SOURCE: Introduction to The Euclidean "Division of the Canon," University of Nebraska Press, 1991, pp. 1-108.

[In the following essay, Barbera examines the evidence and scholarly opinion surrounding the issue of the authorship of Sectio Canonis, concluding that "it would be bold to assert definitely" that Euclid is or is not the author.]

The Division of the Canon … is an ancient Pythagorean treatise on the relationship between mathematical principles and acoustical truths. Composed largely in the style of Euclid's Elements of Geometry, the Division is handed down in three distinct traditions: (1) a semi-independent version in Greek, which is attributed to Euclid or to Cleonides; (2) a Greek version contained in the fifth chapter of Porphyry's commentary on Ptolemy's Harmonics; and (3) a Latin version comprising the first two chapters of the fourth book of Boethius's De institutione musica. Of the three traditions, the semi-independent version is the longest. The other two versions present portions of the long version as well as some significant textual variants.

The long version consists of an introduction, a series of mathematical propositions, a series of acoustical propositions, a passage devoted to the enharmonic genus, and a division of the canon. The Introduction is philosophical in character, connecting the existence of sound to pre-existent motion. Motions are multitudinous and therefore enumerable. Enumerable things of a single kind are related to one another by numerical ratio; thus pitches are so related. The Introduction then presents the fundamental principle of consonance: consonant notes are related to one another by either multiple or superparticular ratios. The mathematical propositions concern multiple, superparticular, and super-partient ratios. The acoustical propositions treat the basic musical intervals of the Greek system. The Enharmonic Passage locates the enharmonic lichanos and demonstrates that the enharmonic pyknon is not divis ible in half. Finally, the Canon locates the standing and movable notes of the immutable system in the diatonic genus.

The following list indicates the portions of the treatise transmitted in each of the three versions.

Euclidean Division

Introduction
Mathematical propositions
Acoustical propositions
Enharmonic Passage Canon

Porphyry's Commentary

Mathematical propositions
Acoustical propositions

Boethius's De institutione musica

Introduction
Mathematical propositions

Clearly, the Division is divided into sections, which almost certainly betrays the treatise's protracted composition. This sectional nature, along with the Euclidean style of presentation—statement, exposition, specification, machinery, proof, and conclusion—, have made the treatise susceptible to quotation by other music theorists since antiquity.

Over the centuries, the Division has attracted the attention of a remarkable number of scholars, including musicologists, classicists, mathematicians, and historians of science. This is so in part because the treatise, although brief, presents the essence of Pythagorean music theory, a theory that served as foundation for music theorists until the Renaissance. The attachment of Euclid's name to the treatise is doubtless also part of the reason for a good deal of this attention. In addition to numerous commentaries on the Division, there have been a series of editions and translations, beginning with Georgio Valla's translation of 1497 and Jean Pena's edition of 1557 and extending to the present.

There are several reasons for yet another study of the Division. First of all, there is a need for an edition of the long version that takes into account all of the sources. For example, previous editors of the treatise were ignorant of the manuscript Vaticanus gr. 2338 (hereafter Vc), which presents a text for the long version that is as early and as authoritative as that contained in any other manuscript. Furthermore, in establishing a text, previous editors have relegated the shorter versions of the treatise to a status inferior to that of the long version. By so doing, they have allowed their understanding of the very nature of the text to be skewed by medieval Byzantine redactions of the Division. In this edition, each of the three traditions has initially been placed on an equal footing, and three texts have been established rather than one. It has then been possible to determine that the Latin version provides important clues regarding the Greek text of the long version. Porphyry's text is of less help in this regard, although the transmission of that text displays intermittent influence from the text of the long version.

This edition presents not only the texts for all three traditions with all significant variant readings from all currently known manuscript sources but also all the diagrams and scholia that accompany these texts. A comparison of the three texts of this edition with previous editions reveals that the Greek texts of this edition differ significantly from the earlier ones, although the Latin text largely parallels that of the most recent editor of Boethius's De institutione musica, Godofred Friedlein.¹ New texts are deserving of new translations, and these have been provided for all three traditions, as well as for the diagrams and scholia. A running commentary accompanies the translations; in the long version, this provides not only a systematic study of the parts of the arguments but also the mathematical and acoustical background for the propositions. For example, students of the Division have long acknowledged the dependence of Proposition 2 on Euclid's Elements of Geometry 8.7, although no one has noted the dependence of Proposition 3 on Elements 7.33. In the shorter versions, the commentary provides cross references to the long version, as well as additional annotation.

The purpose of this edition is not limited to establishing and translating texts. Indeed, some of the most intriguing aspects of the Division are its complex, bilingual transmission, its malleable length and text, and its uncertain authorial attribution. By treating the question of authorship, we are eventually drawn into matters concerning length of text and transmission. In addition, it has been the partial aim of this study to chronicle the learning of basic mathematical and acoustical truths from antiquity to the Renaissance. Although the text is brief, its protracted composition and widespread dissemination in well over two hundred manuscripts affords the opportunity to discern several stages in the study of mathematics and acoustics. Perhaps the most interesting stage, as revealed by the Division, occurred in late medieval Byzantium. The reception and redaction of the Division provides a fairly precise measure of how that culture learned ancient science. Scholars of late Byzantium refashioned the text—and probably added diagrams—in order to make it comprehensible to them and perhaps to justify the attachment of Euclid's name to it. Attention of this sort at such a late date attests to the intellectual vitality of the material contained within the treatise.

Authorship

Who wrote the Division of the Canon? There exists a considerable amount of evidence in antiquity and the Middle Ages pertaining to this question. There also exists a wealth of scholarly opinion on the matter. Taken together, the evidence and opinion form a morass of clues and contradictions that address the subject of authorship. Documentary evidence, a survey of secondary literature on the subject, and a discussion of the sources that shed light on the authorial question are presented below.

The surviving texts represent in each case a final stage in the compositional process, although this process has continued to some degree since the Renaissance with the succession of editions and translations of the Division. The earliest stages of the development of this text stretch back into deep antiquity. They consist of early experiments with sound and early attempts to comprehend the visible and audible world. The figure who looms largest at these early stages is Pythagoras. It is hardly necessary here to add to the bulk of writing on that ephemeral figure,² although it is noteworthy that the Division may begin with a quotation from Pythagoras himself.³ Throughout the fifth century B.C., the followers of Pythagoras pursued acoustical matters that were originally "discovered" by their leader. To these pursuits, the Pythagoreans applied—or perhaps discovered in these pursuits—a numerical way of comprehending the world, for which they became famous. Preserved in much later treatises are remarks and writings attributed to Philolaus and Archytas (fifth and fourth century B.C. respectively) that relate these early mathematical-acoustical discoveries.

The stages leading up to these fragmentary remarks presumably consisted of observation and then reflection. At some subsequent stage, these discoveries were written down, although it is difficult to determine when that stage occurred. This difficulty draws us into the heart of the question regarding the authorship of the Division. With the texts of the Division, we have a relatively late stage in the compositional process, although even these underwent revision through the fourteenth century. Commitment to print, however, is not the only way this information could have been conveyed from person to person and generation to generation. Indeed, the Pythagoreans were famous for their sayings, their secret doctrines, and their adherence to a way of life. It is from this world of observation, reflection, comprehension, secrecy, and obedience that the Division emerges.

The treatise is short, even in its longest version, and yet it contains within itself the crosscurrents of its origins.⁴ We find in the Division a statement on the nature of sound, a qualitative yet numerical definition of consonance, a presentation of mathematical truths argued in strict Euclidean style, a series of acoustical truths containing a paralogism and an attack on the acoustical errors contained in the musical theory of a rival, and appendages that relate to the actual musical system of the ancient Greeks. An attempt to consider the authorship of such a treatise should take into account these crosscurrents. This is not a matter of scholarly lassitude, imprecision where precision may be possible, but rather a necessarily comprehensive view of composition.

Documentary evidence

A survey of the persons involved with the Division may provide a historical context for the question of authorship, if not a single answer to the question: Who wrote the treatise? Let us consider the Latin version first, for in this respect, it is the most straightforward. The Division appears at the beginning of the fourth book of Boethius's De institutione musica without authorial ascription. Boethius (480-524 A.D.) introduces it simply by exhorting the reader to review some matters before continuing. We might naively assume at this point that Boethius was the author of the Division, although it is doubtful that Boethius can be considered the author of much that appears in his treatise.⁵ Furthermore, the appearance of the Division in Porphyry's commentary on Ptolemy's Harmonics eliminates Boethius from consideration, since Porphyry lived two hundred years before Boethius. Nevertheless, the Latin version of the Division exists in the oldest physical documents, twenty-three ninth- and tenth-century manuscripts. The Latin version was also the most widely circulated: it exists in over 130 manuscript copies of the De institutione musica.

The situation is more complicated with the semi-independent version in Greek. Of thirty-three appearances, the treatise is directly or indirectly ascribed to Euclid twenty-one times. This is the case in the manuscript Venetus Marcianus app. cl. VI/3 (hereafter Mm), one of the two oldest codices containing this version of the Division. The ascription to Euclid in seventeen manuscripts, a group we shall call mu, stems from Mm. The other three manuscripts, a group we shall call za, present a different version of the treatise that is related to the first hand of Mm, Mm¹, by a hyparchetype γ,⁶ and za attributes the treatise to Euclid. In eleven codices, the Division is attributed indirectly to Cleonides, about whom we know nothing.⁷ Of these eleven attributions, one occurs in Vc, the other relatively old codex, which is the direct source for the attributions in Bononiensis gr. 2432 (hereafter Bb), Cantabrigiensis gr. 1464 (Gg.II.34) (hereafter Cb), and Neapolitanus gr. 260 (III.C.2) (hereafter Nn), and the source once removed for the attributions in Florentinus Riccardianus gr. 41 (K.II.2) (hereafter Fr), Parisinus gr. 2535 (hereafter Pe), and Parisinus gr. 3027 (hereafter Pf). The Division in the group of manuscripts we shall call jh represents a more complicated lineage, although the Cleonides attribution in jh may stem ultimately from Vc. Finally, the appearance of the Division in Vaticanus gr. 191, ff. 295r-296v (hereafter Vv) carries no authorial ascription.

Porphyry (ca. 232-ca. 305 A.D.), in his commentary, mentions Euclid as the author of a Division of the Canon … and also quotes two statements from the mathematical propositions of the Division (Propositions 3 [124.2-3] and 6 [134.2-3]).⁸ Later, Porphyry sets out principles sufficient to demonstrate the connection of the consonances with the multiple and superparticular ratios of the quartemary {1, 2, 3, 4}, and he cites Euclid's Division of the Canon as the source for a series of acoustical propositions.⁹ He lists the propositions, and then the Division appears. Absent are the Introduction, the Enharmonic Passage, and most important, the Canon itself, whence the treatise presumably receives its name. Several discrepancies exist between this presentation of the mathematical and acoustical propositions and the presentation of the long version of the Division, which with one exception always follows the Harmonic Introduction … attributed to Cleonides.¹⁰ For our present concern with authorship, Porphyry is an important source because he explicitly connects Euclid's name with the Division, and he presents enough of the Division to convince us that it shares a common ancestor with the longer text handed down in the codices. There still exists, of course, a six-hundred-year gap between Euclid, who lived around 300 B.C., and Porphyry.

Some uncertainty exists as to whether or not Porphyry is the author of the commentary on Ptolemy's Harmonics. In Vaticanus gr. 176 (hereafter Va) and Parisinus Supplementarius gr. 449 (hereafter Pj), the fifth and following chapters are ascribed to Pappus, the distinguished mathematician who flourished in Alexandria around 320 A.D.¹¹ Pappus also receives credit for the second appearance of the Harmonic Introduction in Vc, but the uncertainty between Porphyry and Pappus has little bearing on our deliberations here, since the two authors were nearly contemporaries.

The next mention of Euclid's name in connection with music comes from the late neoplatonic philosopher Proclus (d. 485 A.D.) and from his student Marinus of Neapolis.¹² Proclus, in his commentary on the first book of Euclid's Elements of Geometry, tells us that Euclid wrote several other works besides the Elements of Geometry, including an Elements of Music. …¹³ Marinus succeeded Proclus as head of the New Academy in Athens in 485, and in Marinus's commentary on Euclid's Data, he repeats Proclus's reference to an Elements of Music.¹⁴ Neither of these authors mentions a division of the canon, but the title Elements may be appropriate for the Division, since the treatise's presentation of the propositions is similar to the propositions of Euclid's Elements of Geometry.

Many Greek writings were preserved in Arabic during late antiquity and the early Middle Ages. Ikhwân al-Safâ', a tenth-century writer, lists Nicomachus, Ptolemy, and Euclid as musical successors to Pythagoras. The context of the remark convinced Henry George Farmer that the order from Pythagoras to Euclid was meant to be chronological, and he asserted: "it might be assumed that the Ikhwân recognized the late composition of the treatise ascribed to Euclid."¹⁵ The Ikhwân does not specify a Euclidean treatise, but another tenth-century source, the Fihrist of Ibn al-Nadîm, lists two such works: a Book on the Notes, also known as Book on Music (Kitâb al-nagham wa yu'raf bi'1-mûsîkî) and a Book on the Canon (Kitâb al-kânûn). Nearly three centuries later, Ibn al-Kiftî (d. 1248) mentions in his History of the Learned two works on music by Euclid: a Book on Music (Kitâb al-mûsîkî) and a Book on the Canon (Kitâb al-kânûn). Finally, Ibn Abî Usaibi'a (d. 1270) lists a book by Pythagoras entitled The Canon of Arithmetic and Music (Sharî'at al-arithmetîkî wa'lmûsîkî).¹⁶

A final reference to Euclid as author of a work on music, contemporary with our earliest manuscripts, comes from the waning years of the Byzantine empire. Theodorus Metochites (1270-1332)—chief minister to the emperor Andronicus II Palæologus, student of Manuel Bryennius, and teacher of Nicephoros Gregoras—mentions in his essays that Euclid touched on musical inquiries.¹⁷

Based on the documentary evidence presented thus far, we can conclude the following. The tradition deriving from γ ascribed the Division to Euclid, while that from a hyparchetype δ tacitly ascribed it to Cleonides.¹⁸ A third tradition deriving from the hyparchetype β apparently did not make an authorial attribution. Porphyry and later writers relate that Euclid wrote on music, although there are some discrepancies in these testimonies regarding titles and contents. Porphyry's testimony occurs in the fifth chapter of his commentary, the earliest source for which is the fourteenth-century codex Vaticanus gr. 198 (hereafter Vg). Study of the manuscript tradition of Porphyry's commentary in the present edition has been limited to the appearance of the Division in the commentary. From this perspective, the transmission of the commentary appears to be a quagmire, and testimony in any of the books after the Certainly first four must there be considered evidence, with existed some hearsay, perhaps suspicion.¹⁹ in the late ancient and early medieval Greek and Arabic worlds that Euclid wrote on music. Based on the documentary evidence, however, the connection of our treatise to the great geometer is tenuous. The transmission of the long version of the Division, the version that contains in some manuscripts an ascription to Euclid, paired it with the Harmonic Introduction now attributed to Cleonides. With the exception of za, the manuscripts that transmit the two treatises do so under one name: Cleonides, Euclid, or Anonymous.²⁰ Finally, there is the evidence, or rather the lack of evidence provided by the Latin version, where no authorial ascription is made.

Before turning to interpretations of the documentary evidence, we must consider the person or character known as Zosimus. A colophon inserted by the original scribe at the end of the treatise in Mm reports that Zosimus corrected the Division while prospering in Constantinople.²¹ This might lead us to conclude that Zosimus was the scribe.²² Matritensis gr. 4678 (here-after Za), which assigns the Division to Euclid, attributes the Harmonic Introduction to none other than Zosimus. These two citations invite the student of the Division to connect the two persons named Zosimus, but identifying this character has proven to be frustrating. Antiquity is littered with figures named Zosimus, including the famous historian, a sophist, a neoplatonist, a pope, a heretic, and a martyr. A list appears below, based principally on Fabricius's Bibliotheca graece.²³

1. Zosimus of Thasos, a writer of epigrams, some of which are still extant, who may have lived during the first century B.C.

2. An engraver from around the time of Jesus.

3. A freedman from the house of the younger Pliny (61-112 A.D.) who possessed talent as a comedian and musician.

4. The tutor of the rhetorician Aelius Aristides (117/128-181 A.D.).

5. A physician mentioned by Galen (129-199 A.D.).

6. Zosimus of Tharas, a third-century Bishop in Numidia.

7. A fourth-century Bishop of Naples.

8. Zosimus the Deacon, a heretic of the fourth century, whom both Stephen the Deacon and Nicephorus Callistus condemned for idolatry.

9. A martyr mentioned in a letter by St. Polycarp.

10. M. Canuleius, from Egyptian Panopolis, who taught and practiced alchemy around the turn of the fourth century.

11. A Prefect of Epeirus mentioned in connection with laws promulgated in 373 A.D.

12. Pope Zosimus, whose short reign (417-418) was marked by his defense of the controversial Caelestius and Pelagius against the Carthaginian synod and his subsequent denunciation of these two as heretics.

13. An abbot who lived in Palestine around 430 A.D. and wrote some [dialogismoi / arguments or calculations].

14. Zosimus of Alexandria, a philosopher mentioned by the Suda. Apparently he wrote a work on metallurgy, a life of Plato, and twenty-eight other books, none of which is extant.

15. Zosimus of Ascalon, known also as Zosimus of Gaza, a sophist and grammarian, who wrote a principles of rhetoric and a commentary on Demosthenes.

16. The famous Greek historian, whose history of the Roman Empire runs from the time of Augustus (d. 14 A.D.) to 410 A.D.

17. Zosimus the presbyter, late fourth and early fifth centuries, whom St. Isidore of Pelusium often admon ishes in his letters.

18. The recipient of a letter from Isidore of Pelusium (d. ca. 449 A.D.), who calls him a [bibliophoros], i.e., a [book] messenger or clerk, and who warns this Zosimus to take care with the arrangement of books, guarding against loss of damage.

19. Zosimus the hermit, recipient of a letter from Photius. This monk lived alone in the mountains.

From the group, two deserve further mention. The Zosimus of item 18 was connected in some way with books, and he is therefore a possible candidate for our redactor of the Division. But Isidore of Pelusium lived in Egypt, and we must wonder why he should exhort a messenger or bookkeeper in Constantinople to mind his work.²⁴ The Zosimus of item 10 taught and practiced alchemy around the turn of the fourth century and was apparently known to Michael Psellos in the eleventh century. Writings by this Zosimus occur in Venetus Marcianus gr. 299, an eleventh-century codex devoted primarily to alchemy and physics, which was part of Cardinal Bessarion's original bequest of 1468 to St. Mark's.²⁵ Several treatises or expansions of treatises that contain no specific authorial ascription also exist in this manuscript, one of which (ff. 90r-93r) makes a number of connections between the principles of alchemy and Pythagorean music theory. This portion of the codex was at one time thought to be part of the treatise of Zosimus of Panopolis, although Otto Gombosi has argued that it is independent and dates more likely from the seventh and not the fourth century.²⁶ The musical treatise, whether by Zosimus or Pseudo-Zosimus, contains no material from the pair of the Harmonic Introduction and the Division. Nevertheless, the connections are enticing. In fact, the anonymous treatise on music concerns itself with elements … of both alchemy and music, and Proclus and his student Marinus both noted that Euclid wrote an elements of music. This codex may be worthy of additional study for what it reveals about ancient music theory, although the relationship of its Zosimus to the Division's Zosimus is strained at best.

Although we may not be able to further identify Zosimus, we can speculate as to what he might have corrected. The pair of the Harmonic Introduction and the Division is highly sectional. The treatise or treatises consist of several short segments, some of which fit together better than others. The late Byzantine penchant for correcting ancient texts is well attested in scholia and letters, and it is conceivable that Zosimus pieced together any or all of the segments that we know.²⁷ Passages exist in the Division where we might desire more correction or at least more information²⁸, but rather than deal with fine points of meaning, let us look at an obvious mistake, a gaping hole in the version of the treatise as it appears in Vaticanus gr. 191, ff. 395v-397r (hereafter Vw). The text breaks off in this codex with two essential words to go in the first acoustical proposition (Proposition 10 [152.2]) and picks up again without break at the Canon (178.1). This same break occurs in Za, but there it is final. Zosimus the redactor may have "corrected" the lacuna of Vw, although we do not know if these are the very manuscripts in which the correction took place. The actual circumstances of this correction are unknown, and there are problems: in particular, we note that Vw contains the Canon, whereas Za does not. More puzzling is the manner in which Zosimus might have gone about correcting the faulty tradition. If he consulted the long version, he did not correct that version but repeated it. We can only speculate on what might have needed correction based on the corrected version. We can compare Mm with Vc where the treatises are ascribed to Cleonides, but there is very little difference in text, and we could hardly call one version a correction of the other, unless Zosimus rejected the ascription to Cleonides in favor of Euclid.²⁹ In any event, the only solid evidence we possess that indicates a need for correction is the lacuna in Za and Vw.

A second issue is the relationship between author of the Harmonic Introduction in Za and redactor of the Division in Mm. We might argue that the scribe of Za, upon seeing the colophon of Mm, mistook Zosimus for the author and wrote in his name at the beginning of the Harmonic Introduction.³⁰ The correlate to this codex, Vw, as well as Mm¹ do not attribute the Harmonic Introduction at the beginning of the treatise to an author nor do they give a title. Furthermore, the title of the Harmonic Introduction in Za is provided by a second hand. The colophon in Mm, however, gives credit to Euclid, not to Zosimus, for having written the Division. The scribe in Za might have consulted the beginning of Mm where there was no title or authorial attribution. It is difficult to believe, however, that the scribe of Za had Mm in front of him, since the latter codex contains the long version of the Division, whereas Za breaks off with the first acoustical proposition. It is possible that a similar circumstance took place prior to the execution of the manuscripts that we now possess, especially Za and Vw. Thus, the scribe not of Za but perhaps of γ made these corrections, and γ was subsequently mutilated to produce γ¹, from which Za and Vw descended. Even under this hypothesis, there must be additional steps to account for the resumption of the treatise in the Canon (178.1) in Vw.

A related matter concerns Zosimus the scribe as opposed to Zosimus the redactor. Some scholars have claimed that Zosimus was actually the scribe of Mm. Although this is possible, the colophon provides no clear evidence. Subsequent copying of the colophon by Italian scribes during the sixteenth century emphasizes the danger of such a claim.³¹ We must wonder, further, why the scribe would choose to sign his name at the end of the pair of the Harmonic Introduction and the Division rather than at the end of his work, the codex. Such a signature would make sense only if the Division were the only portion of the codex corrected by Zosimus.

Survey of secondary literature

With this examination of the documentary evidence on authorship, we are in a position to evaluate the arguments advanced over the centuries on the subject. In 1497, Georgio Valla published a Latin translation of the Harmonic Introduction together with the Division, all attributed to Cleonides, but Valla derived this attribution simply from the manuscript, Nn, that he used for his translation.³² Sixty years later, Jean Pena published his Latin translation of the same pair of treatises along with a Greek text, all ascribed to Euclid.³³ Pena's ascription, like Valla's, came simply from the manuscript employed.³⁴ Since the time of Pena, there have been many editions and translations of the Division, no doubt owing to the attachment of Euclid's name to the treatise.³⁵

Around the end of the sixteenth century, the father of Hugo de Groot was one of the first persons to raise the question of authorship, not so much for the Division but rather for the Harmonic Introduction.³⁶ Marcus Meibom related this uncertainty regarding authorship in the introduction to his edition and Latin translation of the Harmonic Introduction and Division.³⁷ In fact, Meibom reviewed the previous editors and translators of both texts, decrying their tendency to follow blindly the authorial ascription of a single manuscript. Meibom attributed both works to Euclid, based in part on his study of the sources. Moreover, he felt that the propositions of the Elements of Geometry provided the mathematical apparatus necessary to work out the propositions of the Division. Meibom was especially proud of his editorial work on the diagrams that accompany the Division, and no doubt saw these, as did later editors, as a significant connection between the music treatise and the great geometer.

David Gregory, who edited the works of Euclid around the turn of the eighteenth century, doubted that Euclid was the author of the Division and thought perhaps Ptolemy was the author.³⁸ A new edition of the works of Euclid was begun in 1883, and the editor, Johan L. Heiberg, was persuaded on the basis of style that the Division was truly composed by the author-compiler of the Elements of Geometry.³⁹ With Heiberg and his co-editor, Heinrich Menge, we arrive at the modern controversy regarding the author of the Division.

At the end of the nineteenth century, Karl von Jan published his monumental edition and study of ancient Greek music theory, the Musici scriptores graeci, and in this edition, he addressed the matter of authorship of the Division.⁴⁰ He noted that the Division contains expressions introduced by [esto / lit it be]; that variables ranging over numbers were indicated as [ho α, ho β, / the α, the β] and so forth; that [phemi de / I say in truth] occurs often; and that [ara / then, therefore] is ubiquitous. These four expressions stylistically connect the Division to the Elements of Geometry, where the same expressions occur. In addition, Jan noted that like the seventh book of the Elements, the Division assigned numbers to lines or, more precisely, represented numbers by line segments. The line drawings that accompany the Division had also engaged Meibom. As we shall see, however, the line drawings that accompany the Division were added to the oldest and most authoritative manuscripts after the treatise had been copied. In Mm, the later hand we shall call Mm² added the lines; in Vc, the fourteenth-century hand Vc³ added the drawings; and the line drawings in all other manuscripts containing them are derived from these two manuscripts. There are no such drawings in the tradition descending from γ or in the Porphyry tradition. Although diagrams do accompany the Division in the De institutione musica, these accompany the numerical examples that were added to the text and thus do not accompany the version of the text that also exists in Greek.

Jan also took up the matter of the other traditions of the Division, that is, in the commentary of Porphyry and the De institutione musica of Boethius. Noting the appearance of the mathematical and acoustical propositions in Porphyry as well as the style of their presentation, Jan concluded that this much of the Division was authentically attributed to Euclid.⁴¹ The Introduction, however, Jan doubted as part of the original work by Euclid. As an alternative, Jan speculated that Euclid extracted the propositions per se from a long work, an Elements of Music.

About ten years after Jan's treatment of the subject of authorship, Paul Tannery launched an extended attack on the ascription of the treatise to Euclid.⁴² Rather than dealing with issues of the three traditions or the connection of the Division to the Harmonic Introduction, Tannery based his argument on the content of the treatise. He determined that the mathematical quality, the sectional nature, and the historical location of the treatise indicated an author or authors other than Euclid.

Tannery's first criticism was that the Division contained "une contradiction flagrante" between the diatonic and enharmonic genera. The Canon is undertaken in the diatonic genus, whereas the Enharmonic Passage (172.1-176.2) presupposes the enharmonic genus. Tannery felt that the entire treatise except for the Canon was based on the enharmonic genus, but in fact the Introduction, the mathematical propositions, and most of the acoustical propositions are sufficiently general to apply to all three genera. The treatment of the whole tone (Proposition 8 and especially Proposition 11 [170.1-3]) has special application to the chromatic genus. Only with the introduction of the pyknon in the Enharmonic Passage must the enharmonic genus be assumed. Tannery saw in this discrepancy a conflict unworthy not only of the geometer but also of a single author. He relied on Aristoxenus's claim that the enharmonic genus had been favored in earlier times, that is, before the time of Aristotle and Aristoxenus (ca. 330 B.C.). This is a time well before Euclid, who flourished around 300 B.C. Tannery then speculated that the diatonic division of the canon was attached to the rest of the treatise a couple of generations after the time of Aristoxenus, perhaps around the time of Eratosthenes. Although this latter part of Tannery's argument is pure conjecture, his central criticism is valid. The assumption of the enharmonic genus is initially startling when it appears in the Enharmonic Passage, although it is possible to see the entire treatise as applying to the enharmonic genus. Equally surprising is the change in style within the Enharmonic Passage. From a series of propositions and demonstrations, the treatise shifts abruptly to the style of a manual describing the process for locating the enharmonic paranete and the lichanos. Abrupt shifts between proof and construction or demonstration also occur in Euclid's Elements of Geometry. The Canon, too, is a presentation of a method rather than a proof of a proposition. On the basis of tonal disposition, if not style alone, the end of Proposition 11 and the beginning of the Enharmonic Passage seem to be a likely point for a change in authorship.

Tannery next attacked the Introduction, noting that its claims rest solidly in the particular world of Pythagorean arithmetic. He noted that the fourth-century development of geometry and the accompanying generalization of mathematics is completely absent from the deliberations of the Division. Since this fourth-century development culminated in Euclid, Tannery concluded that Euclid could not have authored an Introduction devoid of the new, Eudoxian proportional theory. At this stage of his argument, however, Tannery conveniently overlooked the Elements of Geometry 7-9, which present in detail arithmetic proportional theory completely bound to the natural numbers and thus devoid of the advances in geometry. That these books were composed by Euclid as part of the Elements has never been seriously questioned. Historians of mathematics have long been aware that Books VII-IX contain knowledge of an earlier time, perhaps from around the time of Plato and Archytas. Nevertheless, we have no reason to doubt that Euclid was the mathematician who compiled and formulated this knowledge as part of his Elements. By analogy, the appearance of particular Pythagorean arithmetic in the Division cannot be used to exclude Euclid from consideration as author-compiler of our treatise. Tannery, of course, was aware of the nature of the Elements of Geometry 7-9 as well as of the dependence of the Division upon these books,⁴³ but he noted that these books are considered to be older than Euclid. The dependence of the Division upon them was therefore not an argument in favor of the Elements preceding the Division. Indeed, the Division never refers to the Elements by name, nor does it locate the propositions that it employs from the Elements by book or number. Tannery is therefore correct in asserting that the propositions of the Division need not have been composed after the composition of the Elements. But no part of this argument, which depends upon the age of knowledge, rules out Euclid as the possible and eventual compiler of the material.

Tannery proceeded to attack Proposition 11 (154.2-156.1), which he claimed contained a paralogism. This is true in a strict sense, for it argues that since an interval is dissonant, it is therefore not a multiple. This argument presumably rests on the claim in the Introduction that all consonant intervals are either multiple or superparticular. Of course, the Introduction never claims that all multiple intervals are consonant. The Division, however, tacitly relies on the quarternary {1, 2, 3, 4} for its discussion of consonance, and under this constriction, all multiples are consonant. A limitation to this quarternary rules out all dissonant intervals, since the four numbers can be combined only to form consonant intervals. In fact, these four numbers produce the ratios that represent all the consonances recognized by the Pythagoreans: 2:1 and 4:2 represent the octave; 3:1 represents the twelfth; 4:1 represents the double octave; 3:2 represents the fifth; and 4:3 represents the fourth.

Tannery concluded his argument by noting that the mathematical sophistication of the Division probably does not predate Plato and Archytas. Here Tannery is referring to the style of language and argument in the propositions, which is similar to that of the Elements. In conclusion, he suggested Eudoxus as a possible author for all of the Division except for the actual division according to the diatonic genus, which Tannery assigned to the third century B.C.

Charles-Émile Ruelle, who had earlier translated the Division into French,⁴⁴ responded briefly to Tannery's criticism.⁴⁵ Ruelle's response consisted of a terse summary of the contents of the long version of the Division, from which he argued that the treatise had integrity and was worthy of ascription to Euclid. Ruelle noted that the acoustical propositions are an application of the mathematical propositions to sound, and that this application is truly what the Introduction promises. In consequence, Ruelle criticized Jan for concluding that the mathematical and acoustical propositions were by Euclid, whereas the Introduction was not. Ruelle's argument makes sense as far as the acoustical propositions are concerned. Whether this much of the Division is by Euclid is another question, but this much of the Division can be read to have integrity. Ruelle never satisfactorily answered Tannery's argument regarding the juxtaposition of genera, nor did he address the abrupt shift in purpose in the Enharmonic Passage.

A decade later, Heinrich Menge published the last volume of Euclid's Opera omnia containing the Division, followed, as in Jan's edition, by the Harmonic Introduction. In the manuscripts, however, the Division follows the Harmonic Introduction. Although Menge's edition is essentially the same as Jan's, Menge addressed anew the subject of authorship.⁴⁶ Collecting the testimony of Proclus, Marinus, and Theodorus Metochites, Menge concluded that Euclid probably wrote on music, but he noted that none of these authors attributed a division of the canon to Euclid.⁴⁷ Menge concluded that the Division may be an extract from a work on music by Euclid, perhaps an Elements. Unlike Jan, who proposed that Euclid himself extracted the Division from his longer work on music, Menge felt that the Division was extracted by an author less able than the famous geometer.

The issue of authorship has continued to attract the attention of scholars throughout this century. Thomas Heath, the English translator of the Elements of Geometry, repeated the opinions of Jan, Tannery, and Menge without offering an opinion of his own on the subject.⁴⁸ Ingemar Düring, the most recent editor of the text of Porphyry's commentary, briefly reviewed the matter, noting that the position taken by the Introduction on the addition and subtraction of parts to alter pitch was old Pythagorean.⁴⁹ He also cited Heiberg's conclusion that the language of the Division was that of Euclid. Düring concluded, however, that the authorship of the Division was uncertain.

Ivor Bulmer-Thomas reviewed the testimony of Porphyry, Proclus, and Marinus.⁵⁰ He concluded that Euclid wrote an Elements of Music and that the Division had some connection with it. But he felt that the Division gave "such a trite exposition of the Pythagorean theory of musical intervals" that it "is hardly worthy to be dignified with the name Elements of Music. " Bulmer-Thomas was convinced that the Harmonic Introduction and the Division had different authors because one treatise contained Aristoxenian theory whereas the other contained Pythagorean doctrine.⁵¹ He rightly rejected, however, Tannery's argument that Euclid could not have been the author on the basis of the antiquity of the musical theory. As Bulmer-Thomas noted, Euclid preserved old-fashioned arithmetic in his Elements of Geometry and may have done the same with music theory.

Walter Burkert, in his review of the subject, focused attention on the invention of the canon, determining that the instrument was probably invented after the time of Aristotle.⁵² Contrary to Tannery's position, Burkert felt that the juxtaposition of enharmonic and diatonic genera was "comprehersible." He observed that the enharmonic genus may have been basic to musical practice, but the diatonic immutable system was a necessary basis for modulations … into other genera. Cleonides, however, seems to directly contradict this conclusion by noting that modulation can take place from any one of the genera to another.⁵³ Thus, the juxtaposition of genera, coupled with the change of purpose in the Enharmonic Passage, seems to indicate the piecing together of the Division. Burkert wisely noted in this passage his suspicion of the diagrams that accompany the Division.

In his translation of the Division into English, Thomas Mathiesen chose not to deal with the matter of authorial ascription.⁵⁴ Mathiesen reviewed the pertinent bibliography and then noted that the reliance of the Division on earlier works obviates the question regarding Euclid's authorship. This is true to the extent that the historian seeks to record the invention and development of ideas. In some cases, however, the processes of synthesis and compilation are the necessary and great steps of intellectual history. The Elements of Geometry attests to the importance of coordinating and unifying the discoveries of others.

Jon Solomon, in an appendix to his study of the Harmonic Introduction, addressed the authorial question regarding the Division.⁵⁵ Solomon had already grappled with the authorship of the Harmonic Introduction and decided to attribute it to the otherwise unknown Cleonides. Viewing this decision solely from the point of view of the Division, we might expect the same author for our treatise, since the Division always follows the Harmonic Introduction in the manuscripts.⁵⁶ The matter is complicated, however, by the appearance of the Harmonic Introduction independent of the Division, and in some of these appearances, the Harmonic Introduction is attributed to Pappus. Regarding the Division, Solomon reviewed much of the documentary evidence presented here. Although he wisely made no firm decision regarding the Division, he did determine that the Division and the Harmonic Introduction have different authors based on the disparity of style and content in the two works.

Citing the antiquity of much of the acoustical knowledge and many of the principles contained in the Division, Andrew Barker assigned the entire work to the late fourth century B.C.⁵⁷ He chose not to determine the role that Euclid might have played in the composition of the treatise but rather determined that there was no good reason to assign the propositions and the Enharmonic Passage to a date later than Euclid. In fact, Barker viewed the entire treatise with the exception of the Introduction as a single work.⁵⁸ His reasons for rejecting the Introduction rest largely on what he perceived as a discrepancy between the acoustical principles stated there and those worked out in the propositions. He noted that the Introduction treats sound as percussion in the air, with pitch dependent upon the density or rarity of that percussion. In the particular world of Pythagorean music theory, high numbers would be assigned to high pitches and low numbers to low pitches. Barker felt that this perception of sound was at odds with the one presented in the propositions, but he was misled by the line drawings that accompany these propositions. If we were to regard the line segments in the manuscripts as lengths of string, high numbers would be assigned to low pitches and low numbers to high pitches. Of course, we need not view the line drawings as such, and the invitation to do so comes only from the attachment of the Canon to the rest of the treatise. Rather than this, the lines can be viewed as representing the numbers or alphabetic variables that appear in the text, much like the line drawings that accompany Euclid's Elements of Geometry 7-9. But the entire question is obviated by the suspicious nature of the diagrams themselves. These drawings are not part of the arithmetic version of the Division and were added to our manuscripts by later hands.⁵⁹ The late addition of these drawings does not entirely rule out the possibility of their being old, but certainly they cannot be used to argue against the connection of the Introduction with the propositions.

Barker provided two reasons for viewing the Enharmonic Passage as integral to the other propositions. His primary reason was that the Enharmonic Passage along with Proposition 11 (164.3-170.3) formed a polemica 1 passage directed against the harmonikoi, who, according to Aristoxenus, had concentrated their energies solely on the enharmonic genus.⁶⁰ This is a novel interpretation of the Enharmonic Passage, but it does not account for the passage's change in style from that of mathematical and acoustical proofs to that of a manual. Barker may be correct in his interpretation of the Enharmonic Passage, but such an interpretation does not argue for its integral attachment to the preceding propositions. Proposition 11 (164.3-170.3), in the style of the other acoustical claims, is directed against Aristoxenus, if it is directed against anyone. Barker's second reason for treating the Enharmonic Passage as part of the preceding material is the special attention devoted by this section to notes—enharmonic paranete, lichanos, parhypate, and trite—that are not located from fixed notes at any of the intervals presented thus far by the Division. Such an argument, however, does not account for the attachment of the Enharmonic Passage to the previous propositions, since we would expect in such a passage a treatment of the other intervals, especially the chromatic paranete and lichanos, which are not located from the fixed notes by previously established intervals. Finally, Barker admitted that his argument for considering the Canon itself as an integral part of the treatise rested on the acceptance of the enharmonic sections as integral.⁶¹ Since the integrity of the Enharmonic Passage is suspicious at best, so is the Canon.

To conclude this survey of secondary literature on the subject of authorship, we should note two recently published translations of the Division. András Kárpáti, in his Hungarian translation of the Division, treated some of the matters presented here.⁶² Kárpáti concentrated in part on the development of the canon. He noted that many of the acoustical truths presented by the Division predate Euclid, but like Ruelle, he concluded that the long version was an integral work, leading up to the mathematical construction of the immutable system. Kárpáti also suggested Euclid as author of the Division. Luisa Zanoncelli, in her Italian translation, provided an even-handed survey of positions regarding the authorship of the Division without firmly deciding in favor of Euclid or against him as author.⁶³ Finally, Flora Levin has recently argued for the historical integrity of the Introduction with the ensuing propositions and for the attribution of the entire treatise to Euclid.⁶⁴

Interpretation

As this survey of secondary literature indicates, the authorship of the Division has attracted considerable scholarly attention over the centuries, due in large part to the ascription of the treatise to the famous geometer. A comparison of this attention with the minimal attention bestowed, for example, on the Harmonic Introduction of Gaudentius underscores this point. The act of attaching a name to a work or group of works—whether the name is Cleonides, Euclid, or Gregory the Great—is an important matter. Assigning the name of a person rather than a numbered or unnumbered anonymous makes the work seem more real, more palpable. Our attitude toward the work is psychologically transformed after we identify it with a "real" person with a name. In the case of the Division, there is much at stake in the authorial determination. First, there is the matter of dating. Second, Euclid's reputation as a mathematician is on the line. Third, the context in which we read the treatise is an issue.

Although many of the mathematical and musical principles embodied in the treatise seem to be old Pythagorean, perhaps dating from as early as the fifth century B.C., the compilation and partial synthesis of these principles may be considerably more recent. As noted above, the act of synthesis is an important intellectual event that the historian should not allow to be overshadowed by either the quest for a new idea or the age of the material synthesized. Some form of the treatise existed by the time of Porphyry, and this may have been the Division of the Canon that he attributed to Euclid. As Tannery observed, the style of mathematical presentation seems to be sufficiently sophisticated so as not to predate Archytas and Plato's Academy. Furthermore, the acoustical theory is an improvement over that of Archytas. A time after the turn of the fourth century B.C. therefore seems to be the earliest possible date for the composition of the Division. We are left, nevertheless, with a considerable amount of leeway. At least a half millennium separates Eudoxus, for example, from Porphyry, the apparent chronological limits for authorship.

Scholars have expended much effort over the years in attempts either to view the Division as an intelligent, unified treatise or to remove Euclid's name from it. These attempts attest to the importance of the second issue, Euclid's reputation. Tannery, Menge, and Bulmer-Thomas were quite simply embarrassed by the ascription of the Division to the great geometer, and each scholar developed his own argument why Euclid could not have composed the treatise. Ruelle and Kárpáti, on the other hand, saw in the Division a unified work with its ultimate goal the mathematical construction of the immutable system. Thus, they were willing to attribute the treatise to Euclid. But for historical, stylistic, and musical reasons, it seems that the unity of the Division ends in Proposition 11 (170.3), the point where Porphyry's quotation of the treatise ends. Beginning with the Enharmonic Passage, the long version becomes a manual rather than a logical treatise on the foundation of acoustics. And at this point the treatise abandons its general treatment of intervals—fourths, fifths, octaves, and tones, the ratios for which had been established in the mathematical propositions—to pursue the enharmonic genus. If the purpose of the Introduction and mathematical propositions had been to treat the intervals of the enharmonic genus such as the ditone and the semitone (256:243), we should find the necessary mathematics to represent these intervals with ratios. For example, in Proposition 11 (166.2-170.1), the Division demonstrates that the fourth is smaller than two-and-one-half tones, and the fifth is smaller than three-andone-half tones. This demonstration relies on an earlier part of Proposition 11 (164.3-166.2), which affirms that the diapason is less than six tones. This claim rests ultimately on the ninth mathematical proposition, which demonstrates that six sesquioctave ratios exceed the duple. It would seem proper that the next interval considered should be the semitone, that is, the amount by which a fourth exceeds two tones. The first part of the Enharmonic Passage (172.1-6) locates the enharmonic lichanos at this interval above the hypate, but the expected characterization of this interval as 256:243 (i.e., 4:3-[9:8 + 9:8]) is absent. The author of the Enharmonic Passage must have wanted to continue the treatment of successively smaller intervals begun in Proposition 11 (i.e., octave, fifth, fourth, tone [164.3]) with the semitone and enharmonic diesis. The absence of the appropriate mathematical propositions indicates, along with the other evidence cited above, that the Enharmonic Passage was attached to the preceding part of the Division at a time after the composition of the original treatise.

The third issue resulting from the question of authorship is the context in which we read the treatise. The divorce of mathematics from the corporeal and particular world of Pythagorean philosophy was one of the great achievements of the fourth century, which culminated in the general and abstract science of the Elements of Geometry.⁶⁵ Our perception of the Division varies considerably according to whether we perceive it as abstract theory to be applied impartially to the physical world or as the material display of philosophical and qualitative arithmetic. It is little wonder that a scholar such as Jan, having decided that the mathematical and acoustical propositions were by Euclid, was eager to separate the Pythagorean Introduction from the remainder of the treatise.

The documentary evidence concerning the authorship of the Division comprises an entanglement of clues and contradictions that invites a variety of interpretations and evaluations of the treatise. Some of these interpretations and evaluations presented by other writers on the subject have now been discussed. There are, however, additional matters that deserve some treatment.

The question whether Euclid or some other author could have written both the Harmonic Introduction and the Division is not as easily answered as might seem possible from the writings of earlier scholars. The empirical evidence, the manuscripts and ancient testimony, favors treating the two works as two parts of one treatise, as the scribe does overtly in Nn.⁶⁶ The long version of the Division always follows the Harmonic Introduction, and with the exception of Vv, which does not provide an author for either part, both parts are invariably attributed to the same author. Even in Za where the Harmonic Introduction is attributed to Zosimus at the beginning, it is later ascribed to Euclid by the original scribe. Arabic citations in the Fihrist of Ibn al-Nadîm and in the History of the Learned of Ibn al-Kiftî also ascribe a pair of works on music to Euclid, a Book on Music and a Book on the Canon. It is reasonable to assume that this Book on Music is our Harmonic Introduction. Against the association of the Harmonic Introduction with the Division, we can point to the independent appearance of the former in manuscripts and the appearance of the Division alone in the works of Porphyry and Boethius. Furthermore, Porphyry refers to a Division of the Canon by Euclid, and Proclus and Marinus to an Elements of Music, but none of these three authors mentions a Harmonic Introduction or a Book on Music. This absence of testimony linking the two works and the isolated appearance of either the Harmonic Introduction or the Division is insufficient evidence to overcome the union of these two works in the manuscripts containing the long version of the Division and the Arabic testimony. Indeed, most scholars who have argued for separate authors for these two works have proceeded from an entirely different position.

The Harmonic Introduction and the Division present significantly different music theories and seem to do so in contrasting styles. Many writers on the subject have found the Aristoxenian theory of the Harmonic Introduction to be incompatible with the Pythagorean theory of the Division. On several theoretical points (e.g., the size of the fourth), the two theories are simply contradictory. The Harmonic Introduction methodically presents the Aristoxenian system logically and peripatetically, in apparent stylistic contrast to the propositions of the Division. These propositions, similar to those of the Elements of Geometry, are unique in ancient music theory. The Introduction to the Division, however, which belongs with the propositions, is perhaps closer in style to the Harmonic Introduction than it is to the ensuing propositions. Solomon, who argued for separate authors on the basis of content, noted that the two works "are stylistically and lexically quite similar."⁶⁷ Thus the burden of proof for separating the two works seems to fall heavily on the contrasting musical theories contained in them.

This is not the place to reconcile Aristoxenian and Pythagorean theory. The later part of Proposition 11 (164.3-170.3), with its demonstrations that the octave is smaller than six tones, the fifth smaller than three and a half tones, the fourth smaller than two and a half tones, and the interval of the tone indivisible into two equal parts, seems to be a direct attack on Aristoxenian musical principles. The difficulty with separating the two works solely on the basis of content stems from the fact that the conjunction of these incompatible theories is hardly unique in antiquity. The juxtaposition of Aristoxenian and Pythagorean theories occurs in the musical treatises of Nicomachus, Aristides Quintilianus, and Gaudentius. In the latter two works, we find the explicit contradiction regarding the size of intervals, fourth, fifth, and so on.⁶⁸ Furthermore, the entire first section of the treatise by Gaudentius presents the Aristoxenian system, whereas the second section presents the elements of Pythagorean musical theory.⁶⁹ As for Nicomachus, he tacitly assumes the Aristoxenian system for his presentation of the genera, but then notes that the octave equals five tones and two semitones rather than six tones.⁷⁰ The embodiment of contrasting theories in these three works is certainly more integrated than in the pairing of the Harmonic Introduction with the Division, but this integration does not necessarily argue more strongly for one author or for a more capable author. Perhaps we should have higher esteem for an author who, when compiling contradictory musical theories, separates them into discrete parts than for one who jumbles them together. This argument, of course, does not necessitate that we view the Harmonic Introduction and Division as having one author. The argument, rather, can be turned around to call into question the integrity of the musical works by Nicomachus, Aristides Quintilianus, and Gaudentius. The combination of conflicting musical theories in a single treatise seems to confuse the issue of authorship, but in a more positive light, we can view the combination as a reflection of the compilatory and synthetic nature of our treatises. These treatises (considering here at least the pair of the Harmonic Introduction and the Division) were put together at a relatively late date when the animosity of the Aristoxenian-Pythagorean conflict had subsided for most writers.

We venture precariously into an argument based on the absence of evidence, but we must do so in the case of Ptolemy's testimony on the subject of authorship. In the fifth and sixth chapters of the first book of Ptolemy's Harmonics, he reviews the Pythagorean stand on consonance and dissonance.⁷¹ His treatment is thorough, and he takes up in the course of these two chapters the fundamental principle of consonance of the Division's Introduction and the ensuing treatment of musical intervals. Ptolemy even directs sharp criticism at the Pythagoreans for their seemingly arbitrary restriction of consonant ratios to the quarternary {1, 2, 3, 4} and for their rejection of the eleventh from the category of consonance.⁷² It is clear from Ptolemy's remarks that he had knowledge—and we are even tempted to say that he had a treatise—nearly identical to that presented by the Division but without the Enharmonic Passage and the Canon.⁷³ And yet at no point in the Harmonics does Ptolemy mention Euclid or a Division of the Canon. These omissions would figure less significantly were not Ptolemy careful to assign the theories that he presents to the appropriate persons and groups. Archytas, Aristoxenus, Eratosthenes, Didymus, and the Pythagoreans receive specific citation and criticism, but not Euclid.

On the basis of Ptolemy's remarks, we can conclude that he was well aware of the Pythagorean position on music theory that is presented in the Division and that he did not associate this position with Euclid or Cleonides. We can plausibly conclude that Ptolemy was likewise unaware of the specific Euclidean pre sentation of the material we find in the Division. It is therefore possible that the Division, as we know it up to Proposition 11 (170.3), had not been compiled by the time of Ptolemy. This late date of composition may be corroborated by the tenth-century Arabic treatise Rasâ'il Ikhwân al-Safâ', which identifies the musical successors to Pythagoras in apparent chronological order: Nicomachus, Ptolemy, Euclid. If these names refer to the Enchiridion, Harmonics, and Division, we have evidence for a late date of composition for the Division.

Additional evidence, although hardly confirmation, for late composition is provided by the connection of the Division to the Harmonic Introduction. Although we know nothing about Cleonides, we are familiar with a series of Aristoxenian treatises, all of which seem to date from late antiquity. It is not unreasonable to group the Harmonic Introduction with those treatises and to propose a date of second century A.D. or later for it. Furthermore, as Solomon noted, we know of no treatise entitled Introduction … before the first century B.C.⁷⁴ We can at least speculate that an author after the time of Ptolemy compiled musical knowledge that was undoubtedly old and representative of two different philosophical traditions. The result of this compilation may have been a peripatetic introduction to harmonics and a Euclidean formulation of the basic mathematical and acoustical principles of the Pythagoreans.

A consideration of the three texts of the Division has some bearing on the question of authorship. The most common interpretation of the documentary evidence has been that the long version of the Division was composed at a time a good deal earlier than Porphyry and Boethius and that the latter two authors then quoted from the longer work. Although this hypothesis may be correct, we cannot logically infer it from the texts as they have been handed down in the manuscripts. To confirm textually the traditional hypothesis, we should be able to derive the texts of Porphyry and Boethius from the long version. But only the mathematical propositions occur in all three traditions, and we shall see below that any two traditions can be read against the third.⁷⁵ A comparison of Porphyry's version and the long version of Propositions 10 and 11 provides little insight into the matter of authorship. Boethius's version of the Introduction, however, differs in important ways from the Greek text.⁷⁶ Of course, these textual analyses do not favor one author over another or one historical sequence over another. Rather, the textual tradition as preserved in the manuscripts cannot confirm any of the proposed hypotheses….

Conclusion

It would be bold to assert definitely that the Division was or was not written by Euclid. Certainly the at tachment of Euclid's name—the teacher of the Elements of Geometry⁹⁷—to this treatise indicates much greater esteem for its contents than ascribing it to Cleonides. This matter of authorship, of course, relates to dating, and this in turn relates to the sections of the treatise.

The various possible forms of the treatise and the contrast between Aristoxenian and Pythagorean musical theories are insufficient to rule out the possibility that both the Harmonic Introduction and the Division received more or less their final formulation by a single author. It is even possible that this formulation took place in late antiquity. There is evidence that the source for the first three books of Boethius's De institutione musica is a lost treatise by Nicomachus,⁹⁸ and we must wonder if the fourth book, which begins with the Division, might not have the same source. It is even remotely possible that Nicomachus was drawing from another work of his own on music.

The sectional nature of the Division argues for a protracted composition. After all, we are dealing here with basic truths of mathematics and acoustics, the kinds of truths that can and should be learned or at least memorized. And we are also dealing with Pythagoreans or neopythagoreans, who were famous for their slogans, secret societies, and adherence to … a rule or way of life. This bears on the composition of the Division: we can imagine a seed of acoustical truths and beliefs that sprouts propositions over a period of time, propositions that are instances in a musical system of the essential truths. This much musical theory could have been transmitted orally, although it was eventually formulated in Euclidean style and written down. But the commitment to print need not kill the seedling. Composition continues through recension and redaction, and the muddled state of affairs surrounding the Division attests to the longevity and dynamism of Pythagorean musical theory. The final stage of this intellectually dynamic theory took place in Constantinople during the waning years of the Byzantine empire. First, line drawings were added, perhaps from an old written source no longer extant, to supplement the propositions. Then the text was modified to accommodate the diagrams. We can only speculate whether the line drawings were added to lend credibility to the attachment of the geometer's name to the treatise.

Notes

¹ Boethius, De institutione arithmetica libri duo. De institutione musica libri quinque, ed. Godofred Friedlein (Leipzig: B. G. Teubner, 1867; reprint ed., Frankfurt: Minerva, 1966).

² The literature on Pythagoras and the Pythagoreans is enormous, but one might begin with Walter Burkert's Lore and Science in Ancient Pythagoreanism, trans. Edwin L. Minar, Jr. (Cambridge: Harvard University Press, 1972), the sources cited by Burkert, and the sources cited by Burkert's sources.

³ See pp. 48-49 infra.

⁴ One such crosscurrent within Pythagoreanism pitted the acusmatici against the mathematici. See Burkert, Lore and Science, pp. 192-207.

⁵ See p. 38 and n. 107 infra.

⁶ All these manuscripts and groupings of manuscripts are discussed on pp. 62-79 infra.

⁷ See pp. 36-37 infra.

⁸ The bracketed numbers refer to the page and line numbers of this edition.

⁹ Ingemar Düring, ed., Porphyrios Kommentar zur Harmonielehre des Ptolemaios, Göteborgs Högskolas Årsskrift, vol. 38/2 (Göteborg: Elanders, 1932; reprint ed., New York: Garland, 1980), pp. 92.29-93.2; 98.19.

¹⁰ See pp. 37-38 and 44-48 infra.

¹¹ On the question of authorship, see During, Porphyrios, pp. xxxvii-xxxix.

¹² Heinrich Menge provides a summary of attributions of the Division to Euclid in his edition (Euclidis Opera Omnia, vol. 8, Phaenomena et scripta musica, ed. Henricus Menge [Leipzig: B. G. Teubner, 1916]), pp. xxxvii-xxxviii.

¹³Procli Diadochi in primum Euclidis elementarum librum commentarii, ed. Godofred Friedlein (Leipzig: B. G. Teubner, 1873), p. 69.3.

¹⁴ Marinus of Neapolis, "Commentarius in Euclidis data," in Euclidis opera omnia, vol. 6, Data, ed. Henricus Menge (Leipzig: B. G. Teubner, 1896), p. 254.19.

¹⁵ "Greek Theorists of Music in Arabic Translation," Isis 13 (1929-1930): 330.

¹⁶ For the Arabic sources on music, see Amnon Shiloah, The Theory of Music in Arabic Writings (c. 900-1900), Répertoire Internationale des Sources Musicales, B/X (München-Duisberg: G. Henle, 1979). These sources are also listed in Henry George Farmer, The Sources of Arabian Music (Leiden: E. J. Brill, 1965). The sources appear below with their RISM number and with Farmer's number in brackets.

Ibn Abî Usaibi'a, Tawâ-lîf fî arithmâtîqî wa'l-mûsîqî (The Canon of Arithmetic and Music), no. 085 [p. xii].

Ibn al-Kiftî, Ta'rîkh al-hukamâ' (History of the Learned), no. 126 [239].

Ibn al-Nadîm, Kitâb al-fihrist (Book of the Index [to Literature]), no. 132 [188].

Ikhwân al-Safâ', Rasâ' il Ikhwân al-Safâ' (Treatises of the Brothers of Sincerity), no. 154 [192].

¹⁷Theodori Metochitae miscellanea philosophica et historica, ed. Christian Gottfried Müller and Theophilus [Gottlieb] Kiessling (Leipzig, F. C. G. Vogel, 1821; reprint ed., Amsterdam: A. M. Hakkert, 1966 [1967]), p. 108.

¹⁸ See pp. 62-64 infra.

¹⁹ I base my suspicion of fourteenth-century Byzantine sources on my findings about the emendations and recomposition of ancient writings by scribes and scholars in Constantinople. See André Barbera, "Reconstructing Lost Byzantine Sources for MSS Vat. BAV 2338 and Ven. BNM gr. VI.3: What Is an Ancient Music Treatise?" in Music Theory and Its Sources: Antiquity and the Middle Ages, ed. André Barbera (Notre Dame, Ind.: University of Notre Dame Press, 1990), pp. 38-67; and pp. 40-44 infra.

²⁰ Mm¹ presents a special case; see pp. 36-37 infra.

²¹ See pp. 37 and 185, n. 73 infra.

²² For example, Henry S. Macran, ed. and trans., The Harmonics of Aristoxenus (Oxford: Clarendon Press, 1902), p. 90, remarks that Mm was written by "one Zosimus in Constantinople in the twelfth century."

²³ Johann Albert Fabricius, Bibliotheca graece, 14 vols. (Hamburg: C. Liebezeit, 1707-1728).

²⁴Sancti Isidori Pelusiotœ … Epistolarum libri quinque, Patrologiae cursus completus, series graeca, ed. J.P. Migne, vol. 78 (Paris: L. Migne, 1860), Book III, Epistle 86.

²⁵ For a description of the manuscript, see Elpidio Mioni, Bibliothecae Divi Marci Venetiarum codices graeci manuscripti, vol. 1, Thesaurus antiquus: codices 1-299, Indici e cataloghi, n.s. VI (Rome: Istituto Poligrafico dello Stato, 1981), pp. 427-33. Venetus Marcianus gr. 322 and probably Mm, also part of Bessarion's collection, came to St. Mark's in 1472 after the Cardinal's death. On the various stages of the acquisition by the Marciana of Bessarion's library, see Lotte Labowsky, Bessarion 's Library and the Biblioteca Marciana: Six Early Inventories, Sussidi eruditi, vol. 31 (Rome: Edizioni di Storia e Letteratura, 1979).

²⁶ Otto Gombosi, "Studien zur Tonartenlehre des friihen Mittelalters III," Acta musicologica 12 (1940): 29-52.

²⁷ See, for example, the letter of Nicephoros Gregoras to Sebastian Kaloeidas in St. Bezdeki, "Nicephori Gregorae Epistulae XC," Ephemeris dacoromana. Annuario della scuola romena di Roma 2 (1924): xxm. See also Rodolphe Guilland, Correspondance de Nicéphore Grégoras (Paris: Société d'Édition "Less Belles Letters", 1927), no. 51.

²⁸ For example, the matter of "one name" (see pp. 55-58 infra).

²⁹ Mm and Vc are related in the fourteenth century, when hyparchetype α, in the form of Mm3 and Vc3, alters both codices. See Barbera, "Reconstructing," pp. 38-67.

³⁰ See, for example, the comments of Karl von Jan in Musici scriptores graeci (Leipzig: B. G. Teubner, 1895; reprint ed., Hildesheim: G. Olms, 1962), pp. xliii-xliv.

³¹ See p. 185, n. 73 infra.

³²Hoc in volumine hœc opera continentur. Cleonidœ harmonicum introductorium interprete Georgio Valla Placentino. L. Vitruvii Pollionis de architectura libri decemi. Sexti Iulii Frontini de aquœductibus liber unus. Angeli Policiani opusculum: quod panepistemon inscribitur. Angeli Policiani in priora analytica prœlectio. Cui Titulus est Lamia (Venice: Simon Papiens dictus Bivilaqua: Die Tertio Augusti, 1497).

³³Euclidis rudimenta musices, ed. and trans. Jean Pena (Paris: Andreas Wechelus, 1557).

³⁴ Pena used Upsaliensis gr. 52, Vaticanus Reginensis gr. 169, or their source k² (see rmr.K, pp. 72-74 infra).

³⁵ See pp. 98-101 infra. I shall consider here only those scholars who explicitly address the issue of authorship.

³⁶ Hugo Grotius, Martiani Minei Felicis Capellae … De nuptiis Philologiae & Mercurij … (Leiden: Christophorus Raphelengius, 1599), p. 316.

³⁷ Marcus Meibom, Antiquae musicae auctores septem graece et latine, 2 vols. (Amsterdam: Ludovicus Elzevirius, 1652; reprint ed. in Monuments of Music and Music Literature in Facsimile, 11/51, New York: Broude Brothers, 1977), preface. Each treatise in volume 1 of this publication is separately paginated.

³⁸Euclidis quae supersunt omnia, ed. David Gregorius (Oxford: Theatro Sheldoniano, 1703), preface.

³⁹ Johan L. Heiberg, Litterargeschichtliche Studien iiber Euklid (Leipzig: B. G. Teubner, 1882), pp. 52-55.

⁴⁰ Jan, pp. 115-20.

⁴¹ Jan numbers these propositions 1-16, as had Meibom, following the subdivision of the treatise that was de-veloped during the Renaissance (see p. 119, n. 7 infro).

⁴² "Inauthenticité de la «Division du canon» attribuée à Euclide," Comptes rendus des séances de l'académie des inscriptions et belles-lettres 4 (1904): 439-45; also in Paul Tannery, Sciences exactes dans l'antiquité, ed. J. L. Heiberg and H. G. Zeuthen, Mémoires scientifiques, vol. 3 (Paris: Gauthier-Villars, 1915), pp. 213-19.

⁴³ Proposition 2 (122.6-8), for example, appeals to Elements 8.7.

⁴⁴L'Introduction harmonique de Cléonide. La division du canon d'Euclide le géomètre. Canons harmoniques de Florence, Collection des auteurs grecs relatifs à la musique, vol. 3 (Paris: Firmin-Didot, 1884).

⁴⁵ "Sur l'authenticité probable de la division du canon musical attribuée à Euclide," Revue des études grecques 19 (1906): 318-20.

⁴⁶ Menge, Phaenomena et scripta musica, pp. xxxviiliv.

⁴⁷ The one ancient author who attributes a division of the canon to Euclid, Porphyry, presents a portion of the treatise, but not the Canon.

⁴⁸The Thirteen Books of Euclid's Elements, trans. Thomas L. Heath, 2d ed., 3 vols. (Cambridge: Cambridge University Press, 1926; reprint ed., New York: Dover, 1956), p. 17.

⁴⁹ Ingemar During, Ptolemaios und Porphyrios über die Musik, Göteborgs Högskolas Årsskrift, vol. 40/1 (Göteborg: Elanders, 1934; reprint ed., New York: Garland, 1980), p. 177.

⁵⁰ "Euclid," Dictionary of Scientific Biography, ed. Charles C. Gillispie (New York: Scribner's Sons, 1971).

⁵¹ See pp. 25-27 infra.

⁵² Burkert, Lore and Science, pp. 374-75, especially n. 22.

⁵³ Cleonides, "Isagoge," in Musici scriptores graeci, pp. 204.19-205.4.

⁵⁴ Thomas J. Mathiesen, "An Annotated Translation of Euclid's Division of a Monochord," Journal of Music Theory 19 (1975): 253, n. 1.

⁵⁵ Jon Solomon, "Cleonides: EIΣAΓΩΓH APMONIKH [EISAGOGE ARMON1KE / MUSICAL, HARMONIC INTRODUCTION]; Critical Edition, Translation, and Commentary" (Ph.D. dissertation, University of North Carolina at Chapel Hill, 1980), pp. 368-73.

⁵⁶ The one exception to this succession is the isolated appearance of the first four propositions in Oxoniensis Bodleianus Langbainius gr. et lat. 3, a seventeenth-century source.

⁵⁷ Andrew Barker, "Methods and Aims in the Euclidean Sectio Canonis," Journal of Hellenic Studies 101 (1981): 1.

⁵⁸ See also Barker's translation of the Division in his Greek Musical Writings, vol. 2, Harmonic and Acoustic Theory (Cambridge: Cambridge University Press, 1989), p. 190, where he makes similar claims about the authorship and integrity of the treatise.

⁵⁹ See p. 15 supra and pp. 40-44 infra.

⁶⁰ Barker, "Methods and Aims," p. 12.

⁶¹ Ibid., p. 13.

⁶² "Eukleidés: A Kanón Beosztàsa. Tanulmàny es fordítàs," Zenetudomànyi dolgozatok (1987): 7-27.

⁶³ "Sectio canonis," in La manualistica musicale greca (Milan: Angelo Guerini, 1990), pp. 31-36.

⁶⁴ Flora R. Levin, "Unity in Euclid's 'Sectio canonis,'" Hermes 118 (1990): 430-43.

⁶⁵ For more information on the generalization of mathematics during the fourth century B.C., see Part I of André Barbera, "The Persistence of Pythagorean Mathematics in Ancient Musical Thought" (Ph.D. dissertation, University of North Carolina at Chapel Hill, 1980), pp. 1-146.

⁶⁶ See p. 37 infra.

⁶⁷ Solomon, "Cleonides," p. 111 and n. 9.

⁶⁸ Cf. 1.8 with 3.1 in Aristides Quintilianus, De musica libri très, ed. R. P. Winnington-Ingram (Leipzig: B. G. Teubner, 1963); and p. 339.7-16 with p. 342.7-8 in Gaudentius, "Harmonica Introductio," in Musici scriptores graeci, pp. 327-55.

⁶⁹ The first section in Musici scriptores graeci, pp. 327-339.20; the second section, pp. 339.21-344.24.

⁷⁰ Nicomachus, "'Aρμονικòν ὲγχειρίδιον [Harmonikon Enchiridion /Harmonic Handbook]," in Musici scriptores graeci, pp. 262.7-264.5.

⁷¹ Ingemar Düring, ed., Die Harmonielehre des Klaudios Ptolemaios, Göteborgs Högskolas Årsskrift, vol. 36/1 (Göteborg: Elanders, 1930; reprint ed., New York: Garland, 1980).

⁷² See André Barbera, "The Consonant Eleventh and the Expansion of the Musical Tetractys: A Study in Ancient Pythagoreanism," Journal of Music Theory 28 (1984): 201-3.

⁷³ Ptolemy's subsequent treatment of the canon and his presentation of the enharmonic genus, which he attributes to the Pythagorean Archytas, does not parallel the final passages of the long version of the Division.

⁷⁴ Solomon, "Cleonides," p. 169.

⁷⁵ See pp. 44-48 infra.

⁷⁶ See pp. 55-58 and 228-33 infra….

⁹⁶ The sources for the De institutione musica are of no help in determining the author of the Division.

⁹⁷ Porphyry refers to Euclid as [stoicheiotes / author of the Elements] (During, Porphyrios, p. 92.29).

⁹⁸ For studies of the sources of the De institutione musica, see Calvin Bower, "Boethius and Nicomachus: An Essay Concerning the Sources of De institutione musica" Vivarium 16 (1978): 1-45.