Dante's Historical Arithmetics: The Numbers Six and Twenty-eight as 'numeri perfecti secundum partium aggregationem' in Inferno XXVIII

by Otfried Lieberknecht

Paper given at the 32nd International Congress on Medieval Studies, 8-11 May 1997, Western Michigan Unviersity (Kalamazoo), during the session n. 322 (Problems in Dante's Inferno, dir. Christopher Kleinhenz, sponsored by the Dante Society of America)

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As philologists, we are entitled to feel uncomfortable with allegory and number exegesis: words are our business, not things, not numbers. But when dealing with medieval texts, we are dealing with a cultural context in biblical exegesis where things, especially but not exclusively things as described in biblical texts, were interpreted as signs in their own right, as signs pointing to other, partly similar things; and the number of these things was one of their properties which used to be regarded as siginificant, as constituting similarity betwen the signifyer and the signified. When dealing with a text which originated within this context, whe should therefor try to make ourselves familiar with the exegetical practice of the time, if the author himself was familiar with this practice and might have expected his readers, or at least some of them, to give the things and numbers in his work an understanding of this kind. Dante's Commedia can count as such a text, with regard to the intellectual formation of its author, and with regard to the intended audience, for this work did address not only a majority of readers still in need of doctrine and doctrinal guidance, but also a minority of 'happy few' who, like Dante himself, had reached out early for the 'bread of the angels' and had acquired an uncommon mastery of theology and other sciences.

In the following I will try to present a small piece of this 'bread of the angels', a very limited element of medieval doctrine, as relevant for the composition of Dante's work. The doctrinal element in question is the arithmetic notion of 'perfect numbers'. It shall be introduced first in its arithmetic understanding and in its adoption for biblical exegesis, to be pointed out then as a clue for the thematic composition of a single canto of Dante's Commedia, the 28th canto of the Inferno. Within the given limits of time and oral presentation, my demonstration has to be brief and selective, and will have to rely heavily both on your indulgence and on your willingness to make sense of the diagrams in the additional handout.

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Arithmetic, as taught in the tradition of Boethius, was not operative arithmetic teaching how to add and subtract, divide and multiply numbers, but was a theory of the properties of numbers and of their relations with each other. The properties of numbers are treated in the first book of the Institutio arithmetica, which explains the differences between even, odd and prime numbers and also further subdivisions. One of these further subdivisions is the distinction of even numbers as 'imperfect', 'perfect' or 'abundant' numbers, distinguished according to how a given number relates to the sum of its possible divisors ('aggregatio partium'). If the sum of the divisors is smaller than the number itself, the number is an 'imperfect number'. Example: Four, the first imperfect number, can be divided by one and by two, and the sum of one and two is three. If the sum of the divisors is greater, the number is an 'abundant' number. Example: Twelve, the first abundant number, can be divided by one, by two, by three, by four and by six, and the sum of these divisors is 18, greater than 12. Only those numbers which equal the sum of their divisors are 'perfect numbers' in the arithmetical sense of the word. The first number to fulfill this condition is six, which can be divided by one, by two and by three, the sum of these divisors amounting again to six; and the second perfect number is twenty-eight, with the divisors one, two, four, seven and fourteen, which add up again to 28. Boethius stresses that 'imperfect' and 'abundant' numbers, like moral vices, are very frequent and do not obey a certain order, whereas 'perfect' numbers, like moral virtues, are extremely rare, occur in a certain order and can be found with the help of a rule which derives them from the series of prime numbers. Since prime numbers themselves are laborious to identify, Boethius and his medieval followers were acquainted only with the first four perfect numbers: six, twenty eight, 496 and 8128. From this limited knowledge it was inferred that in the decadic system each rank of units, tens, hundreds and so on contains exactly one and only one 'perfect number', and that the last digit of perfect numbers is always either 6 or 8 (the fifth perfect number, 33 550 336, which proved the decadic rule to be wrong, was only discovered in the 15th century).

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In more or less simplified form, this understanding of 'perfect numbers' became common lore in medieval arithmetic, as a rather basic element of number theory. Unlike other elements of arithmetical doctrine it was not essential for one of the three other quadrivial sciences (geometry, astronomy and music), and it also had no relevance for operative arithmetic. But it found a proper domain of practical utilization in the exegesis of the Bible, where commentators throughout the middle ages came to apply it to more or less every biblical occurence of the numbers 6 and 28, and especially to the number of the six days of creation, to the six ages of the world derived from these six days, and to the week day and daytime when Christ consummated his passion.

Not the first, but for the Latin world the most influential father to apply this arithmetic understanding to the six days of creation was Augustine. He treated the subject in several of his writings, but with regard to Dante one passage in De genesi ad litteram deserves special attention (Table I). In this passage, Augustine does not only associate the perfection of divine creation with the arithmetical 'perfection' of the number six, but uses the numerical properties also as a starting point to detect a certain order in the six days, an order which reveals them to be a 'perfect' and gradually ascending series of one, two and three: on the first day God created the light; on the second and third day he created the 'fabrica mundi', first the 'superior part', that is, the firmament, and then the 'inferior part', the seas and the dry land with the plants. The works of the three remaining days are summed up by Augustine as being the visible beings created to move in this 'fabrica mundi': on the fourth day, beginning again with the 'superior part', God created the celestial bodies, that is, the sun, the moon and the stars; and on the fifth and sixth day he created the inhabitants of the 'inferior part', first the animals of the sea and air, and then the animate beings living on the land, including man.

This explanation is still part of Augustine's litteral commentary and does not yet interpret the six days as signs for something else. To illustrate his allegorical understanding, we may quote a passage from De trinitate (Table II), and this passage too is of particular interest with regard to Dante. In the fourth book of De trinitate Augustine adduces the biblical account of creation as the first of several biblical testimonies to the 'perfection' of the number six, stressing how in this account the perfection of the number corresponds to the perfection of man, created "ad imaginem dei" on the sixth day. This time Augustine also reaches for an allegorical understanding, explaining the six days as a figure of the six ages of the world, that is, the ages of Adam, Noah, Abraham, David, the Babylonian captivity and the actual age of Christ, who, like another Adam, was incarnated in the sixth age to reform or re-create humanity 'after the image of God'. And relating this distinction of the six ages to the threefold distinction of the times 'before the law', 'under the law' and 'under grace', Augustine finally discovers a tripartite subdivision of the six ages, which again exhibits an arithmetically 'perfect' composition of six, occuring this time as two plus three plus one: the two first ages of Adam and Noah were the time 'before the law', the three following ages until the arrival of Christ were the time 'under the law', and the sixth and present age of Christ is the time of grace.

It should be noted that the same tripartite subdivision of the six ages as two plus three plus one could also be reached without the arithmetical concept in question, and withouth the distinction of the 'tria tempora', but with the help of a genealogical speculation which Augustine had already set out earlier in De genesi ad manichaeos: the two first ages of the world comprised ten generations each, whereas the three following ages corresponded to the three times fourteen generations in the genealogy of Christ as given in the Gospel of Matthew. And this subdivision, according to Augustine, is in accordance also with the six ages of man, which are equally prefigured in the six days of creation: at the age of 'infantia' and 'pueritia', man is dedicated to his five corporal senses, and the number of these senses duplicated with the two sexes of man gives the number of the each time ten biblical generations of the first two ages of the world; whereas at the ages of 'adolescentia', 'juventus' and 'senium', the sensitive number five is completed by the prevailing faculties of 'cogitio' and 'actio', and these seven, again duplicated with the two sexes, give the each time fourteen biblical generations of the next three ages of the world. The sixth age of man, 'senectus', has no specific number, because it is only the undetermined span between the fifth age and death, as the sixth age of the world, the Christian aera, has no predetermined number of generations, because it is God's will not to let us know in advance when the last day of the world will arrive. It was this genealogical distinction of the two plus three plus one ages of the world which came to be widely diffused in medieval tradition, whereas the explanations given in De trinitate seem to have been of more limited influence. Similarly, his elaborate distinction of one plus two plus three days of creation in De genesi ad litteram remained an option for subsequent exegetes, but it did not become an exegetical standard. More often we find brief summarizing explanations, which explain the sum of the six days as a 'perfect' number indicating the perfection of God's work, but without a more specific distinction of these six days as one, two and three.

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Mostly by reason of their association with divine creation and with moral perfection, 'perfect' numbers also had a certain tradition to be used for purposes of literary composition. There are several examples, escpecially in Carolingian times, where poets point out in explicit form that they have used the numbers six or twenty-eight as 'perfect' numbers to structure their works. The most famous example are the 28 figural poems of Hrabanus Maurus' Liber de laudibus sanctae crucis, where Hrabanus explains to have chosen the 'perfect' number 28 with regard to his subject, the form of the Cross, as the "consummatrix et perfectio rerum". It was Burkhard Taeger who discovered that the numerical concept in question has informed the composition of this work even more strongly, because the 28 poems, if analyzed in their number of letters per line, can be distinguished as being one, two, four, seven and fourteen (Table VIII), thus manifesting in the formal composition of the cycle the 'perfect' fulfillment of the number 28 by its possible divisors. A very instructive example, because even if we did not have the additional explicit explanation of the author pointing us to the arithmetic concept of perfect numbers, the formal subdivision of the 28 poems nevertheless would still allow us to identify this arithmetic concept as the one on which the composition was based. There is a certain possiblity that Dante himself might have known this work, but in any case he could become acquainted with the arithmetic notion and exegetic tradition of 'perfect' numbers in numerous doctrinal writings, and did not necessarily need additional poetic sources or models to adopt this notion for his own purposes of literary composition. In his own writings, Dante never mentions this concept in explicit form. He once uses the term 'numero perfetto' in his Vita nuova, but in this case he applies it to the number ten and by consequence uses it in a different understanding which was no less diffused than the specific understanding we are discussing here. If he adopted this specific understanding anywhere in his works, he adopted it tacitly, and we would have to find the evidence in the making of the work itself.

Not yet hard evidence, but a good possibility deserving further investigation is the general timeline of Dante's journey: having spent two days as described in the first canticle, that is, first in the "selva oscura" and then on his way through Hell, and three more days climbing the mount of Purgatory, Dante-pilgrim on the morning of the sixth day finally reaches the earthly paradise, to be re-created as another Adam. By consequence the six days of his journey to earthly paradise can be seen as a 'perfect' series of two plus three plus one. Furthermore, the description of his arrival in the earthly paradise begins with the 28th canto of the Purgatorio, which at the same time is the first of six cantos closing the Purgatorio with this final episode. There are also several shorter passages which seem of interest here, and one of them comes close to an explicit statement of the doctrine in question and may reassure us that Dante in fact was familiar with this doctrine. In the 32nd canto of the Paradiso St. Bernard explains how the conditions for salvation had changed in the course of salvation history: during the first time of mankind, called "le prime etadi", innocence and faith where sufficient for future salvation; during the next period, circumcision became necessary; but for those who lived in the Christian aera, in the "tempo della grazia", there was no way to salvation "sanza battesmo perfetto di Cristo" (Pd 32,76-84). At first sight, this explanation adopts only the common threefold distinction of the three times before the law, under the law, and under grace. But the plural form "le prime etadi" for the time before the law, and the phrasing "battesmo perfetto" in the description of the time of grace both seem to be significant: taken together they indicate that within this threefold distinction there is also the 'perfect' subdivision of the six as two plus three plus one ages at work, precisely as it had been taught by Augustine in De trinitate. (Table II)

However, Dante's most systematic and most striking adoption can be found in the 28th canto of his Inferno, a canto structured as a whole and in its parts in accordance with the concept of the 'numerus perfectus'. I will have to confine my presentation to very few aspects of this highly complex composition, and will also follow, at least for now, the example of Augustine in De genesi ad litteram, that is, I will confine my interpretation to the litteral sense, but will not try to interpret how the numerically 'perfect' arrangements of events or persons treated in this canto are designed to make them function as allegorical signs for something else.

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Inferno XXVIII relates the encounter with the "seminator di scandalo e di scisma", sowers of religious, political and familial discord who are punished in the ninth bolgia of the eighth circle of Hell by a devil's sword. As the pilgrim is told by the first of these damned, the devil is standing somewhere in the background at a fixed point of the circular bolgia, thrusting with his sword at the damned who have to pass in front of him. The wounds which they receive heal while they proceed on their way, but only to be stricken again when, at the end of their way, they have to face the devil again. Dante-pilgrim himself, standing first on the rim of the bolgia and then crossing it on one of the small bridges, views only a limited section of this place. He neither sees the devil at work nor does he see the miracoulous healing of the wounds, but he sees, and the narrator reports with precise details, 'the blood and the wounds' of the damned.

The canto opens with a prologue where the narrator insists on the non-narrability of the 'blood and wounds' seen in this infernal place, stressing first the insufficence of human language and memory to relate this experience 'a pieno', and drawing then an elaborate comparison between the sight of the damned and an imaginary reunion of the wounded of several battles or wars in the history of South Italy; and these wars are: (1) the wars of Turnus and his followers against the Trojans of Aeneas, (2) the second Punic War of the Romans against Hannibal with the Roman defeat at Cannae, (3) the military defeats of those who tried to resist the Norman Invader Robert Guiscard, and of the battles at (4) Ceprano and (5) Tagliacozzo, where Manfred and Konradin were defeated by Charles of Anjou. If all those who shed their blood at these events came together and exhibited their wounds, the comparison goes, this still would be 'nothing' to equal the 'modo sozzo' of the ninth bolgia. Given that 28, the second 'perfect' number, is the number of this canto in the Inferno, we have reason to notice that the groups of wounded evoked in this prologue are a 'perfect' number of six (Table III): two groups of wounded from the pre-Christian aera and three groups of wounded from the Christian aera, evoked in their chronological order and contrasted to the single group of damned of all times, wounded eternally in hell. This sixfold structure of two plus three plus one in the prologue mirrors and anticipates one of the basic compositional arrangements of the ensuing narrative part of the canto.

The narrative part itself relates a series of encounters with invidividuals punished in this place, who exhibit their wounds to the visitor, describe and explain to him the mode of their punishment, bring to his memory their names and the deeds for which they are punished there, or prophesy future events of murder and discord. Although Vergil at the beginning of the next canto warns Dante-pilgrim not even to think of counting the "ombre triste smozzicate" in this bolgia, we can nevertheless safely count at least those which are presented with their names and with their individual variant of the bodily punishment: they are six, and they form 'perfect' groups of one, two and three under various aspects, some of which you can find listed on Table V. The most obvious arrangement of this kind again, as in the prologue, is a sequence of two plus three plus one, resulting from the individual form of the bodily mutilitations -- vertical splittings of the body in the case of Mohammed and Ali, minor mutilations in the case of Pier da Medicina, Curio and Mosca, and a horizontal splitting in the case of Bertran de Born --; in correspondence also with the predominant aspect of their sin -- splitting religious unity in the case of Mohammed and Ali, political unity in the case of Pier da Medicina, Curio and Mosca, and familial unitiy in the unique case of Bertran de Born --; and corresponding also to their geographic origins from Arabia in the case of Mohammed and Ali, from Italy in the case of the three sowers of political discord, and from South France in the case of Bertran de Born. But in addition to this recurrent subdivision as two plus three plus one there is also another, on first sight less obvious 'perfect' arrangement to be noted, where Curio takes over the singular properties and Bertran joins Pier and Mosca to form together with them a group of three: as, for instance, with regard to the religious properties of the six, which are two Muslims or, more generally, followers of circumcision, three baptised Christians and one ancient Pagan or polytheist intersecting these three. Further variants of numerically 'perfect' arrangements of six can be found in the gestures exhibiting their wounds (Table VI), and in the six speeches they address to Dante-pilgrim (Table IV). I have to refer you for these series of six to the tables in my handout, but want to close this brief overview by pointing out that also the entire series of 14 historical matters treated in this canto, all of them events of war and discord, seems to be arranged in accordance with the concept of the 'numerus perfectus' (Table VII):

Grouped like this they can be found as being one, two, four and seven events of discord, which together with their sum fourteen are exactly the divisors constituting the arithmetical 'perfection' of the number 28.

This arrangement can remind us of the composition of the 28 figural poems in Harabanus Maurus' De laudibus sanctae crucis. But the subdivision of Hraban's 28 poems as one, two, four, seven and fourteen remains purely formal, a subdivision based on the number of their letters and, as Burkhard Taeger has shown, without further relevance for the content of these poems. Whereas Dante's numerical composition arranges the things or historical events treated in his text, reveals aspects as for instance the geographic or temporal origins of the six damned as potentially significant which we would not even have noticed without regard to their number, and thus imitates more closely the way in which divine providence in history, even where history seems at its worst, still coordinates the apparently unrelated and fulfills her plan.

Preliminary HTML version, 11 June 1997
Copyright 1997, Otfried Lieberknecht