Early Christian Sources of Platonic Geometry: the Latin Encyclopedists

In the foregoing sections, it has been shown that the passage of Platonic and Neoplatonic thought from Alexandria to Asia Minor and the Greek Fathers produced writings of sufficient importance that they were immediately to find their way to the West, reaching not only Ambrose in Milan but Rome as well, whence the teachings of Origen were to spread in the translations of Rufinus. In the meantime, however, Neoplatonism had already reached Rome with the arrival of Plotinus in 244.

It was Plotinus’s pupil, Porphyry (233-c.300), a leading Neoplatonist himself, who edited the work of his master and whose own writings included commentaries on Timaeus and apparently the Elementa as well. Writing in Greek though living in Rome, he was to be highly regarded by Augustine as a pagan philosopher, though this was a tribute Augustine was to qualify because of Porphyry’s anti-Christian stance, manifest for example in his treatise Adversus Christianos.

Marius Victorinus, who taught rhetoric in Rome and became a Christian convert in the middle of the fourth century, translated writings of Plotinus and other Neoplatonists into Latin which may have numbered among the works of Victorinus recorded by Alcuin at York. He was in touch with
Simplicianus, the priest in Milan who prepared Ambrose for baptism the year before the latter became bishop of that city in 374.

At about the time of Victorinus’s conversion, Chalcidius was producing his celebrated Timaeus translation and commentary, copies of which, as already stated, were a necessary possession for all medieval libraries of note.
This was in spite of the text ending prematurely at a point which immediately precedes Plato’s treatment of the elements and the regular solids. Nevertheless, the work remained the most important of all Platonic sources for the Latin middle ages.

This was followed by two more Platonic works of hardly less importance, namely the Commentarii in Ciceronis Somnium Scipionis of Macrobius and Martianus’s De nuptiis Philologiae et Mercurii. Macrobius’s Commentarii,
second only to Chalcidius’s, was written late in the fourth century or early in the fifth based on a lost commentary on Timaeus by Porphyry. Yet it is less a commentary than an encyclopedia of Neoplatonism illustrated with diagrams. Its starting-point is Plato’s Republic,
Cicero’s original work also being entitled De republica,
and it is with Scipio’s Dream that he ends it as an obvious counterpart to Plato’s Vision of Er. Martianus’s De nuptiis was approximately contemporary, being written between 410 and 439, and uses the allegorical marriage between Philology and Mercury as a setting for summarizing the seven liberal arts, in which each appears personified as a bridesmaid at the wedding.

Whilst Somnium Scipionis is among the works most frequently referred to in early medieval manuscripts and is itself among the most common manuscripts from that time, so De nuptiis was perhaps the most widely used schoolbook, its popularity during the ninth and tenth centuries being matched by that of Somnium Scipionis possibly from early in the tenth. Their influence in transmitting Plato’s cosmology was second only to Chalcidius partly as a result of expositions of number theory that deal not only with numerical relationships but with their powers as well, the attributes of the Pythagorean decad for example and the discovery of his musical ratios being relayed extensively in medieval literature. This is not to say that the transmission was exact and unvarying since tradition was always open to interpretation and development. For example, Plato’s and Clement’s concept of 7 planets revolving around an eighth, which is Earth, becomes in Somnium Scipionis 7
planets revolving within an eighth which is an all-encompassing celestial sphere. This is composed of 5 zones. Because justice is even-handed, to Clement it was represented by 4, by Martianus 2 and by Macrobius 8. 7, being a virgin number, is identified with Pallas Athene. In De nuptiis, 9 is also a perfect number and signifies the Muses. Nevertheless, it seems fair to say that these are additions to, not an undermining of, the basic precepts, for these remained those of Pythagoras and Plato as recognized by Martianus.

Meanwhile the august company of the gods… acknowledged [Arithmetic] herself… to be in very truth the procreator of the gods. And the host of philosophers, too, who stood nearby – in particular, Pythagoras, with all his disciples, and Plato,
expounding the cryptic doctrines of his Timaeus – worshipped the lady with words of mystic praise….

Martianus, De nuptiis 803

According to this whole tradition as transmitted, 1 is confirmed as the monad and the generator of numbers. 2, being the first departure from unity, represents discord and is the female number because it lacks a middle term.
3 is male because it possesses a middle term and is therefore the first number that is wholly odd. In other words, because,

1 + 1 + 1=3,

it is the first number comprising a mean and two extremes; it also stands for the triangle and the three divisions of the soul.
By the same reasoning, 4 is the first number wholly even because it is the first consisting of two means; it is the terminal number of the tetrad as well as that of the geometric elements of point, line, plane and solid; it represents the quadrangle and the
4 elements and seasons.

The pentad comes next, the number assigned to the universe. This identification is reasonable, for after the four elements, the universe is a fifth body of a different nature.

Martianus, De nuptiis 735

To this Macrobius adds that 5,

. . . alone embraces all things that are and seem to be

Macrobius, Commentarii, 1.6.19

Yet in conveying Plato’s association of the macrocosm with the human microcosm, Martianus adds that 5 also stands for marriage, being the sum of the male and female numbers, as well as the sum of the human senses. 6 is a perfect number because it is the sum of its parts. In other words,

1 x 2 x 3 = 1 + 2 + 3 = 6

Moreover, it is the product of the male and female numbers and so signifies creation. Because 7 begets no numbers in the decad,
it is virgin; as the sum of 3 + 4, it is the number by which the World-Soul is generated, according to Timaeus;
and, the Moon being the seventh planet, it also relates to the phases of the Moon measured in 7-day periods and the lunar stages of each month.
8 is the first cube and is perfect because it has 6 surfaces.

(10 is) the highest degree of perfection of all numbers…

Macrobius, Commentarii, 1.6.76

It contains within itself all numbers with their varied attributes and degrees of perfection…

De nuptiis 742

Interestingly, it is important to note that Martianus has Geometry preceding Arithmetic at the wedding. Her Book is the longest in the work and contains more geography than geometry, yet it does incorporate a ten-page summary of Euclid’s Elementa. Here a classification of angles, planes and solids leads to a description of how solids are generated from planes. Having described the basic solids, among which are found the pyramid and cube, Martianus concludes with the ‘noble’ figures of the octahedron, dodecahedron and icosahedron. Not surprisingly, geometric thinking finds its counterpart in arithmetic for, just as the sphere is recognized as containing all other figures, and in particular the regular solids, so 10 contains all numbers. And since geometric solids are recognized as being based upon plane figures, this would explain Macrobius’s allusion to 5 embracing all things.
Because 5 is the first figurate number of the pentagon, which is the plane figure of the dodecahedron which signifies the universe, Macrobius seems clearly to be associating 5 directly with the macrocosm, as indeed does Martianus.

Had Augustine written his text-books on the liberal arts, he would undoubtedly have belonged to the encyclopedic tradition of his near-contemporaries Macrobius and Martianus. As it is, his importance for the present study is arguably even greater, not only for transmitting Platonic thought within a theological framework but also for securing thereby its acceptance by the Church. Consequently, his contribution will be considered next.

Autore: Nigel Hiscock
The Wise Master Builder. Platonic Geometry in Plans of Medieval Abbeys and Cathedralsl
: Ashgate
Luogo: Aldershot
Anno: 2000
Pagine: 61-64

Early Christian Sources of Platonic Geometry: Clement of Alexandria (1)

It was the cultural crucible of Alexandria that initiated early Christianity to Platonic thought. In the first century of the new era, the writings of Philo Judaeus, a Greek-speaking Jew, attempted the conciliation of the Hellenic and Judaic traditions of learning, a process continued by Clement (c.150-c.215) in the next century. As a Greek philosopher and convert to Christianity, Clement compared both traditions with each other and with the emergent teachings of Christianity. As a result, he was the first to appreciate how much the writings of Plato and the evangelists John and Paul had in common. Recognized as the leading Christian scholar of his day and the foremost exponent of natural philosophy from the Christian point of view prior to Augustine, Clement was able to produce the synthesis of classical and Christian thought which gave birth to Christian Platonism. This was developed by his most famous pupil, Origen, and provided the theological foundations for the writings in the East of the Cappadocian Fathers as well as for Augustine in the West. Although it has to be admitted that Clement’s own work might not have been read much in the middle ages, his teaching was nevertheless to reach the West through Cassian’s adaptations in the fifth century which were soon to be consulted by Benedict himself. Through Origen’s writing, it was also to reach Ambrose in Greek and thence Augustine, along with translations by Rufinus, some of which were also known to Augustine. The result of this was to be the permanent acceptance of Christian Platonism by the Latin Church. In these early centuries, it not only answered the pagan reaction of Porphyry and Julian, it also gave Christian faith its intellectual content and, such was its enduring appeal, it was to reach its culmination in the cathedral school of Chartres nine hundred years after Clement died.

Clement’s forum was the Didascaleon, or Catechetical School, in Alexandria of which he became head. It had only recently been opened by Pantaenus, also a Christian convert and a Pythagorean, with the purpose of promoting Christian studies for educated converts in opposition to the paganism of Alexandria’s Museum and the esoteric cabalism of the Gnostics. Accordingly, the methods of classical philosophy were applied to a curriculum
which included philosophy, mathematics and scripture. Education was the path to knowledge, or gnosis,
which led to freedom. The union of knowledge with ‘right reason’ led to virtue just as the union between the human and divine spirit resulted in love. Thus Plato’s three-part division of rational, moral and natural philosophy found its place in Clement’s school.

For the uneducated, a state of grace was still possible through the acceptance of faith but from them, however, gnosis should be concealed:

For Plato also thought it not lawful for ‘the impure to touch the pure.’ Thence the prophecies and oracles are spoken in enigmas, and the mysteries are not exhibited incontinently to all and sundry, but only after certain purifications and previous instructions.

Clement, Stromateis V.4

… even those myths in Plato . . . are to be expounded allegorically, not absolutely in all their expressions, but in those which express the general sense. And these we shall find indicated by symbols under the veil of allegory.

Stromateis V.9

Clement found as much authority for obfuscation in the scriptures:

But since this tradition is not published alone for him who perceives the magnificence of the word; it is requisite, therefore, to hide in a mystery the wisdom spoken, which the Son of God taught. . . . because, ‘even now I fear,’ as it is said, ‘to cast the pearls before swine, lest they tread them underfoot, and turn and rend us’ (Matthew VII.6). For it is difficult to exhibit the really pure and transparent words respecting the true light, to swinish and untrained hearers. For scarcely could anything which they could hear be more ludicrous than these to the multitude; nor any subjects on the other hand more admirable or more inspiring to those of noble nature.

Stromateis 1.12

In retrospect, such determined concealment of gnosis from the uninitiated might arguably be confused with the secret societies of the Gnostics themselves, particularly since Clement often refers to the followers of ‘the true philosophy’ as Gnostics. However, this would be a modern misperception, since Clement had simply decided to combat Gnosticism with his own invention of Christian Gnosticism. When he writes:

Then [the Preaching of Peter, an apocryphal book] adds: ‘Worship this God not as the Greeks’ – signifying plainly, that the excellent among the Greeks worshipped the same God as we, but that they had not learned by perfect knowledge that which was delivered by the Son.’

Stromateis VI.5 [my italics]

Clement makes clear his regard for gnosis as ‘perfect knowledge’, as opposed to the arcane superstitions of the Gnostics.

Clement’s three main works constitute a progression in which the acquiring of gnosis leads to an understanding of Logos, the Word. Protreptikos exhorts the reader to renounce paganism; Paedagogus instructs him in Christian ethics; whilst the major part is Stromateis, a miscellany of essays devoted to a higher knowledge of God and his creation. In these works he repeatedly refers to and quotes from Timaeus and other Dialogues as well as scripture. Of the Protreptikos and Paedagogus,
a tenth-century manuscript has been noted in the Bibliotheque Nationale in Paris together with an eleventh-century manuscript of Stromateis in the Laurentian Library in Florence.

The origins of philosophy are succinctly stated by Clement:

From Pythagoras Plato derived the immortality of the soul; and he from the Egyptians.

Stromateis VI.2

However, the composition of the universe and the nature of the 4 elements had, even since Aetius, become somewhat muddled. Although he writes:

And indeed the most elementary instruction of children embraces the interpretation of the four elements ….

Stromateis V.8

And Athamas the Pythagorean having said, ‘Thus was produced the beginning of the universe; and there are four roots – fire, water, air, earth: for from these is the origination of what is produced’. . . .

Stromateis VI.2

he continues,

Empedocles of Agrigentum wrote:

‘The four roots of all things first do thou hear – Fire, water, earth, and ether’s boundless height: For of these all that was, is, shall be, comes.’

Stromateis VI.2

Nevertheless, despite an apparent confusion between ether and air here, Clement himself seemed clear enough in his previous chapter when he repeated the colours associated with the 4 elements – blue for air, purple for water, scarlet for fire and linen for earth.
Interestingly, this reveals that the atmospheric elements are chromatically – as well as physically and geometrically – related to each other and distinct from the element earth. For just as fire can cause water to evaporate into air and cooling can cause water to condense in air, so the purple of water is a synthesis of the red and blue of fire and air and their geometric solids are also relations of each other in that they are each enclosed by the regular triangle, quite distinct therefore from the colour and cube of earth. Plato’s relation of the macrocosm of the universe to the microcosm of the human also seems preserved by Clement, particularly when it is remembered that the number 10 was equated with perfection.

And the perfect inheritance belongs to those who attain to ‘a perfect man,’ according to the image of the Lord.

And there is a ten in man himself.

Stromateis VI.14, 16; see also V.6

In this passage, Clement then refers to the 5 senses and adds to them another 5, namely power of speech, power of reproduction, spirit received through creation, rule of the soul, rule of the Holy Spirit through faith.

Thus the Platonic Christian appears complete, the conjunction of the two traditions seeming to be effortless.

If then we consider, virtue is, in power, one. But it is the case, that when exhibited in some things, it is called prudence, in others temperance, and in others manliness or righteousness. By the same analogy, while truth is one, in geometry there is truth of geometry; in music, that of music; and in the right philosophy, there will be Hellenic truth. But that is the only authentic truth, unassailable, in which we are instructed by the Son of God.

Stromateis 1.20

The synthesis of the two traditions was evidently derived at least partly from the belief that Plato himself had had sight of certain scriptures. Several of Clement’s essays are devoted to the theme of Greeks borrowing from Hebrews, or of the two traditions at least coinciding. Nevertheless they were still distinguishable:

Rightly, then, to the Jews belonged the Law, and to the Greeks Philosophy, until the Advent ….

Stromateis VI. 17

That scripture associated the Law with 10 is evident above all in the Decalogue, or Ten Commandments. More than being a mere list of rules, however, the Commandments were regarded as an image of heaven and, in this aspect, they are to be identified with the same Pythagorean number of perfection. As Clement writes:

But law is the opinion which is good, and what is good is that which is true, and that which is true is that which finds ‘true being,’ and attains to it. … In accordance with which, namely good opinion, some have called law, right reason, which enjoins what is to be done and forbids what is not to be done. . . . That ten is a sacred number, it is superfluous to say now.

Stromateis 1.25, VI. 16

Therefore just as the equation of the law with perfection appears safe so, it seems, can references to law and perfection be equated with 10.

Autore: Nigel Hiscock
The Wise Master Builder. Platonic Geometry in Plans of Medieval Abbeys and Cathedralsl
: Ashgate
Luogo: Aldershot
Anno: 2000
Pagine: 50-54

La musica e il sapere enciclopedico: Cassiodoro

L’opera più ampia di Cassiodoro, le Institutiones divinarum et saecularium litterarum, scritta ormai in età avanzata per i monaci di Vivarium e giuntaci in tre diverse redazioni, include un libro, il secondo, interamente dedicato alle arti liberali, noto come De artibus et disciplinis liberalium litterarum. Qui Cassiodoro paragona le sette arti ai sette pilastri del biblico tempio di Salomone, il tempio della Saggezza (Proverbi IX, 1). L’immagine, che sarà universalmente adottata nell’alto Medioevo per inquadrare il rapporto fra la filosofia e le discipline liberali, pone senza dubbio le sette arti a fondamento di ogni umano sapere, ma, così facendo, ne delimita lo scopo e l’utilità. Abbracciando l’ideale di una sapienza tutta fondata sulla Bibbia, i contenuti disciplinari si esaurivano infatti nell’impegno interpretativo di passi biblici che, quanto alla musica, alludono al canto, agli strumenti musicali, alla perfezione dell’opera della creazione e alla rispondenza fra l’armonia del creato e l’armonia interiore. Questa tendenza, che Cassiodoro per primo manifesta, «di rifare l’opera di Varrone ad uso dei cristiani», come dice Gilson, fu a tutti gli effetti obiettivo condiviso dagli ecclesiastici e dai monaci eruditi delle successive generazioni, come Isidoro di Siviglia e Beda, nel “buio” dei secoli VII e VIII.

Un mito storiografico oggi assai dibattuto è quello della cultura boeziana di Cassiodoro. Limitando lo sguardo alla disciplina che qui interessa inquadrare, risulta infatti difficile da spiegare la sostanziale ignoranza che Cassiodoro dimostra nei confronti del De institutione musica, soprattutto se si tiene conto del suo esplicito richiamo alla traduzione boeziana del musicus Pitagora. Boezio non “tradusse” Pitagora se non in senso metaforico, poiché il De institutione musica esordisce nel nome di Pitagora, innestandosi sulla tradizione della musica speculativa platonico-pitagorica trasmessa da Nicomaco; in tal senso possiamo giustificare l’allusione di Cassiodoro. Tuttavia, la distanza fra il contesto teoretico musicale boeziano e l’inquadramento deIla musica offerto da Cassiodoro nel secondo libro delle sue Istituzioni è notevole. Qui il monaco di Vivarium elenca molti musici del passato: i greci Alipio, Euclide, Tolomeo, e soprattutto Gaudenzio, che egli afferma di conoscere nella traduzione latina oggi perduta di Muziano, i latini Albino – il cui trattato sulla musica, citato anche da Boezio, è andato perduto – e Censorino (la sezione musicale nel De die natali), ma anche Varrone e infine Agostino: «Infatti anche Agostino scrisse sei libri sulla musica, nei quali mostrò che la voce umana può naturalmente avere suoni ritmici e un’armonia modulabile in sillabe lunghe e brevi». Boezio non c’è nell’elenco, e le notizie che Cassiodoro trasmette sulla disciplina musicale sono abbastanza scarne. La definizione della musica come scientia bene modulandi deriva da Agostino o da Censorino, così come ne deriva l’accenno all’armonia dell’universo e all’idea che il ritmo cardiaco è regolato da ritmi musicali. Il resto della discussione sulla musica verte sulla triplice divisione in armonica, ritmica e metrica, probabilmente ricavata da Alipio, ma che abbiamo incontrato anche in Marziano e, anche se non come definizione, in Agostino. Da Gaudenzio proviene poi la discussione sulle sei consonanze (Cassiodoro vi include infatti l’intervallo di undicesima, ottava + quarta, che Boezio aveva invece rifiutato), mentre la descrizione dei 15 toni, presentati nello stesso ordine in cui appaiono in Aristide Quintiliano, probabilmente è conosciuta attraverso Albino. La discussione sulla teoria musicale è quindi ridotta a poca cosa, e non sembra organicamente connessa all’esposizione sulla struttura dei saperi, che apre il secondo libro delle Istituzioni. Qui infatti Cassiodoro segue più da vicino il De arithmetica di Boezio, che affianca ad altre fonti, come Rufino, proponendo una distinzione fra le artes, cioè la grammatica, la retorica e la dialettica (il “trivio medievale), e le disciplinae, cioè aritmetica, musica, geometria e astronomia.

La questione della natura speculativa della musica è messa in evidenza con l’inquadrare questa e le altre discipline sorelle quali sottodivisioni della filosofia inspectiva (termine che in Rufino equivale alla boeziana filosofia teoretica), le cui partizioni sono: naturalis, cioè la fisica; doctrinalis,
le discipline matematiche; e divina, la teologia. Nessuna novità, dunque, è prospettata da Cassiodoro, anche se è opportuno sottolineare come egli indirizzi in senso nuovo l’idea di “scienza musicale”, che nelle sue parole si allarga a comprendere tutte le azioni della vita, da quelle biologiche a quelle razionali e relazionali, come il linguaggio, in quanto tutte sottoposte ai numeri musicali (rhythmi):

La scienza della musica è presente in tutte le azioni della nostra vita, soprattutto se osserviamo i comandamenti del Creatore e adempiamo con mente pura alle regole da lui fissate: infatti è dimostrato che ogni parola pronunciata e ogni movimento interiore provocato in noi dalla pulsazione delle vene è collegato mediante i numeri musicali al potere dell’armonia. La musica, infatti, è la scienza dell’esatta modulazione; se viviamo sotto virtù siamo sempre sotto tale disciplina. Quando ci comportiamo in modo ingiusto non abbiamo musica, e anche il cielo e la terra e tutto ciò che si compie per dispensa divina non esiste senza disciplina musicale (Institutiones II, 5, 2).

Autore: Cecilia Panti
Filosofia della Musica. Tarda Antichità e Medioevo
: Carocci (Studi Superiori, 541)
Luogo: Roma
Anno: 2008
Pagine: 106-108