The ‘quadrivium‘ is the name for a part of the medieval curriculum, as it was derived from the seven ‘artes liberales‘. The ‘artes liberales’ were distinguished by bishop and philosopher Augustine (354 – 430 AD) in an effort to give (Latin) education a theoretical framework. The ‘artes‘ were, later, divided in two parts, reflecting the ‘human’ and the ‘natural’ sciences’ (fig. 327 – 329):
the ‘trivium’ the ‘quadrivium’
—————– grammatica arithmetica
—————– rhetorica geometrica
—————– dialectica astronomica
. harmonica (music)
Fig. 327 – The seven liberal arts in the Convenevole da Prato, a poem for Robert of Anjou. 1334 – 1343. Wien, Osterr. Nationalbibliothek, Cod. ser. n. 2639, fol. 30r. Abb. 33 in: TEZMEN-SIEGEL (1985).
Fig. 328 – The seven liberal arts. Left: trivium Right: quadrivium. Thomasin von Zerclaere – Der Welsche Gast. Cod. Ms. B (second half fourteenth century). Erlangen, Universiteits-bibliothek. fol. 19v. First publication. Abb. 13 in: TEZMEN-SIEGEL (1985).
Fig. 329 – Some illustrations of the seven liberal arts in ‘The Mirrour of the World’ by Vincent of Beauvais (Westminster, 1481). The ‘trivium‘ (grammatica, rhetorica, and dialectica) was associated with teaching to a multiplicity of pupils. The ‘quadrivium‘ (arithmetica, geometrica, harmonia and astronomica) showed individual pupils or a duo (music, harmonia). In: VINCENTIUS (Vincent of Beauvais) (1481/1979).
The division of knowledge found its roots in classical times. The Greek oral tradition (the ‘epic cycles’) and Roman rhetorical rules (expressed by Cicero) used divisions in the art of memory. This art was, according to legend, invented by the poet Simonides of Ceos, who realized that orderly arrangement was essential for a good memory. Cicero tells the story of the named poet, who established the identity of mutilated bodies after the roof of the banqueting hall collapsed. He remembered the seating of the guests at a banquet, which he had left just minutes before.
Four operations can be used to improve memory (PILTZ, 1981; p. 223):
1. Use pictures, which resemble what you are trying to remember. These pictures should be slightly out of the ordinary and stand out against a certain background;
2. A systematic attention is necessary and a certain order must be introduced;
3. A selection of the things to remember is of prime importance because the more firmly something is etched in our senses, the more difficult it is to escape our memory;
4. Meditation about the choice is necessary all the time: ‘it is meditation that saves the memory’.
A logical result of the systematic attention (of the second step) is the introduction of a division. The choice of a division introduces, in a philosophical environment, a cognitive environment: the elementary dual, tri- or quadripartitions reveal the frame of mind in which decisions are taken.
‘The actual grouping of the four branches went back to Plato, as well as to Archytas, in the fourth century before Christ’, stated Pearl KIBRE (in: MASI, 1981; p. 69). Their common nominator and purpose were a definition of quantity, expressed in a language of ‘mathematica‘ (or ‘quadriviales‘): ‘And this quantity was either discontinuous or continuous, that is discontinuous either per se as in arithmetic, or in relation to another, as in music or harmony; and continuous either without motion as in geometry, or in motion as in astronomy. Thus the quadripartition was specifically that of quantity’ (MERLAN, 1960; pp. 94 – 95).
Christiane JOOST-GAUGIER (2006) emphasized in her proficient book the link (of the four-division of knowledge) with Pythagoreanism. She also pointed to Archytas the Pythagorean, living in the four century BC, as the initiator of the quadrivium. The concept was later refined by Boethius and Macrobius: ‘Through the works of these men especially, the Pythagorean tradition would be kept alive long after the world of Antiquity had given way to the Middle Ages.’ (JOOST-GAUGIER, 2006, p. 111).
An important contribution to structural thinking came from the Roman orator and man of letters Marcus Tullius Cicero (Tully), born in 106 BC. Cicero’s teacher was Posidonius of Apamea, a man who calculated the earth diameter and had his teachings recorded in the ‘Tusculanae Disputationes’. Cicero’s keen interest in cosmological matters led to the translation of Plato’s ‘Timaeus‘ into Latin (and handed to the West in the Middle Ages an important lead to their ‘Greek-Pythagorean’ past). He was also the first to mention Euclid, although it is unlikely that a Latin translation of this work existed at the time (no record of any Latin translation of Euclid is known before Boethius, c. AD 480; RUSSELL, 1945; p. 212).
One of Cicero’s earlier works was the ‘De inventione’ (or ‘Rhetorici libri duo’) written in 84 BC. He defined the basic four virtues. The book was concerned with the first part (of five) of the rhetoric, the ‘inventio‘: the composing of the subject matter of a speech and the collection of ‘things’ to deal with. The work was often associated with an anonymous work called ‘Ad Herennium‘ (Rhetorica nova), and together they were known in the Middle Ages as the ‘First and Second Rhetorics’ of Tullius (YATES, 1966; p. 36). Cicero personalized the theme with four individuals (Crassus, Marcus Antonius, Quintus Scaevola and Caesar Strabo) in a later work called ‘De Oratore’ (in three books). The book was written in 55 BC and described the positions in a communication:
1. Crassus pointed to knowledge of law and philosophy as a prerequisite for a good communication; Antonius reckoned that natural ability and experience was sufficient;
2. Antonius expounded his ideas (in Book 2) with the ‘inventio‘ (the deliberate choice of items to draw the attention of the audience); Caesar highlighted the importance of humour;
3. Antonius was in favor of order and a clear structure of the material discussed; and
4. Crassus (in Book 3) summarized the preconditions of a competent conversation and considered elegance in style and rhythm as the highest objective.
Cicero’s thoughts were fully developed in his book ‘De Officiis’ (‘On Moral Duties’; MILLER, 1921), written in his ‘days of distraction’ towards the end of his life (46 – 43 BC). The four virtues were presented in three books: 1. Moral goodness; 2. Expediency; and 3. The conflict between the right and the expedient. The book was about duty and morality, and gave practical rules to achieve those goals.
The four cardinal virtues were in the center of attention: wisdom, justice, fortitude (courage) and temperance represented the four stages of communication, aiming at equilibrium in a dynamic environment: ‘the rule of the golden mean is best’ and ‘the whole glory of virtue is in activity’. He also realized that the position taken by an observer in a communication (either voluntary or involuntary) was of prime importance: ‘tanta vis est et loci et temporis‘ (Great is the significance of place and circumstance; Book I, XL, 144). VAN DER ZANDE (1998) vividly described the triumphal reception of Christian Garve’s German translation of Cicero’s work in 1783.
Varro (116 – 27 BC) presented, in his ‘De Novem Disciplinis libri novem’ (Nine Books of the Nine Disciplines; 33 – 31 BC), a general view of the curriculum in ancient times. Unfortunately, only fragments of this work remain. ‘The most learned man of his times’, as Varro was called by Quintilian, added medicine and architecture to the list of primary subjects (KNOWLES, 1962), bringing the total to nine.
The actual (theoretical) division into trivium and quadrivium dated from later than the seventh century (COBBAN, 1975). RAJNA (1928) put the effective introduction of the division during the life of Alcuin (730 – 804; articulated in the ‘Horatius’-commentary of Pseudo-Alcuin). RASHDALL (1895/1936, p. 36) insisted that ‘the real education of the Dark Ages was the trivium‘. The quadrivium was, with retrospective effect, ‘filled up by discoveries or rediscoveries of the twelfth-century Renaissance’.
Calvin BOWER (in MASI, 1981; p. 163) stated that ‘recent studies have shown that the liberal arts played a rather minor educational role in most of Europe between 500 and 850′. A five-fold division prevailed at that time. The actual duties of the monks in their educational quest were formulated in Charlemagne’s ‘Capitular 72’:
———————————— psalmi (or liturgy),
———————————— notae (writing),
———————————— cantus (singing),
———————————— computus (calendric studies) and
———————————— grammatica (reading).
Two writers had a direct influence on the European scholars of the Middle Ages: Martianus Capella and Boethius, both living around 500 AD, at the time when the Roman Empire disappeared from the stage of European cultural history. Both can be seen as vital links between the classical knowledge (and imagery) and the young European culture.
Martianus Capella lived in Cartage and used the classical division of knowledge in his ‘De Nuptiis Philologiae et Mercurii’ (On the Marriage of Philology and Mercury). The book is an encyclopedia in a popular form and became the leading canon of the ‘septem artes liberales‘ from the sixth to the fourteenth century (STAHL, 1971). The tractate was often attributed to ‘Tullius’, for instance, by Hieronymus Stridonensis in his letter (#53) to Paulinus of Nola, and associated with Cicero’s ‘De Inventione’.
The book of Martianus Capella was a tribute to division-thinking in general. Firstly, the marriage, as a unity of two: the allegorical marriage between Mercury and scholarship (Book I-II), the unity of words is three-fold (Philology: Book III – V: Grammatica, Dialectica, Rhetorica) and the unity of things is four-fold (Mercury: Book VII – IX: Geometrica, Arithmetica, Astronomia and Harmonia). Two-hundred-and-forty-one manuscripts of ‘De Nuptiis’ are known to exist. Eight are illustrated (TEZMEN-SIEGEL, 1985).
The Roman philosopher Boethius was born c. 480 AD in Rome and executed in 524 AD at Pavia. He used division-thinking as a guideline in his thoughts (MASI, 1974, 1981; WHITE, 1981). The term ‘tessares methodoi’ (four methods) was rendered as ‘quadrivium‘, or a place where four roads join (STAHL, 1971; HÜBNER, 1989) in Boethius’ translation of Nicomachus of Gerasa’s book ‘De Arithmetica’ (second century AD).
These crossroads marked the four areas of knowledge: ‘it is impossible to achieve the summit of perfection in the disciplines of philosophy, unless one approached this noble wisdom by a kind of fourfold way’ (‘De Arithmetica‘; PL. LXIII, 1079D). The following cerebral processes (De Cons. Phil., Book V; in the translation of WATTS (1969), p. 157) were noted to guide a human communication:
——————————– 1. sense-perception
——————————– 2. imagination
——————————– 3. reason
——————————– 4. intelligence (understanding)
The transmission of the quadripartite image into the European Middle Ages was intensified by Boethius’ ‘De Consolatione Philosophiae’ (The Consolations of Philosophy), written in jail before his execution on October 23, 524, when he was forty-four years old. In this ‘consolatio‘, or manual for mental health, the goddess ‘Philosophia‘ sings of the power of love in the natural world preserving peace and keeping chaos at bay (in the last poem of Book II). Philosophy moves on (in Book IV, poem 6) to the concord of the elements, of the seasons, and of birth and death’s finality (fig. 330).
Fig. 330 – Boethius, Philosophia and Fortuna in a miniature from a French translation of ‘De consolatione philosophiae‘. Fig. 43 in: MERSMANN (1982).
UHLFELDER (in: MASI, 1981; p. 31) noticed thematic bonds in the thirty-nine poems, which intermingle with the same number of passages in prose: ‘Boethius’ explicit identification of divisions of the argument proves that there were two coexistent structural principles, one based on the fivefold division into books, and the other on the fourfold stages of the ‘plot’, with special emphasis on the threefold division of the philosophical argument.’
The world view of Boethius was manifestly put forward in the first poem of Book IV, describing the ascent of the soul to God, the center of light, and its return (fig. 331).
Fig. 331 – This scheme gives a representation of the quadripartite cosmos, as presented by Boethius in his ‘Consolatione Philosophiae’ (Book IV, poem I).
The human mind travels from the earth through the sky to the sphere of the moon. The lightest element (fire) reaches to the moon. Beyond the moon is the fifth element, the quintessence or ether. The soul succeeds through the sphere of the stars to its ultimate destination: God, the source of light. The (cyclic) movement of the (human) invisibility continues, descending from God, back through the ether, to reach the earth and emanate (again) in a human soul.
At birth the soul emanates or descends to the earth from God, and its ascent is an account of its return. The emanation is described as follows (verses 15 – 26; translated by WATTS, 1969; p. 117/118):
. And when the orbit’s path is done
. The furthest heaven it forsakes.
. It treads beneath the ether swift
. Possessing now the holy light,
. For here the King of kings holds sway,
. The reins of all things holding tight,
. Unmoving moves the chariot fast,
. The lord of all things shining bright.
. If there the pathway brings you back –
. The path you lost and seek anew –
. Then, ‘I remember,’ you will say,
. ‘My home, my source, my ending too.’
A ‘descriptio‘ (visual explanation) of Boethius’ cosmic consciousness was given in an eleventh-century copy of the book ‘De Arithmetica’ (Arithmetike eisagoge) by Nicomachus of Gerasa (Jerash, Jordan). This important mathematician lived in the Roman province of Syria from c 60 – 120 AD. Boethius’ ‘De institutione arithmetica’ was a Latin translation of this book (MURDOCH, 1984; p. 102, fig. 97)(fig. 332/333).
Fig. 332 – The world view of Boethius in an eleventh century copy of the translation of ‘De Arithmetica’ by Nicomachus of Gerasa. Boethius provided in this book a philosophy of numbers, which should be used as guidelines to a moral order. The relation of particular numbers (in this case the oddly even numbers) set an example of logic and harmony reign.
Boëthius’ ‘De Institutione arithmetica libri II‘, Bamberg, H.J. IV.12, fol. 28r. In: DIRINGER (1967).
Fig. 333 – The mathematical world view of Boethius, as given in his book ‘De Arithmetica‘ (Paris, 1521; Lib. I, p. 25). This scheme was preceded by a treatment of the odd and even numbers. A sequence (of evenly even numbers; pariter par) with an even number of terms (like 1, 2, 4, 8, 16, 32, 64, 128 = eight terms) don’t have a single middle term, but a double one (8 en 16). A multiplication of these two terms results in the last number of the sequence (128). A sequence (of evenly even numbers) with an odd number of terms has a single middle term. This middle term gives – if multiplied by itself – the last term of the sequence. These properties are illustrated on oddly even numbers by giving four rows of numbers, which are generated by the multiplication of two even term with three (row 1: 2 x 2 = 4 x 3 = 12 etc.), five (row 2: 2 x 2 = 4 x 5 = 20), seven and nine. Two middle terms (‘medietas’) are multiplied and written outside the square in a horizontal (latitudo, addition) and vertical direction (longitudo, multiplication) and performed in the outer and inner sequences. Boethius – ‘De Arithmetica’ (Paris, 1521), Lib. I, p. 25 in the Biblioteca Hermetica, Amsterdam. See also p. 88 in: MASI (1983).
Cassiodorus and Isidore of Seville (‘Etymologiae’) jointed Boethius crucial position in the continuation of classical knowledge into the Middle Ages. They also contributed to the formal use of the system of division of knowledge. Cassiodorus, living towards the end of the sixth century and founder of the monastery of Vivarium in Calabria, wrote a handbook on the liberal arts (the ‘Institutiones’ or ‘Introduction to Divine and Human Readings’).
The ‘quadrivium‘ could be seen as a path to abstract knowledge (WHITE, 1981). In terms of a Pythagorean number-symbolism it means, that arithmetic deals with the numbers itself, geometry with the numbers in space, harmony with the numbers in time and astronomy with numbers in space and time (GUTHRIE, 1987). This division reflected the cognitive ‘visio‘, as it was experienced during the apex of Medieval thinking.
A clarifying article on the possible origin of the term ‘quadrivium‘, was written by HÜBNER (1989). He pointed to the difference in age: the term ‘quadrivium‘ was older than the ‘trivium‘. Boethius never knew the term ‘trivium‘ (MASI, 1981; p. 11). The associated subjects (of the quadrivium) reached prominence only when tetradic thinking itself became visible in a wider sense (from the middle of the eighth century). The general use of the terms ‘trivium‘ and ‘quadrivium‘ dated from the eleventh century (LESNE, 1940; WOLTER, 1959).
The symbolism of the cross-roads was, according to Hübner, more often seen as a metaphor (sometimes in connection with a ‘bridge’) of the opposition between body and soul: ‘this was the way the metaphorical ‘quadrivium’ was understood in the Middle Ages’. A reference to Alcuin had to support his view. To draw the symbolism (of the quadrivium) in such a dualistic environment is, in my view, a simplification. It is true that Alcuin was – in his ‘Retorica’ – a faithful follower of Augustine (HOWELL, 1941), who felt attracted to lower division-thinking. However, Alcuin – the ‘educator of Europe’, originating from York and a teacher of Charlemagne – distinguished himself from Augustine and brought a ‘Celtic’ flavor to his teaching.
Augustine – who had earlier crossed the Channel to England (in 596 AD) – was a representative of the ‘Roman’ interpretation of Christianity, with its emphasis on opposition. Now (some two hundred years later) Alcuin returned the tetradic mood back to the Continent and brought with it a reintroduction of the liberal arts. He referred to the arts in his ‘Grammatica‘ as seven gifts of the Holy Spirit and as the seven steps to the study of philosophy (PL 101, col. 853).
Rhabanus Maurus (c. 784 – 856) showed interest (but little knowledge) in the liberal arts in his ‘De Clericorum Institutione‘. Book III is a reworking of Isidore’s description of the arts with slight additions. A ‘Grammatica‘ attributed to Clemens Scotus, of Irish origin and composed around 800 AD, divided philosophy in three genera (physics, ethics and logic) and divided physics in turn in four principal parts, the four mathematical arts or ‘quadrivium philosophiae‘ (quoting Boethius’ mathematical work for the first time). John Scotus Erigena quoted an even more extended passage from the ‘Proemium‘ of Boethius’ arithmetical treatise in Book I of his ‘De Divisione Naturae’.
‘It is no coincidence’, according to BOWER (in : MASI, 1981; p. 167), ‘that both names citing Boethius contain the term ‘scottus‘, for the revival of the liberal arts and of speculative philosophical thought in the ninth century was largely the result of the work of ‘scotti peregrinantes‘. They brought to the continent, along with their love of learning and speculative thinking, many books that had been basically unknown for several centuries.’
The education at the Carolingian monastery schools was given at three levels (PILTZ, 1981; p. 15), inspired by a practical approach:
1. The first step consisted in the learning of the elementary principles of writing, reading and singing, some grammar and an explanation of the calendar.
2. The next step was a study of the seven liberal arts (‘septem artes liberales‘), divided into the ‘trivium‘, i.e. grammar, rhetoric and dialectic, and the ‘quadrivium‘, consisting of arithmetic, geometry, astronomy and music (fig. 252). The seven liberal arts were mentioned by Plato in his ‘De Republica‘ (The Republic, Book VII; LARSON, 1979) and are part of the Platonian system of ‘planned education’ (COBBAN, 1975).
3. The final step in the Carolingian cathedral and monastery schools was the actual preparation to the task as a priest and the practical familiarity with the skills of priesthood, like the reading and interpreting the Scriptures and teaching the catechism.
The cathedral school of Chartres became in the early twelfth century, under the guidance of Thierry of Chartres, a center of the ‘exact’ sciences of the quadrivium (STODDARD, 1966; MASI, 1983). Plato’s ‘Timaeus‘ (in the adaptation of Chalcidius, living in the fourth century) was the major point of departure. It was thought possible to learn more about God within the structural setting of nature. The study of nature was therefore regarded as a devotion to the almighty God.
‘For three centuries, from the thirteenth century until the revolutionary changes that took place at the beginning of the sixteenth century, all people in Europe with any claim to education at all could make themselves understood to each other. This was not only because they shared a common language in Latin. What is more remarkable is that they shared a common world picture and uniform terminology for describing it’, said Anders PILTZ (1981) in the preface of his book on ‘The World of Medieval Learning’. This unity was mainly due to the Roman Catholic Church and its schools associated with churches and cloisters.
The ever-present current of Neo-Platonism in the European cultural history favored the reciprocity between the Idea and Nature and showed therefore an interest in the quadrivium. The scholar Gemistus (Plethon), for instance, living in Mistra (southern Greece), some two hundred years later, was educated in the trivium and quadrivium (FUCHS, 1926; MASAI, 1956; p. 55)
The transfer of knowledge in the Middle Ages followed an established, trodden path. VERGER (1973, p. 13) noted in his expose of teaching at the universities in the Middle Ages: ‘The method was always the same; the master reads the text which had to be learned (lectio) and interrupts his reading by commentaries, which explain the literal sense (sensus) and reveal the deeper meaning of the excerpt (sententia). VERGER (1973) divided the different universities in their way of origin:
1. spontaneous (from cloister schools), like Paris, Bologna, Oxford and Montpellier;
2. by migration, like Cambridge (1208), Orleans, Padua (1222);
3. planned, like Naples (by Frederick II, 1224), Toulouse (1229), and the Spanish universities Palencia, Salamanca and Valladolid.
The quadrivium remained favorite in the faculties of arts in Padua, Bologna and particularly Oxford. Furthermore, Toledo, in Spain, was the ‘famed city for the teaching of the arts of the quadrivium‘ (GIMPEL, 1979/1988).
Thomas Aquinas (1225 – 1274) continued the further intellectual framework (of division-thinking) in the thirteenth century. His ‘De Unitate Intellectus’ (‘Over the unity of the soul’) was written around 1270. It aimed at specific, but not further identified, philosophers of the University of Paris, which were known as ‘Averroists’. They were named after the Arab scholar Averroes (1126 – 1198), who explained the works of Aristotle. Indirectly, Thomas Aquinas aimed at the scholar Siger of Brabant.
Siger of Brabant was probably born between 1235 and 1240. He started his study at the University of Paris in the middle of the thirteenth century in the ‘artes liberales‘. The writings of Aristotle and its interpretations were in the center of attention at the time. He was summoned for the Inquisition on the 23rd of November 1276, but Siger had already left France. He was murdered in 1284 in Orvieto (Italy) by a mentally disturbed secretary (MANDONNET, 1899/1976) (fig. 334).
Fig. 334 – Wretched man from Orvieto (Photo: Marten Kuilman, 1995).
Siger drafted many books, of which his commentary on Aristotle’s ‘Metaphysica’ (Metaphysics), ‘De nima’ (On the Soul), the ‘Physica’ (Physics) and ‘De Generatione’ (On Growth and Decay) are preserved. In addition, work of the fifth-century Neoplatonist Proclus (‘On Providence‘) and small treatises like ‘De Aeternitate Mundi’ and the ‘Liber de Felicitate’ survived. Dante Alighieri (1265 – 1321), in his ‘Divinia Commedia‘, placed Siger of Brabant in the circle of twelve wise men.
Siger of Brabant’s ‘De Intellectu’ was published in 1270 or 1271, shortly after Thomas Aquinas’ ‘De Unitate Intellectus’ came into circulation. Siger’s work was not a direct answer to the work of Thomas, but it made several references to it. The tension between Thomas Aquinas and Siger of Brabant touched the deeper interpretation of division-thinking, as felt and understood by Aristotle in the book ‘De Anima‘, giving an exposition of the soul.
Siger, as an ‘Averroist’, took the conservative outlook to safeguard the tetradic heritage. He held on to history. Thomas, on the other hand, represented the progressive side, aiming at a synthesis and understanding of lower division-thinking. The first (Siger) believed in a collectivity (of the soul) which could be found in the invisible invisibility of the First Quadrant. The second (Thomas) searched for individuality and unicity to be found in the visible visibility of the Third Quadrant. Such a difference in emphasis can be accommodated in a tetradic program of learning. However, it will become a threat in lower division-thinking, if the choice is reduced to a matter of either-or. In that latter situation, one of the positions has to be abandoned and, subsequently, the communication as a whole looses its richness. It has to be understood, for good measure, that Thomas Aquinas did not choose this way.
Thomas was, in the context of European cultural history, the most important of the two scholars. He was canonized in 1323 AD and became an ecclesiastical authority. The achievements and person of Siger of Brabant were gradually pushed to the background. Only the monument in Dante’s ‘Divina Commedia‘ remained, in which he figured in two terzinen, together with Salomon and Boethius.
The discussion (between Thomas and Siger) was, from the quadralectic point of view, an inquiry into the nature of the First Quadrant (the ‘soul’). The ultimate question was, and probably still is: how do we envisage this (double) invisible part (of a division)? 1. As the harbor of all multiplicity – like it is regarded in higher/fourfold division-thinking – or 2. As a unity in its own right – as in lower/twofold division-thinking. KLÜNKER & SANDKÜHLER (1988; p. 18) pointed to the relevance of the discussion as a dispute on the proper explanation of Aristotle in relation to the soul.
Thomas placed the observer in the environment of the visible visibility (of the Third Quadrant), in which the body (Seele) and soul (Geist) are separate entities (substantia), contributing to the identity (of a human being). Siger positioned, on the other hand, his observer in the setting of the invisible invisibility (of the First Quadrant), a place where no divisions were made (as yet). There is no distinction between body and soul and everything is in everything. Siger’s (and Averroes’) visions were cosmic directed and contained the four main points of the interpretation of Aristotle (WEISHEIPL, 1974):
————————– 1. unicity of the intellect for all men
————————– 2. the denial of free will
————————– 3. the restriction of divine providence
————————– 4. eternity of the world
These points reflect the various stages of a tetradic communication, as they are present – ‘in statu nascendi‘ – in the First Quadrant and can be interpreted as the four stages of communication itself:
1. The First Quadrant (I) contains the ‘unicity of the intellect’ (a One) for ‘all men’ (a Many): a ‘one in all’-situation, a potentiality.
2. The Second Quadrant (II) is specific. Rules are set at the very moment that visibility becomes apparent. Within the framework of a given visibility, there is no free will, once the primary (division) decision takes effect.
3. The Third Quadrant (III) is down-to-earth and out the hands of God. Man is so devoted to the visible visibility of material existence, that there is no place left for divine providence (which implies some sort of invisibility).
4. The Fourth Quadrant (IV) is a synthesis of the previous stages and brings the four kinds of insight (principles) together into the highest ‘sapientia‘ (within this particular cycle of communication): to understand the ‘quattuor causae‘ (finalis, formalis, materialis and efficiens) within the eternity of the world. It creates the new invisibility, which was temporary lost in the Third Quadrant, and gives room for an active intellect, which is immortal and eternal.
Van STEENBERGHEN (1977) disagreed with the views of French historians Renan and Mandonnet, who depicted Siger as an outspoken representative of the ‘averroïsme latin‘. He rejected the opinion, that the ‘double verité‘ – the ‘double truth’ or the opposition between belief and ratio, which was a hallmark of Averroism – was adopted by Siger (p. 242): ‘neither Siger of Brabant, nor the other ‘averroists’ of the Thirteenth Century have adopted the theory of the two truths; in their most radical phrasing, they only declare that the philosophical conclusions, understood on a purely rational level, despite their opposition to the given facts of belief, are true’.
Thomas reached a conclusion with a dual aspect in his description of the soul: ‘two explicit different forms exist: the human soul, connected with the body, and the angelic soul, which is separated from the material’. It is the familiar scheme of oppositional thinking, with a particular item (the soul in this occasion) placed and valued in the realm of the visible and the invisible. The ‘double truth’ of Averrois (and his interpretation of Aristotle) can be recognized. The ‘verité theologique‘ and the ‘verité philosophique‘ are respectively the representations of the ‘truth’ (or ‘revelation chretienne‘, Christian revelation) in the first and third quadrant. Averrois rejected dualistic thinking, as it featured in the celebration of the Eucharist (with bread and wine suddenly changing into the real body of Christ).
The spirit of Averroes lived on in Boethius of Dacia (‘De Aeternitate Mundi‘), Jean de Baconthorp (died 1346), a ‘docteur‘ of the Carmelites, Walter Burleigh and the ‘peripateticiens de Padove‘ (RENAN, 1861): a group of intellectuals around the university of Padua, including Caesalpinus, Cardanus (FIERZ, 1977), Vanini and Berigad. Padua remained a center of European Averroism (DESSOIR, 1925; p. 351).
Julius Cesar Vanini (1585 – 1619) was a typical example of this group of most original scholars. His ‘Opere‘ – accessible in a modern edition by PAPULI & RAIMONDI (1990) – presents him as a campaigner against ‘bad philosophes, atheists, Epicurists, Peripatetici and Stoics’. Averrois was right at the beginning on the stage of the ‘Amphitheatrum Aeternae providentiae divino-magicum, christiano-physicum nec non astrologo-catholicum’ (1615). Vanini treated a wide range of knowledge in around fifty ‘Exercitatio’s’. The ‘pantheistic’ tendencies caused the Roman Catholic Church to ban him. He was burned at the stake in Toulouse in 1619.
The structure and educational aims of the ‘quadrivium‘ were by that time virtually obliterated. KLINKENBERG (1959) put the first signs of this diminishing influence already in the early Middle Ages and blamed the growing authority of theology. Others, like KRISTELLER (1959), saw its downfall in the diversification of sciences at the end of the twelfth century. However, the individual members of the ‘quatuor genera rationum‘ revived – stronger than ever before – under the new, general denomination of ‘science’.
The Ptolemaic (geocentric) cosmic system, which had ruled for almost fourteen hundred years – from the second to the sixteenth century – was replaced by the (heliocentric) system of Copernicus, who had proposed the idea around 1507. His book ‘On the Revolutions of Heavenly Bodies‘ was only published towards the end of his life in 1543. It was meekly provided with a preface addressed to Pope Paul III. The Reformation broke, at the same time, the power of Roman Catholic Church in Europe and gave way to a new approach to material matters, based on empirical knowledge rather than on a belief.
Francis Bacon (1561 – 1626) introduced in his ‘Novum Organum‘ (1620) a ‘new knowledge’ to replace the old quadrivium. His system of induction by the method of exclusions expressed a shift in the position of the observer towards greater control. He was devoted to the ‘double truth’ (that of reason and that of revelation), but not from an intellectual point of view, but for dualistic reasons: to give unlimited freedom for (material) reason: (lower) division was the name of the game. Divide and rule. Knowledge is power (fig. 335).
Fig. 335 – Fortuna on the title page Francis Bacon (1642) – Henry VIII. Etching by Cornelis van Dalen. In: CHEW (1962).
His four classes of ‘idols of the mind’ (or defects of human understanding) were a farewell to the neutral approach of the quadrivium. Bacon provided, in this rare example of tetradic division in his work (together with the division of the knowledge of a man’s body in Health, Beauty, Strength and Pleasure in ‘The Advancement of Learning‘, 1605) a guide how to achieve the power of knowledge by avoiding errors:
. IDOL ERROR
—————- 1. Of the Tribe – inherent in human nature;
—————- 2. Of the Cave – by personal prejudices;
—————- 3. Of the Market Place – through inexactness of language;
—————- 4. Of the Theatre – through the system of thought,
(of the Schools) – believing in blind rule
A closely related subject to the quadrivium is the existence of the ‘Nations‘, a name given to the students (groups) at the early medieval centers of learning. The university as a scientific institution, in which knowledge had an autonomous place, originated in Southern Italy. In Salerno developed – under the influence of Greek, Latin, Jewish and Arabic know-how – a famous medical school, which flowered around the year 1100 AD. The city of Bologna became in the same period a center of law studies, concentrating on a revival of the Roman laws. The gathering of people was called a ‘studium generale‘: a study facility, which could be visited by international students.
Within the early Italian centers of knowledge grew a division in the ‘ultramontane’ (the ‘universitas ultramontanorum‘ for students from outside Italy) and the ‘cismontane’ (the ‘universitas citramontanorum‘, for students from Italy itself). This primary division developed gradually towards the end of the twelfth century – possibly influenced by the rise of guilds – into groups of students originating from the same area. From the early thirteenth century onwards these groups were called ‘nations’, with own rules and administration.
Efforts were made, between 1265 and 1317, to diminish the number of groups in Bologna, but in stead their number grew in the law department from two to seventeen at the end of the fifteenth century. The arts- and medical faculties, established in the second half of the thirteenth century, did not divers in the same way. The faculties kept their four nations (Ultramontane, Lombardy, Tuscany and Rome) until the sixteenth century.
The University of Padua, founded in 1222, had at first a division in four associations (French, Italian, German and Provencal), but followed after 1260 the model of Bologna. The same holds for the other universities, which were later founded in southern Europe: the main division is two-fold, with a subsequent subdivision. Exceptions were the universities in Orange, Dole, Caen, Cahors, Perpignan, Nantes, Bordeaux, Erfurt and Cologne. They had no division in nations (KIBRE, 1948; COBBAN, 1975).
The English universities did not have such a division either: ‘In contrast to the nations at Paris, Bologna and other continental universities, the nations of Oxford and Cambridge had never been deeply rooted in either a governmental or an academic sense. Since there was an insufficiently large cosmopolitan population to accord the nations an importance as defensive groupings for students of widely divergent ethnic origins, they were fairly rapidly phased out as irrelevances on the university scene’ (COBBAN, 1988; p. 402).
In Paris – as the archetypal setting of a university with students from many different geographical areas – were four nations, finding their members in the faculties of arts, theology, law and medicine. The faculty of arts was by far the greatest with about two-third of the total number of students. Until 1219 these groups are more or less of equal standing, but after that year the arts faculty gained a dominance and a ‘corporate identity’ (COBBAN, 1975). The bureaucratic organization also meant the onset of petty power struggle and strife, so vividly recorded in Kibre’s book on the nations in the medieval universities. The four ‘nations’ (at the Paris University) were:
1. The French nation, with students from Paris and its environment, Southern France, Spain, Italy, Greece and further east;
2. The Normandic nation was composed of students from northwestern France (Avranches, Evreux, Bayeux, Coutances, Liseaux and Sens) and Brittany.
3. The Picardic nation consisting of students from the Low Countries and Northern France (Beauvais, Noyon, Amiens, Laon, Cambrai, Liege, Utrecht and Tournai).
4. The English nation, with students from northern, central and north-eastern Europe (the British Isles, Holland, a part of Flanders, Germany, Denmark, Norway, Finland, Hungary and the Slavic countries).
Gerardo de Borgo San Donnino sparked off the crisis at the university in Paris (1252 – 1261) and his ‘Introduction to the Eternal Evangel’. Guillaume de Saint-Amour (‘Sur les perils des temps nouveaux, 1255/56?) saw him as the messenger of a new time. The University of Oxford had a similar intellectual crisis about half a century later (1303 – 1320).
‘The division into only four nations at Paris appeared, more than at Bologna where the large number of nations was more closely representative of the localities from which their members came, the result of a convenient administrative grouping rather the result of any natural affinities of language or homeland’ (KIBRE, 1948; p. 27). The division into ‘nations’ was not an effort to conform to tetradic thinking, but found its origin in an oppositional setting, aiming at the identification of a (national) identity.
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