<fnref n="102" />

Fragment 5 (spurious)

ovan ἀριθμὸς ἔχει δύο μὲν ἴδια εἴδη,
περισσὸν καὶ ἄρτιον, τρίτον δὲ ἀπ'
ἀμφοτέρων μειχθέντων ἀρτιοπέριττον·
ἔκατέρω δὲ τῷ εἴδεος πολλαὶ μορφαί,
ἄς ἔκαστον αὐτὸ σημαίνει.

Number has two distinct forms, odd and
even, and a third compounded of both, the
even-odd. Each of the two forms has many
manifestations which are shown by every
separate thing.<fnref n="103" />

This fragment tells us about the two fundamental types of number and, most
importantly, about the central relation between numbers and things. To rephrase and
thus elaborate on the text: Every single thing<fnref n="105" /> shows, is a sign of, hints at (σημαίνει),
the multiple manifestations or shapes (μορφαί) of the two distinct forms, kinds, species
(εἴδη) of number. That is, every single thing gives signs of its kinship to either even
or odd number, not directly but as if in a second degree, through their manifestations
(numerals, perhaps?). The famous Orphic pun reported in Plato's *Cratylus* (400b9ff.),
σῶμα σῆμα, 'the body is a tomb,' or also 'the body is a token,'<fnref n="106" /> can shed some light
for us on Philolaus' meaning here: things are tokens of the varieties of odd and even,
bespeaking the latent 'forms' as the body reveals the character and movements of the
soul, or as a tomb indicates the resting place of a body. Restricting ourselves to the
wording of the text, this fragment might be said to speak unequivocally of a *semantic*
relation between reality and numbers.

The mention of *Cratylus* is hardly casual, since this dialogue is the prime reference
for the discussion of the relation between a system of signs and reality. It is indeed
remarkable that no Platonic discussion is found regarding the relation between

<footnote>*History of Pythagoreanism*, ed. C. Huffman (Cambridge: Cambridge University Press, 2014), 427.</footnote>

<footnote n="102">102. Whoever wishes to understand the poet/ Must go into the poet's land.</footnote>

<footnote>Goethe, West-östlicher Divan, 241.</footnote>

<footnote n="103">103. My translation follows mostly the interpretations of M. E. Hager, 'Philolaus and the Even-Odd,' *The Classical Review* (Cambridge, UK) 12, no. 1 (March 1962): 1 (not discussed by Huffman), and J. Mansfeld, ed., *Die Vorsokratiker*, 2 vols (Stuttgart: Reclam, 1983/1986), I, 153 (hereafter cited as RV) while making use of Huffman's emendation of *αὐταυτὸ*, *Philolaus*, 192.</footnote>

<footnote n="105">105. This interpretation of ἔκαστον αὐτό seems to agree with at least two similar uses we have found in Arist. SE 181b13, and Proclus *Theol.Plat*. II, 1, and it would corroborate Huffman's emendation.</footnote>

<footnote n="106">106. Cf. A. Bernabé, 'Una etimología platónica: σῶμα-σῆμα,' *Philologus* - Zeitschrift für antike Literatur und ihre Rezeption 139, no. 2 (1995): 204–237; also the careful discussion of σημαίνω in F. Ademollo, *The Cratylus of Plato: A Commentary* (Cambridge University Press, 2011), 157ff.</footnote>
numerical notation and number, and this might be due to the common use of letters
as numerical notation.

Fragment 11 (almost certainly spurious)<fnref n="107" />

From a philological point of view, as Huffman observes, ‘There is a great deal both in the style and content of the fragment that links it to the philosophy of the early Academy and the later tradition of Platonism, while virtually nothing that accords with what one would expect from a Presocratic author.’<fnref n="108" /> This is quite convenient for us as we are interested in tracing the continuity of the doctrines rather than the soundness of the attribution; and indeed there are several doctrinal threads in this long fragment which are worth noting.

Θεωρεῖν δεῖ τὰ ἔργα καὶ τὴν οὐσίαν τῶ
ἀριθμῶ καττὰν δύναμιν ἄτις ἐστὶν ἐν τῷ
δεκάδι· μεγάλα γὰρ καὶ παντελὴς καὶ
παντοεργὸς καὶ θείω καὶ οὐρανίω βίω
καὶ ἀνθρωπίνω ἀρχὰ καὶ ἁγεμῶν κοινω-
νοῦσα [***]<fnref n="109" /> δύναμις καὶ τᾶς δεκάδος.
ἄνευ δὲ τούτας πάντ’ ἄπειρα καὶ ἄδηλα
καὶ ἀφανῆ.

The functions and essence of number must
be considered in the light of the power
inherent in the decad, for it is great, ut-
terly perfect, utterly effective, and it is the
joint principle and directing force of di-
vine, celestial, and human life. Without it,
everything is undefined, unclear, and indis-
cernible.

We had already come across passages in praise of the decad, and what we have here
might be one of the very first and influential examples of this mysterious veneration.
Mysterious because from a strictly mathematical point of view, the decad has no
extraordinary features that would warrant such a high esteem. The common current
mathematical view is that 'the use of ten as the base goes back to the dawn of
civilisation, and is attributed to the fact that we have ten fingers on which to count.'<fnref n="110" />
Now, it may be precisely this kind of cosmic correspondence which is at the basis of
the Pythagorean doctrine, not something arithmetically intrinsic to the number ten,
but rather the accumulation of perceived correspondences in nature and, thinking of
Aristotle's categories, also in the synthetic power found in decadic configurations. The
epistemological importance attributed to number in the previous fragment, is here
particularised to the number ten, and emphasised in the following paragraphs.

In relation to this aspect, fr. 20b<fnref n="111" /> gives an oft-repeated etymology of the decad:
'δεχάδα προσηγόρευσεν, ὡς δεκτικήν τοῦ ἀπείρου'—he [Philolaus] called it a 'receiver',

<footnote n="107">107. On its authenticity, see Huffman, *Philolaus*, 349–50.</footnote>

<footnote n="108">108. See *Philolaus*, 349.</footnote>

<footnote n="109">109. Twelve letters missing here.</footnote>

<footnote n="110">110. A. Slinko, *Algebra for Applications: Cryptography, Secret Sharing, Error-Correcting, Fingerprinting, Compression*, Springer Undergraduate Mathematics Series (Springer International Publishing, 2015), 33.</footnote>

<footnote n="111">111. Not in DK; see Huffman, *Philolaus*, 352.</footnote>
because it is receptive of the unlimited.

γνωμικὰ γὰρ ἁ φύσις ἁ τῶ ἀριθμῶ
καὶ ἡγεμονικὰ καὶ διδασκαλικὰ τῶ
ἀπορουμένω παντὸς καὶ ἀγνοουμένω
παντί. οὐ γὰρ ἦς δῆλον οὐδενὶ οὐδὲν
τῶν πραγμάτων οὖτε αὐτῶν ποθ' αὐτὰ
οὖτε ἄλλω πρὸς ἄλλο, εἰ μὴ ἦς ἀριθμὸς
καὶ ἁ τούτω οὐσία. νῦν δὲ οὖτος κατ-
τὰν ψυχὰν ἀρμόζων αἰσθήσει πάντα
γνωστὰ καὶ ποτάγορα ἀλλάλοις κατὰ
γνώμονος φύσιν ἀπεργάζεται συνά-
πτων καὶ σχίζων τοὺς λόγους χωρὶς
ἐκάστους τῶν πραγμάτων τῶν τε ἀπεί-
ρων καὶ τῶν περαινόντων.

The originating nature<sup>112</sup> of number is con-
ducive to knowledge, it is orientating and
instructing of whatever be unsolvable or un-
known. Indeed, if there were not number and
its essence, nothing whatsoever would be
clear to anyone, either on its own or in its
relation to other things. As it is, by **adapting**
in the soul all things to perception, number
makes things known and commensurate to
one another according to the nature of the cri-
terion (γνώμων),<sup>113</sup> while it joins and cleanly
splits the ratios of things, of the unlimited and
the limiting.

There is in this paragraph a remarkable doctrine of knowledge through a harmoniz-
ation between the human mind and external reality, and the harmonization is effected
by what in this case sounds like a cosmic principle of equilibrium, the γνώμων. This
gnomon means originally the style, or perpendicular, of a sundial, but its meaning ex-
panded in early Greek science to signify any ‘marker around which the intelligibility
of other things is constituted.’<fnref n="114" /> The reader will remember that the very first recor-
ded occurrence of στοιχεῖον in Greek literature has to do with the sundial, and that its
meaning there is ‘the shadow cast by the dial to mark time.’ I had suggested the sym-
bolic implications of this image,<fnref n="115" /> and this is what De Groot confirms, even though
apparently unaware of the στοιχεῖον connection, in his interpretation of this passage:
‘It is interesting that the gnômôn is not the sign; the shadow is... Gnômôn, the word,
functions in much the same way that Aristotle uses the word archê. Gnômôn does not
separate thought and thing. It brings them closer together.’<fnref n="116" />

ἔδοις δέ κα οὐ μόνον ἐν τοῖς δαιμονίοις You will find that the originating nature

<footnote n="112">112. For the interpretation of φύσις as a dynamic whole of origin, process and result, see Viltanioti, ‘Powers,’ 26.</footnote>

<footnote n="113">113. See D. Manolova, ‘Discourses of Science and Philosophy in the Letters of Nikephoros Gregoras’ (PhD diss., Doctoral School of History, Central European University, 2014), 196 and also T. L. Heath, *A History of Greek Mathematics*, 2 vols. (Oxford: Clarendon Press, 1921), i, 78.</footnote>

<footnote n="114">114. J. De Groot, *Aristotle’s Empiricism: Experience and Mechanics in the Fourth Century BC* (Las Vegas, Zurich, Athens: Parmenides Publishing, 2014), 346–8.</footnote>

<footnote n="115">115. Above, p. 29.</footnote>

<footnote n="116">116. De Groot, *Aristotle's Empiricism*, 347. There is a remarkable analogy with the cosmic role of the Islamic balance, *al-mīzān*, which I shall address in later chapters.</footnote>
καὶ θείοις πράγμασι τὰν τῷ ἀριθμῷ φύσιν
καὶ τὰν δύναμιν ἰσχύουσαν, ἀλλὰ καὶ
ἐν τοῖς ἀνθρωπικοῖς ἔργοις καὶ λόγοις
πᾶσι παντᾶ καὶ κατὰ τὰς δημιουργίας τὰς
τεχνικὰς πάσας καὶ κατὰ τὰν μουσικάν.

and power of number are compelling not
only in angelic and divine matters, but
also fully in every human practical and
theoretical affair, in all the arts of the
craftsman and in music.

This determining role played by number in practical creative activities such as the
'arts and crafts,'<fnref n="117" /> is in this context a clear extension of the demiurgic and artistic nature
of the cosmos itself, 'harmonised' as it is with the powers of the soul.

ψεῦδος δὲ οὐδὲν δέχεται ἀ τῷ ἀριθμῷ
φύσις οὐδὲ ἁρμονίᾳ· οὐ γὰρ οἰκεῖον
αὐτοῖς ἐστι. τὰς τῷ ἀπείρω καὶ ἀνο-
ήτω καὶ ἀλόγω φύσιος τὸ ψεῦδος καὶ ὁ
φθόνος ἐστί. ψεῦδος δὲ οὐδαμῶς ἐς ἀρι-
θμὸν ἐπιπνεῖ· πολέμιον γὰρ καὶ ἐχθρὸν
τᾶι φύσει τὸ ψεῦδος, ἁ δ' ἀλήθεια οἰ-
κεῖον καὶ σύμφυτον τᾶι τῷ ἀριθμῷ γε-
νεᾶι.

No falsehood is admitted by the originating
nature of number or by concord (ἀρμονία),
for it is alien to them. Falsehood and envy
belong to what is unlimited, unintelligible,
and absurd. Falsehood can in no way foster
number, which is by its originating nature
contrary and inimical to falsehood, while
truth is akin and cognate to the lineage of
number.

Apart from the specification and repetition of the power of number in its different
applications, the relation with veracity is noteworthy, and indeed related to the idea
of the *gnomon* as a criterion and enabler of true knowledge. ‘Number’ and truth are
kindred, οἰκεῖον καὶ σύμφυτον, connatural to each other, while number and falsehood
are enemies.

Fragment 23 (spurious)<fnref n="118" />

ἀριθμὸν εἶναι τῆς τῶν κοσμικῶν αἰω-
νίας διαμονῆς κρατιστεύοισαν καὶ αὐ-
τογενῆ συνοχήν.

Number is the controlling and self-generating
bond of the eternal continuity of the universe.

Alternatively, ‘number is the conquering and self-subsisting cohesion of the eternal stay of things...’<fnref n="119" /> The commanding or compelling nature of number is once again mentioned here, making of it a sort of universal power. Viltaniotis speaks in this connection (fr. 11) of ‘a power of connecting (κοινωνοῦσα), fitting together or combining (ἀρμόζων, συνάπτων), which is also the function of ἀρμονία.’<fnref n="120" />

<footnote n="117">117. This seems an apt translation for δημιουργία καὶ μουσική.</footnote>

<footnote n="118">118. Huffman, *Philolaus*, 355.</footnote>

<footnote n="119">119. See A. L. Pierris, 'Pythagoreanism in the Meno and Platonic Development,' in *Symposium Platonicum Therense* (Santorini, June 2001), 16.</footnote>

<footnote n="120">120. ‘Powers,’ 25.</footnote>