Arithmetic and geometry in ancient Rome: surveyors, intellectuals and poets

ARTICLES Gesar GeymMosar TÀ. 2. SA ARITHMETIC AND GEOMETRY IN ANCIENT ROME: SURVEYORS, INTELLECTUALS, AND POETS MARIO GEYMONAT ABSTRACT Augustine showed a good knowledge of mathematics but, as the Greek Cicero's rediscovery of Archimedes’ tomb shows the interest for science in Rome when the ruling class used to be educated in Greece. Geometry, practised mainly by the Gromatici, had a more technical character in Italy. language went declining in the West, the cultural and economic exchanges became very rare across the Mediterranean. The perception of this danger induced the best scholars to translate scientific texts into Latin: e.g. Boethius translated Aristotle’s logic, Nichomachus’ Arithmetica, and Euclid’s Elements (a few complex fragments of the latter, with authorial corrections, are found in a palimpsest in Verona). Many poets were also interested in mathematics: Catullus’ 5 and 7 re-echoed the numbers of Archimedes, killed by a Roman soldier during the conquest of Syracuse; Virgil, too, found delight in arithmetic; in Constantine’s times such attention is testified by Optatianus, and Ausonius offers an interesting example of combinatory geometry. Keywords: Roman Ancient Mathematics. (1.2.5): The devaluation of mathematics in ancient Rome begins with a goodnatured self-critical affirmation by Cicero in the Tusculanae Disputationes In summo apud illos [Graecos] honore geometria fuit, itaque nihil mathematicis inlustrius; at nos metiendi ratiocinandique utilitate huius artis terminavimus modum.! 1 “With the Greeks geometry was regarded with the utmost respect, and consequently none were held in greater honor than mathematicians, but we Romans have restricted this art to the practical purposes of measuring and reckoning.” (Tusculanian Disputations, trans. by John Edward King [Cambridge, MA: Harvard University Press, 1950 (1927).])

Lux TRES It was the same Cicero who would show a clear interest in scientific subjects in his efforts to publish Lucretius’ De rerum natura? and who boasted of having rediscovered the tomb of Archimedes when, as a quaestor in Sicily in 75 B.C., he saw on a stele hidden in some bushes the image of a sphere inscribed within a cylinder — the subject of the most important work of the Syracusan mathematician (Tusculanae 5.23.64). In the second book of the Academica Priora (Lucullus 116-117), he seems convinced of the logical superiority of geometry over the more nebulous arguments of the philosophers: non quaero rationes eas quae ex coniectura pendent, quae disputationibus huc et illuc trahuntur, nullam adhibent persuadendi necessitatem; geomettae provideant, qui se profitentur non persuadere sed cogere, et qui omnia vobis quae describunt probant. non quaero ex his illa initia mathematicorum, quibus non concessis digitum progredi non possunt, punctum esse quod magnitudinem nullam habeat, extremitatem et quasi libramentum in quo nulla omnino crassitudo sit, liniamentum <longitudinem> sine ulla latitudine [...] carentem [...] Quod si geometricis rationibus non est crediturus, quae vim adferunt in docendo, vos ipsi ut dicitis, ne ille longe aberit ut argumentis credat philosophorum.? We should not forget that in his youth Cicero prepared a poetic translation of Aratus’ difficult Phaenomena and put an abstract discussion of astronomy in the mouths of intellectuals and political men meeting at the house of Scipio Emilianus (De re publica 1,13-14; 19-22), and made special reference there to Archimedes’ planetaria that Marcellus had brought back as a spoil of war from Syracuse.* In the same period Marcus Terentius Varro dedicated two entire sections of his encyclopedic treatise Disciplinarum libri to arithmetic and geometry. This treatise is probably the source of many mathematical terms later used 2 There are two authoritative sources for this: Jerome, in an addition to Eusebius’s Chronicon, refers to 94 BC (cum aliquot libros per intervallainsaniae conscripsisset, quos postea Cicero emendavit, propria se manu interfecit anno aetatis XLIV) and Cicero himself, when he praised Lucretius’s art in a etter to his brother Quintus written in February 54 BC, QFr. 2.10 (9) (Lucreti poemata ut scribis ita sunt, multis luminibus ingeni, multae tamen artis [“Lucretius’s poems are just as you write, born of his brilliant talent, but also of a very refined art”].) 3 “I am not asking these people about those first principles of mathematics which must be granted before they are able to advance an inch — that a point is a thing without magnitude, that a ‘boundary’ or surfaceis a thing entirely devoid of thickness, a line a thing without any breadth [...] But if heis going to refuse credence to the methods of geometry, whichin their teaching exercise a compelling force, as your school itself asserts, surely he for his part will be far from believing the proofs of the philosophers.” (De Natura Deorum - Academica, trans. by Harris Rackham [Cambridge, MA: Harvard University Press, 1933].) 4 On the admiration that Cicero held for Archimedes, see what I have written in I/ Grande Archimede (Rome: Sandro Teti, 2006), pp. 95 and 105-106. ARITHMETIC AND GEOMETRY IN ANCIENT ROME 13 in Latin, but only very small portions of it are left to us today. Varro himself was the author of two other works dedicated to mathematics, but which have been lost: Atticus de numeris and De principiis numerorum (the latter in nine books!). The efforts of personalities such as Cicero and Varro show the widespread interest in mathematics that developed in Rome in the 1* century B.C., when the young members of the ruling class received their most advanced education in Greek schools, in which scientific subjects had a highly formative value. Indeed, during Cicero’s lifetime, Aratus’ Phaenomena were translated by Varro Atacinus and, a little later, by Germanicus, who belonged to the gens Iulia and was adopted as a son by Emperor Tiberius. An additional version of Aratus was prepared in the mid-4® century by Avienus, a high-ranking member of a senatorial family. At the height of the Roman Empire, Rome itself became an important center of Greek science and culture, e.g. Galen’s long stay in the court of Marcus Aurelius, a Roman emperor who even wrote of philosophy in Greek. But in Rome, as Cicero suggested in the passage cited at the beginning, the separation of science and technology was much less pronounced than in the Greek world, and the specific interest in arithmetic and geometry presented a characteristically technical/practical character. An example from the Augustan age is the funny interrogation of a boy on the division of a coin that we read in Horace, Ars Poetica 325-330 and, at a superior level, Vitruvius’s De Architectura. From the 1" century A.D. we could think to the De aquaeductu urbis Romae of Frontinus (geometricae artis inspector providissimus), who was consul various times during the reigns of Vespasian and Trajan. In late antiquity, such a technical/practical approach is evidenced in particular by the Gromatict, whose technique of measurement and delimitation of agricultural properties is best illustrated in a splendid late-antique manuscript kept in Wolfenbüttel.” The mensores were charged with military, civil and bureaucratic surveying, and, among other things, they took care of the calculation of agrarian taxes, the measurement of communal property, the construction of camps, the division of colonies, the construction of bridges and viaducts, and the inspection of borders. A number of professional schools were founded 5 For the writings and doctrines of the Gromatici see the still fundamental studies Die Schriften der römischen Feldmesser, herausgegeben und erläutert von Friedrich Blume, Karl Lachmann und Adolf August Friedrich Rudorff (Berlin: Reimer, 1848-52, reprint Hildesheim, 1967) and MORITZ CANTOR, Die römischen Agrimensoren und ibre Stellung in der Geschichte der Feldmesskunst; eine isorisch mathematische Untersuchung (Leipzig: Teubner, 1875, reprint Wiesbaden: Sändig,

across the Empire for their training and in the instructional texts they used a few elementary explanations of geometry: the definition of point, line, plane figure, the calculation of the area of a square, of a rhombus, of a parallelogram and of a trapezoid, the calculation of the perpendicular bisector of a triangle, given the sides, and that (though not always exact) of the areas of regular polygons, and even the very approximate relationship between the diameter and the circumference. The presence of their extracts helped to maintain a specific connection between geometry and land surveying for the entire Middle Ages. Of much more significance for the relationship with practical disciplines, however, were the scientific interests that developed in the learned circles of Rome along with the broad influence of neo-Pythagoreanism and that exerted by the captivating personality of one Nigidius Figulus (praetor in 58 B.C.). Even beyond Cicero and Varro the interest of the best Latin intellectuals in science had revealing examples. One need only think of Pliny the Elder, the author of the Naturalis Historia, who died heroically in an effort to help people in distress and to better understand the tragic eruption of Vesuvius in 79 A.D., or Quintilian, who suggests complex problems of geometry as being fit for the formation of the young orator (1.10.34-49). In the and century we remember Apuleius, to whom we owe the first Latin version of the Introduction to Arithmetic of Nichomachus of Gerasa, now lost but recorded together with that of Boethius by Cassiodorus.® Still at the end of antiquity a grammarian of great success such as Priscian wrote a brief treatise dedicated to a specific mathematical subject, the De figuris -numerorum quos antiquissimi babent codices (Grammatici Latini IL, 406417). One must recognize, in any case, that the Latin authors, who cited almost only Euclid and Archimedes (even in works of philosophy, literature, and rhetoric), were never able to get close to the questions discussed by contemporary Greek scientists. This explains why no Latin author is ever mentioned in any analogous Greek work; moreover, not even the Latin grammarians, with famous names such as Varro and Priscian, were ever cited by their Greek counterparts. 6 Institutiones 2.4.7: reliquae vero quae sequuntur, sicut eius iam qualitas ostendit, ut sint atque subsistant, indigent aritbmetica disciplina. quam apud Graecos Nicomachus diligenter exposuit. hunc prius Madaurensis Apuleius, deinde magnificus vir Boehius Latino sermone translatum Romanis contulit lectitandum; quibus, ut aiunt, si quis ius utitur, quantum hominibus fas est, lucidissima procul dubio ratione perfunditur (“indeed the remainders that follow, just as its quality has already shown, as they are and stand, require mathematical discipline, as among the Greeks Nichomachus diligently explained. First Apuleius of Madaura, then the magnificent man Boethius completed a Latin translation of it and it must be read often. As they say, if someone makes use of something as often as it is tight for such men, doubt is washed away by brilliant reason”). ARITHMETIC AND GEOMETRY IN ANCIENT ROME 15 One significant development in mathematical interests is revealed in some Latin authors of the 4Ÿ and 5® centuries, when the influence of neo- Platonism became more tangible. It should be of no surprise that the greatest intellectual of the age, Augustine, makes explicit use of geometry in his intimate conversations with his Reason (Soliloguia 1.4.10-11): Rarıo. Ergo lineam in duas lineas per longum scindi, manifestum tibi est nullo modo posse? AUGUSTINUS. Manifestum. R. Quid, transversim? A. Quid, nisi infinite secari posse? R. Quid, sphaeram ex una qualibet parte a medio, ne duos quidam pares circulos habere posse pariter lucet? A. Pariter omnino. R. Quid linea et sphaera? unumne aliquid tibi videntur esse, an quidquam inter se differunt? A. Quis non videat differire plurimum? R. [...] Sic enim nosti lineam et nosti sphaeram, cum se non sic habeat linea ut se habet sphaera. Quamobrem risponde utrum tibi satis sit sic Deum nosse, ut pilam illam geometricam nosti; hoc est, ita de Deo nihil, ut de illa, dubitare.? Augustine’s interest in arithmetic and geometry is specifically connected to that in astronomy and music in the second book of the De ordine, in chapters 14.39-15-43,® and it is not by chance that Sandro Botticelli 7 “REASON: So it is clear to you that a line cannot be divided in two along its length? AUGUSTINE: It is clear. R: What about transversally? A: Can it not be divided infinitely? R: And is it not equally clear that of all the circles that pass through a part more or less distant from the center of a sphere, not even two of them can be equal to one another? A: It is equally clear. R: What about the line and the sphere? Do they seem the same to you, or is there a different between them? A: Who doesn’t see that they differ greatly? R: [...] So you know the line as you know the sphere, although the line isn’t the same thing as the sphere, Answer me now whether it is enough for you to know God as you know the sphere in geometry; that is, having no doubt about God, as you have none about it?” 8 See, for example, 2.15.42: Hine est profecta in oculorum opes et terram coelumque collustrans, sensit nibil aliud quam pulchritudinem sibi placere, et in pulchritudine figuras, in figuris dimensiones, in dimensionibus numeros; quaesivitque ipsa secum utrum ibi talis linea talisque rotunditas vel quaelibet alia forma et figura esset, qualem intellegentia contineret. Longe deteriorem invenit et nulla ex parte quod viderent oculi cum eo quod mens cerneret comparandum. Haec quoque distincta et disposita in disciplinam redegit appellavitque geometriam. Motus eam caeli multum movebat et ad se diligenter considerandum invitabat. Etiam ibi per constantissimas temporum vices, per astrorum ratos difinitosque cursus, per intervallorum spatia moderata, intellexit nihil aliud quam illam dimensionem numerosque dominari. Quae similiter definiendo ac secernendo in ordinem nectens, astrologiam genuit, magnum religiosis arg torment curiosis (“Thus she

painted Saint Augustine in bis Study with geometric designs and figures in the background (Florence, Uffizi, c. 1490). A specific interest in geometry can be seen in chapters 6.10-13.22 of De quantitate animae, a catechistic dialogue strangely ignored by historians of mathematics, from which PseudoBoethius deduced the principal arguments of the Altercatio duorum geometricorum, a brief treatise quite popular in the Middle Ages, published 25 years ago by Menso Folkerts.” A certain theological Platonism is evident in the De quantitate animae, but it is also full of literal references to definitions and postulates of the first book of Euclid’s Elements, as one can see, for example, in this passage from chapter 8.13: ARITHMETIC AND GEOMETRY IN ANCIENT ROME 17 A. Nunc ergo illud responde, quomodo tribus lineis figuram feceris? E. Cum se finibus iungunt. A. Quid? ubi se iungunt, nonne videtur tibi angulus fieri? i E. Ita est. A. Quot ergo angulis haec figura constat? E. Totidem quot lineis. A. Quid? ipsas lineas pares constituis, an impares? E. Pares. A. Quid? anguli tantumdem omnes patent, an est alius alio contractior vel apertior? E. Etiam ipsos pares esse video. Augustinus. Nam si et quid sit linea, et quid sit figura iam tibi notum est, dic quod abs te quaero, id est utrum putes ullam figuram fieri posse, si linea ex utraque parte, aut ex altera per infinitum ducatur? Evopius. Nullo modo id fieri posse confirmo. A. Quid igitur agendum est ut figuram faciamus? E. Quid, nisi ut illa linea infinita non sit, et ducatur in circulum, ut ex alia parte se contingat? non enim video, quomodo aliter possit ex una linea concludi aliquod spatium; quod nisi fiat, secundum tuam descriptionem, figura non erit. A, Quid, si rectis lineis figuram facere velim? potest fieri ut de una linea fiat, an non potest? E. Nullo modo. A. Quid, duabus? E. Ne hoc quidem. A. Quid, tribus? _E. Video posse. A. Bene igitur nosti ac tenes, cum figura lineis rectis facienda est, minus quam tribus non posse. An si ulla tibi adversetur ratio, de hac te sententia devocabit? E. Plane si quis mihi hoc falsum esse monstraverit, nihil erit quod me scire posse confidam. crossed into realm of the eyes and went over the earth and heaven. She sensed that what pleased was nothing other than the harmony, and in the harmony figures, in the figures the measurements, and in the measurements numbers. And she reflected in herself whether this line or this circle or any other form or figure is similar to that which belongs to the intelligence. She found that they are much more imperfect and that what the eyes sees can absolutely not be compared to what the mind sees. And so she analyzed and systematized these notions, placing them into a discipline that she called geometry. The movement of the heavens attracted her very much and encouraged her to study it attentively. She understood that also here, through the constant successions of times, the fixed and defined course of the stars though exactly established distances, was the realm of none other than the measurement and numbers. She systematized these notions as well with definitions and divisions, and thus inaugurated astronomy, which is a great subject of study to the religious and a torment to the superstitious.”) 9 MENSO FOLKERTS, Die Altercatio in der Geometrie I des Pseudo-Boethius. Ein Beitrag zu Geometrie im mittelalterlichen Quadrivium, in Fachprosa-Studien, edited by Gundolf Keil et al. (Berlin: E. Schmidt, 1982), pp. 84-114. À. Potestne fieri, ut in figura, quae tribus rectis paribus lineis facta sit, impares anguli sint; an non potest? E. Nullo prorsus modo. A. Quid? si rectis lineis tribus, sed imparibus figura constet, possunt etiam in ista pares esse anguli, an aliud intellegis? E. Omnino non possunt. A. Recte dicis: sed dic, quaeso, quaenam tibi figura melior videatur et pulchrior? eane quae paribus, an quae imparibus lineis constat? E. Quis dubitet eam esse meliorem in qua aequalitas praevalet? !° 10 AUGUSTINE: If you know what line is and whata figure is, answer my question i that a figure can exist if a line extends infinitely from nen or even from,just me youthing Evonius: I affirm that it cannot happen in any way. A: So what must be done in order to make a figure? . E: What, if nor that the line be made finite and extended into a circle so that it reconnects to itself from the other side? For I do not see how a space can be marked out with a single line; if this is not done, according to your explanation, it will not be a figure. ’ from A: And iftine a sind I should want to make€ aa fifigure out of straight or not ght lines? l i that one be made Is it possible E: Impossible, A: With two, then? E: Note even with two. A: With three? È ge har it is possible, : So you have learnt well, and you remember that when a figure is to be made f i male lines, itye cannot wh be formed ed with with Jess less than than ıthree. If an argument seemed wrong toen you, would it os : early, Clearly, if someone showed me that at it iss false, false, ththere would be nothing i that I would trust : So answer me now: how do you make a figure from three lines? E: The lines are connected at their ends. sane om eee oes A: And where they are connected, does it seem to you that angles are formed?

e subject of MARIO GEYMONAT cularly rdans Capella, Cassiodorus, and Isidore. Parti ; y acquired a weig rds the end of antiquity, science graduall metic and geometry, both par pores in Latin schools, special arith vi ed Platonically as the basis mea ingful is the interest in numbers — consider ci the rad and scholar from therst obius, a politician (ver clarissimus et s, the highest level of the senatorial class) of geometry _ that one can see in Macr Commentarii in sommare of the De re pi ic user influence on the who e me eval half of the 5° century, who had a huge as an example, ss 7-1 and culture of magic and wisdom. Let us read, 18 of chapter five of the first book of the Scipionis, which he dedicated to the final chapters oe philosophy conserve or ki Cicero’s most important work of political some Greek terms are wicely thanks to Macrobius. Note how in this text the language used in used in a very natural manner; evidently, it was . period for scientific discourse, even in the West inal,a est corporum terminus, ita lineis term [7] haec superficies, sicut e finiuntur. er! ace nt corpore nomine ypappág Graecia nominavit: punctis linea industria geometriae | sp urB quae mathematica vocantur, de quibus sollerti ris cogitatur, pro orme sublect ergo haec superficies cum ex aliqua parte corpo trum ut trigonum seuans corporis accipit numerum linearum. nam seu em lineis sese a = me tang ut quadratum, seu plurium sit angulorum, totid nendi sumus quod o tibus planities eius includitur. [9] hoc loco admo . : Yes. e? È How many angles pi up this figur E: There are as many angles as lines. al? A: And are the lines equal in length, or unequ is one more dosed or more open than the jr ET the angles have the same degree, or E: These too I see as equal. others? made of three equal straight lines, of which A: Is it possible or impossible to make a figure the angles are unequal? A straight lines, can the angles in it be equal? À oe Geese is formed by three unequal Or do you see things differently? E: It is absolutely impossible. t0 be better and more figure seems to you ones? A: You are correct; but tell me, È please, which equal lines or the one made of unequal lines de of s is better? a youll doubtthe one in which equality prevail i beau ARITHMETIC AND GEOMETRY IN ANCIENT ROME 19 ductu una dimensio est — longitudo est enim sine latitudine — planities vero quam longitudinis, latitudinis et altitudinis dimensionibus constat. ex his tribus in lineae Greci änıpüveiav vocant, longo latoque distenditur, alto caret, et haec planities sum sua plenitudine nitatur, iure plenus vocatur.!! quantis lineis contineatur expressimus, soliditas autem corporum constat cum his duabus additur altitudo; fit enim tribus dimensionibus impletis corpus solidum quod otepedv vocant, qualis est tessera quae xBog vocatur. [10] si vero non unius partis sed totius velis corporis superficiem cogitare, quod proponamus esse quadratum, ut de uno quod exemplo sufficiet disputemus, iam non quattuor sed acto anguli colliguntur. quod animadvertis si super unum quadratum quale prius diximus alterum tale altius impositum mente conspicias ut altitudo quae illi plano deerat adiciatur fiatque tribus dimensionibus impletis corpus solidum quod otepeóv vocant ad imitationem tesserae quae KbBoc vocatur, [11] ex his apparet octonarium numerum solidum corpus et esse et haberi. si quidem unum apud geometras puncti locum obtinet, duo lineae ductum faciunt quae duobus punctis ut supra diximus coercetur, quattuor vero puncta adversum se in duobus ordinibus bina per ordinem posita exprimunt quadri speciem, a singulis punctis in adversum punctum eiecta linea. haec quattuor ut diximus, duplicata et octo facta, duo quadra similia describunt, quae sibi superposita additaque altitudine formam cybi quod est solidum corpus efficiunt, [12] ex his apparet antiquiorem esse numerum superficie et lineis ex quibus illam constare memoravimus formisque omnibus. a lineis enim ascenditur ad numerum tamquam ad priorem, ut intellegatur ex diversis numeris linearum, quae formae geometricae describantur [...] [17] Pythagorici vero hunc numerum iustitiam vocaverunt, quia primus omnium ita solvitur in numeros pariter pares, hoc est in bis quaterna, ut nihilo minus in numeros aeque pariter pares divisio quoque ipsa solvatur, id est in bis bina. eadem quoque qualitate contexitur id est bis bina bis. [18] cum ergo et contextio ipsius pari aequalitate procedat, et resolutio aequaliter redeat usque ad monadem, quae divisionem arithmetica ratione non recipit, merito propter aequalem divisionem iustitiae nomen accepit, et quia ex supra dictis omnibus apparet quanta et partium suarum et seor- 11 [7] As the terminus of a body is the surface, so the termini of the surface are the lines, grammaî in Greek; and lines terminate in points. Now these are what are known as mathematical bodies, about which geometricians dispute with skill and zeal. [8] When we consider the surface of one side of a body, we find that the number of its lines depends upon the form of the underlying body: whether it be triangular, quadrangular, or polygonal, its surface is enclosed by the same number of lines as it has angles. [9] At this point it would be well to keep in mind that all bodies have three dimensions: longitude, latitude, and altitude. Of the three we obtain one dimension by drawing a line (in longitude we do not have latitude); a surface, called by the Greeks epiphaneia, is marked by length and breadth and lacks thickness (we have indicated how many lines confine this surface); bodies acquire solidity by adding altitude to the other two dimensions (a solid body is produced by filling the space enclosed by three dimensions, and it is called stereon, an example being the cubical die). [10] If you should wish to consider not the surface of one side, but all the surfaces of a solid figure, which we may assume for the purpose of illustration to be rectangular and equilateral, there are now not four but eight angles to be reckoned with. This you will recognize if you imagine that above one quadrate (a surface such as

A little later, however, in the years that follow the formal end of the Western Empire, knowledge of Greek becomes ever more uncertain, and these cultural and commercial exchanges across the shores of the Mediterranean become rare. It was a sense of the danger this represented for the entire Latin culture that pushed the most responsible of the learned men of Italy to endeavor not to lose the knowledge accumulated in previous centuries by translating some of the most important texts of Greek science and philosophy into Latin. Such was the generous attempt of the aristocrat Serverinus Boethius, who translated Aristotle’s major works on logic (the Categories, the De interpretatione, and the Analytica priora which we have along with his version of Porphyry's Isagoge). Boethius also published in Latin the De inszitutione arithmetica, which was actually a re-elaboration of the Introduction to Arithmetic of Nichomachus of Gerasa (1*-2°4 c. A.D.), a version of which had already been prepared by Apuleius.'? Up until his last, most difficult years, when he wrote the De consolatione Philosophiae in prison, Boethius remained convinced of the fundamentally numeric structure of the world (Tu numeris elementa ligas, “You bind the elements with numbers”: book 3, metr. 9, v. 10). This great intellectual also wrote a De institutione musicae to complete the was described above) you have placed another exactly like it, so that altitude, which was lacking in the plane, is now added: with the three dimensions filled up a solid body is produced which geometricians call a die or cube. [11] Hence it is apparent that the number eight both is and is considered a solid body, if indeed one is represented by a point, two by the drawing of a line (which, as we said above, is limited by two points), and four by points arranged at right angles to * each other, with lines extending between the points to form a square. When these four are duplicated and made eight, forming two equal squares, and one is superimposed upon the other, giving the figure altitude, the result is a cubical figure, which is a solid body. [12] Thus it becomes clear that numbers precede surfaces and lines (of which surfaces consist), and in fact come before all physical objects. From lines we progress to numbers, to something more essential, as it were, so that from the various numbers of lines we understand what geometrical figures are being represented [...] [17] The Pythagoreans, indeed, called the number eight Justice because it is the first number that may be divided into two equal even numbers and divided again into two more equal even numbers. It is also the product of equals: two times two times two. [18] Since it is the product of equal even numbers and may be divided equally, even down to the unit, which does not admit of division in mathematical computation, it deserves to receive the name Justice, And since it is clear, from what has previously been said, to what extent it depends both upon the fullness of its parts and upon its own fullness, it deserves to be called full.” (Commentary on the Dream of Scipio, trans. by William Harris Stahl [New York: Columbia University Press, 1952], pp. 96-99.) 12 As Jean-Yves Guillaumin affirms in his introduction to his edition of the De institutione aritmetica (Paris: Les Belles Lettres, 1995), p. xxxix: “Telle est la substance de l'enseignement pythagoricien que Boéce, armé de son excellente connaissance du grec, entreprend de transmettre au monde latin dans son Institution Arithmétique. Titre justifié: le caractère méthodique et progressif de l’ouvrage fait bien du traité de Boèce une institutio au sens exact du terme, ‘exposé méthodique’, à la fois ‘manuel scolaire’ et ouvrage de référence, qui se propose de mettre la science arithmétique à portée des débutants, et en même temps de fournir au professeur un aide-mémoire et un exposé systématyque”. ARITHMETIC AND GEOMETRY IN ANCIENT ROME 21 quadrivium, along with a treatise on astronomy, of which the former has been preserved to us whereas the latter is now lost. What the ancients admired him for most, however, was his version of the entirety of Euclid’s Elements, a translation that confronted very complex logic and semantic values, that never had been tackled in the language of the West. Let us read what Cassiodorus writes of this great innovation in his Institutiones (2.6.3): Cuius disciplinae [i.e. geometriae] apud Graecos Euclides, Apollonius, Archimedes necnon et alti scriptores probabiles extiterunt; ex quibus Euclidem translatum Romanae linguae idem vir magnificus Boethius edidit. Qui si diligenti cura relegatur, hoc quod praedictis divisionibus apertum est manifestae intellegentiae claritate cognoscitur.!3 For the definitions, the postulates, and the propositions of the first theorems, Boethius’s translation was preserved to some degree during the Middle Ages, and two small works that synthesize and resume it were attributed to him (the so-called Geometria I and Geometria II, in five and two short books respectively). In some of the manuscripts that conserve these works one can find the first western notations of Arabic numerals, which Boethius obviously did not know. Boethius's translation of Euclid was dedicated to Symmachus (according toa9™-century gloss in the ms. Vaticanus Lat. 3123, f° 54v), who held the title of patricius.!* In my opinion, when treating the subject of Latin geometry, it is not very fruitful to stop on the Geometriae of Pseudo-Boethius in medieval translation (the Geometria I was produced in Corbie at the end of the 8% century and the Geometria II was probably made in Lorraine at the beginning of the 11® century). They display scant interest in proofs, which are the true fulcrum of Euclid’s work, and they preserve only the very elementary ones in theorems 1-3 of the first book of the Elements. In order to study the good level that Latin scientists reached in the field of geometry in late antiquity, it would be better to reflect on a few fragments of Latin translations of Euclid, which contain much more complex proofs and which were the last product in that field from the moment when 13 “Of this discipline [i.e. geometry], among the Greeks, there were Euclid, Apolloni Archimedes and probably other writers as well. Of these writers, the magnificent aan Boethius published a translation of Euclid into the Roman language. Whereby, if it is reread with diligent care, that which is opened up by the aforementioned divisions becomes known by the brilliance of clear intelligence.” 14 This form is echoed by the Patricius to whom the Geometria is dedicated, published und Boethius’s name by Gottfried Friedlein in 1867 (Leipzig: Teubner), p. 373, AA

knowledge of Greek went through its greatest crisis in the West, i.e. in the last years of the 5“ century and the first of the 6. Six folios (three bifolios) of Euclidean propositions and proofs are preserved in a palimpsest of Verona’s Biblioteca Capitolare. Perhaps originally from Ravenna, the Theodoric’s capital (as Guglielmo Cavallo suggests), these fragments also hold evidence for a series of authorial corrections, possibly produced by the same Boethius. Two very different folios, however, were copied in Corbie at the beginning of the 9* century and, according to Bernhard Bischoff, originate in the court of Charlemagne. Let us begin with one of these last folios (Fig. 1), £. 2v of the second codex 23 ARITHMETIC AND GEOMETRY IN ANCIENT ROME dep Tur . “rerwur que 4, gelo equecliy ft 1 Accra fimo tear imo Kun dy-inSOMA fm Ferro zum Tee DI ee pau fuer fer prime ra que reengulo chuabre ugaprime eer uc To Neg eft quodapromo sfeeundo aferunde Sr no ficurab cm que fuif umufdiferibro queed ran me one fic ‘eda gele firs deriewe pwra cu fimo &tricerap fera ficur copuenıe ao ves arc gent feb omem of 757 of the library of the University of Munich, which I published in 1967.!5 fimo que ringen It completely lacks punctuation, which was evidently absent also in the. antigraph made in continuous writing, perhaps in capitals (I add a few marks to orient us in transcription). The page contains a completely literal, though incomprehensible, translation of the end of the proof of the eighth and the beginning of the proposition of the ninth theorem, in the second book of the Elements: ny'fimo ferro quo {xu Aumar ff für circum KTP fimocdpone To dem triangulr zum ft quo prime fergueminuf ferfian que [column I] <.......... / ....> adversum / p<rimum> et tertium qua/dr<an>gulo equalis / [51 est que ducentissimo / et tricentissimo et qua/dringentissimo scito / enim quo sexuagissi/mo et nono. sed ille du/[10]centissimo et tricentis/simo et quadringen/tissimo scito quo sexu/agissimo et nono to/tum est quo primo / [15] et quinto septimo et / quarto quatrangu/lo, quod est a primo / et quarto: quod autem / quadragenis que sub / [20] primo et secundo se/cundo et quarto ad/versum primum et ter/tium equalis est quod / a primo et quarto / [25] qua_ drangulo; equa/lis autem illa secunda / et quarta que secun/da tertia. quod autem / quadragies que sub / [30] primo et secundo, se[column II] <cundo et tertio circumda> / tum directis angul<is> / adversus primo et ter<tio> / quadrangulo equalis es<t> / [5] que a primo et quarto, hoc / est quod a primo et secundo / et secundo et tertio sicut ab / unius discribto quadran/gulo. si enim dericta picta / [10] scisa, sicut convenit, quo / quadragenis sub totum / et unius scissuris circum/datum triangulis adver/sus quominus scisum qua/[15]drangulum equalis est que / autem ad totum et que dic/tum scisum sicut ab unius / describtum quadrangulo; / quod oportet ostendere. / [20] figura geometrica / Cap. NONO / Si directa pincta scissa. 15 “Nuovi frammenti della geometria ‘boeziana’ in un codice del IX secolo”, Scriptorium, 1967, 21, pp. 3-16. 16 In Greek (Johan Ludvig Heiberg-Euangelos S. Stamatis I, pp. 80.14-81.4): repıexöpevov ÖpBoydviov peta tod dnd AT tetpayovov Toov tori 1 LTY yvdpovi xai TÔ ZO. AA 6 ETY yvopov kai tò ZO Giov éoti tò AEZA terpáyovov, 6 Éonv dnd 16 AA tò äpa terpäkıg drò tüv AB, BA però tod nd AT toov éoti tò and AA tetpaydve ton dè fi BA ti BI. tò äpa tetpékis Sd töv AB, BT repiexduevov òpBoydviov però toi dnd AT terpaydvov toov eoti 1 dnd tig AA, toutéon TÔ darò tis AB rai BI dc ano mäc dvaypapévii terpayave. Eav Gpa edbeïa ypappù tundi, dg Ervyev, 16 ue AMINO fepu mog pce? to due Trcengu lo aod coprimo euver mr Bly uche um ferflern fear ecbomuar «4 were quoi . rino Kficun do fe cunde Bj wwe ‘ to cod ver fim prime aa num go r'efiquod prime Squar To È gt Fes as als ° nd NDA A la #41 J In La quadrengelorqua deere na quodccure waadraguy que fib promo Blfecando fe A v quiet rec quefexun von i AAN pei wu a, 3 drreezee pine Teo fer

T + = N q NIN ZH M “A “A I Ho \ I E P u Yin XL XX U xx LX \ LX|/ c| .; L N LXX / Vv 4 N\ EpcectLal 2 25 spoke limited Latin, and it could only have acted as a crutch for a true translation, such as that completed by the 20-year-old Boethius, evidently assisted by secretaries and assistants. 115 N Le N/ vi mm u un I A B T A ARITHMETIC AND GEOMETRY IN ANCIENT ROME The scientific and linguistic level of the six palimpsest folios in Verona is much higher. They are preserved in the same codex XL (38) with 51 folios of a an important manuscript of Virgil (collated by Otto Ribbeck in 1853 and more recently by me),'* with the precious Scholia discovered and published by Angelo Mai in 1818, as well as with some books of the first decade of Titus Livy (collated by Theodor Mommsen in 1867). The mathematical folios contain various theorems of books XI (24-25), XII (2-3 and 8), and E 6 .A v zZ vum XXX VII XIII (2-3 and 7) of the Elements, with their complex proofs and figures still annotated with Greek letters, similar to those that Johan Ludvic Fig.2 CAP. Nono A geometricfigure is drawn in the second column and the title The Gr ink, red in traced are (S) aph and the initial letter of the new paragr in the Theon by hed publis one the is n versio Latin text that is the basis for this as is uncial, in second half of the 4® century, read probably in a manuscript rie geomet the In A. proved by the frequent confusions between A and 1 ate trans are points single e figure (Fig. 2) the Greek letters that denot o instead III by d denote is T nt: with Latin numerals, but errors are freque did not CCC and E by V instead of CC - all evidence that the translator points te indica that letters Greek The n. notatio of the Greek system know are also translated with Latin numerals in the text of the proof. Thus, o et quarto ‘AEZA (line 2 in the Greek) becomes primo ei quinto septim ds line procee tion transla The (lines 14-16 in the first column of the Latin). for line and slavishly maintains the succession of Greek words, as in show a ree modern scholastic interlinear translations, and does not can Ka This at. gramm Latin of or s matic mathe knowledge of Greek originally explained, I think, only by the fact that this Latin translation was written between the lines of a Greek codex of the Elements, similar to the Greek versions seen in various bilingual papyri of Virgil.” of Euclid, as evidenced by these fragments, was the work Heiberg printed in his modern edition (IV, p. 143). The manuscript originally included more than 70 gwaterniones (as can be seen at the bottom of f. 4v): the number of book XI has been omitted, but those of books XII and XIII have been changed respectively to LIB. XIII and LIB. XV, which could indicate an original division in three parts of Euclid’s broad and complex book X. The folio I reproduce here (Fig. 3) is my transcription of f. 3 (3419), for the completion of which I used Angelo Mai’s manuscript notes in the Vatican Library. It is the most readable of the Verona palimpsests and in the first column it carries the translation of the last lines of the lemma at the end of the second theorem of book XII with its figures (Fig. 4) and in the second column the beginning of the third theorem (pp. 148-150 Heiberg). Let us read the proposition in the first fifteen lines: Omnis pyramis quae habet triangulam sedem dividitur tam in duas pyramidas aequales, triconas sedes habentes ac similes sibi, adque in duo secmenta aequalia; scilicet ex arte discedet ac duo recisamenta totius pyramidis quae forte contigerint maiora sunt ab eo quod dimidium esse cernitur.!? It is a difficult subject, and in the interlinear translation authorial The translation reconsiderations and corrections are frequent. Maria Timpanaro Cardini of a Greek who wrote me in 1963 about these authorial interventions as follows: evov dpboydviov perd hj ne md tetparig ind tig SAng Kai Evög Tv TUNHÉTOV nepiexop Kai 105 sonen ei uahe Tunkarog tetpayavov Toov tori tH and TE tis dang x Er à sition is on Tani dvaypapévr tetpaybvo* òrep Eder deikar. 0. Eav ebbeia voeg ia published y Fri theorem is available in Latin also in Pseudo-Boethius’s Geomeir per ; een inaeg ac p. 386, 3-6, where it begins differently: Si recta linea per aeq lia reprint of he to preface the see Virgil of papyri the of r characte and 17 On the value, number , Edizioni di Storia e my critical edition of his works: P. VERGILI MARONIS, Opera (Rome: II. 2008), pp. Xml and XXIV-XXV 18 I wrote on the Verona palimpsest in Euclidis Latine facti fragmenta Veronensia (Milan: Cisalpino, 1964), pp. 53-65 and in “I codici G e V di Virgilio”, MIL, 1966, 29, pp. 326-335. 19 In Greek (Heiberg IV, p. 148, 21-26): Maca mupapis tpiyovov Eyouse Baow Srarpsîtor eig dbo rupanidag ious te Kai dpoiag GAMMA 1g Kai [ôpoiac] rH Aq tprydvoug Époboas Béoeic Kai sic Sio xpicpata ica Kai tà bo rpiouara usitova gotw à tò fuiov tig Ens mvpopidos, [Every pyramid that has a triangular base is divided into two equal pyramids, both having triangular bases and similar to each other, equally divided into two sections; and these two prisms together are greater than the half part of the whole pyramid”).

Fol. 3 (341r) XIII ARITHMETIC AND GEOMETRY IN ANCIENT ROME E Onatidpy RA MISQUAE MR DEETTAGDUÓRIËCISAMEN (tAtOTiDSd RDS QUARFORTECONTIGERN DIMIDIUMESSECERNI I(rURSITRYRAMISIÙ)NA CUMNPROIPRDIUMEX ARTPRODUGTACHIUS (B)AS(1S)(EAYDEMQUAE meter se «&BTsIrTRIANGULÙM SITQUEAPEKEIUSÌX A: LICORUIADITIDITÙÀ TAMINDUASPYRAM | DASAEQUÄLESTRIAGE Fig. 3 In particolare alle righe 10-11 il traduttore si trova davanti a una parola, npiopata, che non sa come rendere; anche alla riga 8 era in dubbio: aveva scritto sectiones poi corretto in segmenta; andando oltre ha trovato la parola che gli pareva più fedele al testo [...] La differenza tra segmentum e sectio sembra essere quella tra tufjua e xatatopn. Il rpioua era qualcosa di diverso; era effetto di una risezione, ma era poi anche un solido con caratteristiche ben definite, quantunque tutto ciò non si possa affermare senz’altro.”° 20 “Especially at lines 10-11 the translator finds himself faced by a word, rpiopara, that he does not know how to render. He was also in doubt at line 8: he had written sectiones, then corrected it with segrzenta. Proceeding, he had later found the word that he though more faithful to the original text [...] The difference between segmentum and sectio seems to be that between tpfipa and xoratopi. The npiopa was something different; it was the effect of a resection, but was also a solid with well-defined characteristics, although all of this cannot be confirmed for certain.” P 4 d riti SECEIO (AB) AEQUALES A B [GNDOASPYRAROIDAS <CONAS> RERATIONES" nu dè © N 1, ADEUMIGITURQUI IZHO- 27 TI F Fig. 4 Fabio Troncarelli offers confirmation that the author of these corrections was Boethius himself in a recent lecture given in Los Angeles?! With convincing evidence, he builds on Bernhard Bischoffs opinion that the fragments of XL (38), rewritten in Luxeuil script, likely came to Verona from Bobbio (“several monks of Luxeuil used to live in the sister abbey of Bobbio during the 7" century — four of them became Bobbio’s abbots!”). These manuscripts must have belonged to “a great scholar, acquainted with very rare commentaries to Virgil and a very rare version of Livy. He was the only one at the end of the 5® century to have a Latin translation of Euclid, revised by the author himself... The name of Boethius seems so obvious...”. Troncarelli’s original conclusions about the provenance of ms. Verona XL (38) are that: “We must remember that Boethius was kept prisoner and executed in Pavia or nearby. During his imprisonment he certainly had some books, because he quotes different authors literally in his Consolatio. Two years after his death, Boethius was rehabilitated by queen Amalasunta and his books were probably preserved in Pavia, whose bishop had been till 521 Ennodius, a relative of Boethius, a town very important and later the capital of the Lombard’s kingdom. Pavia is only 30 miles from Bobbio, and after its foundation in 612 the Monastery of St Columban obtained a lot of ancient manuscripts from Pavia and the Lombard’s kings: it could have obtained also the remains of the books that Boethius owned during his imprisonment. Not by chance, Bobbio’s library owned the only extant manuscript of Boethius of the 6th century: a fragment of the De Institutione Arithmetica, now in Turin (CLA IV, 450), a simple and professional work copied in uncial, in two columns, similar to the other manuscripts, possibly related to Boethius”. 21 Fazio TRONCARELLI, Thrice-born Boethius: the Last of the Romans from Late Antiquity to Renaissance, summer 2007, now in print in BRIAN P. COPENHAVER, Thrice-born Latinity (Los Angeles: University of California).

that Troncarelli’s paleographic observations are revealing: “It is noteworthy a not althoug space, save to s column the manuscript [of Euclid], in two ancient noblest the rustica, s capitali in calligraphic masterpiece, is written script, typical of the oldest copies of Virgil (the use of the capitals in Rome A separate chapter in the knowledge of arithmetic and geometry grande I/ 4, note at, cited volume the is represented by the writings of poets. In in the extent, what to and whether of m Archimede, I raised the proble an poetry their in hid Virgil and Catullus , numbers attention they gave to Roman soldier provoked frustration and shame in the most sensitive Latin ee ARITHMETIC AND GEOMETRY IN ANCIENT ROME 29 conturbabimus illa, ne sciamus, aut ne quis malus invidere possit, cum tantum sciat esse basiorum.?4 or such as translation of Euclid means that Euclid was considered a classic, ions in the old Latin classics). It is remarkable also that the textual correct by no are letters, an elegant, calligraphic small capital, with some uncial on based is n opinio My | means the author’s autograph corrections. as elegant, and fluent very is palaeographical evidence: the corrector’s hand been”. have could writer only the hand of an upper class, learned amiable memory of Archimedes, whose death at the hands of a e] 28 ri (Yappitnc) Archimedes had constructed a series of huge numbers. It was the most well-known of his works in Rome, still cited by Hyginus Gromaticus at the end of the 1* century AD: em, feorem, inventor Nam et Archimedem, virum preclariiiingeniii et magnarum rerum inv r. repleretu si mundus, posset runt scripsisse, quantum arenarum capere n the mid-1* In the realm of poetry the Arenari2us had been openly exalted int 14.350-351): (Punica it of heard had century by SiliusItalicus, whose readers Non illum mundi numerasse capacis harenas vana fides.” So it seems very probable to me that Catullus is winks at this highly original work on numbers in lines 7-13 of poem 5: Da mi basia mille, deinde centum, dein mille altera, dein secunda centum, deinde usque altera mille, deinde centum; dein, cum milia multa fecerimus, Perhaps there is also an allusion here to the columns of the abacus, the instrument for mathematical calculations commonly used in the market place. Certainly, however, the references to mathematics are substantiated as learned ones in poem 7: Quaeris quot mihi basiationes tuae, Lesbia, sint satis superque. Quam magnus numerus Libyssae arenae lasarpiciferis iacet Cyrenis, oraclum Iovis inter aestuosi et Batti veteris sacrum sepulcrum, aut quam sidera multa, cum tacet nox, furtivos hominum vident amores, tam te basia multa basiare vesano satis et super Catullo est; quae nec pernumerare curiosi possint nec mala fascinare lingua.” Here, in line 5, the oracle ofJove is the temple of Ammon in the African oasis of Siwa, in line 4 Cyrenis is the region of birth of the admired poet Callimachus, and Battus, in line 6, is the mythical founder of Cyrene, whereas magnus numerus in line 3 and pernumerare in line 11 are terms characteristic of mathematics. If the number of kisses requested of Lesbia is compared to the enormous number of grains of sand on the African coast, it seems that the Latin poet truly had in mind the beginning of the Pappitns, where Archimedes says that he intends on working out the number of grains of sand “not only of those around Syracuse and in the rest of Sicily, but also of those in every region, both inhabited and uninhabited,” 24 “Give me a thousand kisses, then a hundred, then another thousand, then a second hundred, then yet another thousand, then a hundred. Then, when we have made up many thousands, we will confuse our counting, that we may not know the reckoning, nor any malicious person blight them with an evil eye, when he knows that our kisses are so many” (Poems, in Catullus, Tibullus, Pervigilium Veneris, 2°° edition, trans. Francis Warre Cornish, rev, George Patrick Goold [Cambridge, MA: Harvard University Press, 1988 (1962)]. 22 LOF THULIN, Corpus Agrimensorum Romanorum (Leipzig: Teubner, 19 13), 25 “You ask how many kissings of you, Lesbia, are enough for me and more than enough. As great as the number of Libyan sand that lies on silphium-bearing Cyrenaica, between the oracle of sultry Jove and the sacred tomb of old Barrus; or as many as are the stars, when night is silent, that see the sands of chie great your mad Catullus; kisses, which neither curious eyes shall count up nor an evil tongue bewitch” (Poems, cit. note 24). enius ad p. 148, ed Olof Thulin (adeed they say that Archimedes, a man of brilliant up). be inventor of great things, wrote how much sand the world can contain, if it should 23 “Not without reason men believed that Archimedes had counted globe” (Punica, trans. by James D. Duff [Cambridge, MA: Harvard University Press, the stolen loves of men, — to kiss you with so many kisses, Lesbia, is enough and more than enough for

The reference to the grains of sand in Africa returns in epithalamium 61 (206-210), where the quantity of love games that he wishes on the newlyweds is a palpable poetic exaggeration, even if we consider the teachings of the Kamasutra: Ile pulveris Africi siderumque micantium subducat numerum prius, qui vestri numerare vult ARITHMETIC AND GEOMETRY IN ANCIENT ROME Proceeding chronologically, it is known that in the first part of the Tiberian age the poet Manilius was interested in astronomy and astrology, both subjects a combination of mythology and science, but it is striking that in his poetry there are even a few specific references to geometry, as in lines 545-547 of the first book of the Astronomica, where he gives a measurement (albeit a generic one) of the relationship between the circumference and diameter of a circle, the famous x that Archimedes had measured with precision in his treatise on the Measurement of the Circle: multa milia ludi.?9 In the generation that followed Catullus, Virgil found delight in numbers as well, He - according to the vita Donatiana (§ 15): inter cetera studia medicinae quoque ac maxime mathematicae operam dedit. e» . o . 27 Lines 73-75 of Eclogue VIII should be read from this perspective, where the conclusive affirmation deus … gaudet indicates how certain numbers please divinities: Terna tibi haec primum triplici diversa colore licia circumdo, terque haec altaria circum > effigiem duco; numero deus impare gaudet. In the Georgics (2, 103-106) as well, Virgil refers to the numbers of grains of sand in Africa in relation to the qualities of wine known at the time (today they would be even more!): Sed neque quam multae species nec nomina quae sint est numerus, neque enim numero comprendere refert; quem qui scire velit, Libyci velit aequoris idem discere quam multae Zephyro trurbentur harenae.?? 26 “Let him first count up the number of the dust of Africa and of the glittering stars, who would number the many thousands of your joys.” (Poerzs, cit. note 24). 27 “Among other studies he also interested himself in medicine and especially mathematics.” 28 “Three threads here I first tie round you, marked with three different hues, and three times this altar I draw your image, In an uneven number heaven delights.” (Eclogues - Georgics Aeneid I-VI. trans. by Henry Rushton Fairclough. revised by Goold [Cambridge MA: Harvard l University Press, 1999 (1935)]. 29 “But for the many kinds, or the names they bear, there is no numbering — nor, indeed, is the numbering worth the pains. He who would have knowledge of this would likewise want to learn how many grains of sand on the Libyan plain are stirred by the West Wind (Eclogues, cit. note 28). 31 Quacumque inciditur orbis per medium, pars efficitur tum tertia gyri exiguo dirimens solidam discrimine summam.° Continuing almost three centuries later, a cryptic attention for numbers and geometric figures shows up Publilius Optatianus Porfyrius, a sort of Apollinaire of the age of Constantine. Take as an example his poem VI, an hermetic composition (Fig. 5), the geometrical system of which is explained in §§ 7-9 of the scholium that accompanies it: [7] Idem versus et per amfractus varios sursum ac deorsum discurrunt. [8] In hac eadem pagina quattuor quadrata in angulis sunt senarum litterarum, duo trigona, et duo octogona maiora dimidiata, et unum hexagonum in medio, et scalena per angulos quattuor, omnia pari numero litterarum et pari ordine crescentia vel decrescentia; [9] id est omnia similia, trigona trigonis, octogona octogonis, quadrata quadratis, scalena scelenis.?! To conclude this list, and to be certain of the real interest in geometry and arithmetic on the part of the best Latin intellectuals, a final example may be found in the small lesson of combinatory geometry from the second half of the 4® century offered by Decimus Magnus Ausonius in the introduction to his Cento Nuptialis (XVIII, lines 37-48), where he explains with the help of Greek terms the many possible combinations of the 14 tiles into which a square had been proportionally divided — evidently Archimedes’s stomachion. Its single parts had the form of isosceles and scalene triangles, 30 “Where a circle is cut through the middle, the lined formed amounts to a third of the circumference, a line so dividing the whole as to leave a small difference” (Astronomica, trans. Goold [Cambridge, MA: Harvard University Press, 1977]). 31 “[7] The same verses also run along various broken lines upwards and downwards. [8] In this page alone in the corners there are four squares of six letters each, two triangles, and two larger halved octagons, and an hexagon in the middle and four trapezoids pointing toward the comers, all with the same number of letters and with the same increasing and decreasing functions; [9] that is, everything is equal, the triangles to the triangles, octagons to octagons, squares to squares, trapezoids to trapezoids.”

ARITHMETIC AND GEOMETRY IN ANCIENT ROME 33 rum variis coagmentis simulantur species mille formarum: helephantus belua aut aper bestia, anser volans et mirmillo in armis, subsidens venator et latrans canis, NORIS MARTIAGESTAMODISAVDAXIMITATAISIO EXIT MVSAPERERFFIGIEMTVRMARVMCARMINAT ECTVM ITER BLIM NOSV TQYI NAGI ÄGME YNCA ETNV TANS MVS1IGENOSPATIVMSEPTENOMILITEDIS MEATV BNYNCEADEMVERSORELEGENSVTOVMQVIE ERVAS KITTITIJNAMFRACTVSNONYVNALEGECAT DISSONACOMPONIDIVE SFLEXVDOC No less interesting from our point of view is Ausonius’ short poem Griphus ternarii numeri (“The enigma of the number three”), dedicated to describing all of the functions of the number three. The first six lines read: A MFINISOCI ORSISNVE 10 P av guin et turris et cantharus et alia huiusmodi innumerabilium figurarum, quae alius alio scientius variegant. Sed peritorum concinnatio miraculum est, imperitorum iunctura ridiculum.*? Ter bibe vel totiens ternos: sic mystica lex est, vel tria potanti vel ter tria multiplicanti, inparibus novies ternis contexere coebum. Turis idem tribus est, quod ter tribus, omnia in istis: forma hominis coepti plenique exactio partus quique novem novies fati tenet ultima finis.?? R o T 7 7 è Y QVAMV OSTEN 15SARMAT VOTAPRE FACTORVMG IAMTOTIENS HOSTILIPO 20CORPORA Let us continue with lines 50-60: À 3/A 4 PLYRIM y MARGR TI RAREBON Per trinas species trigonorum regula currit, aequilatus vel crure pari vel in omnibus inpar. Tris coit in partes numerus perfectus, ut idem congrege ter terno per ter tria dissolvatur. Tris primis par, impar habet mediumque: sed ipse ut tris, sic quinque et septem quoque, dividit unus, 8 A 8 LLALOQVIP ER À 25QV PONACRVORE eens R à EROCEM S mens VDEN AG rurrormasoxoNN et numero in toto positus sub acumine centri ORVMVICINABONO À A KPOSEXCISAQVEAGM PTIVISETDYCATCETERAY CTORIMOLIMVRPROELIAPLECTRO distinguit solidos coebo pergente trientes, aequipares dirimens partes ex inpare terno, Et paribus triplex medium, cum quattuor et sex bisque quaternorum secernitur omphalos idem. 30 D 1 CERENECSATISESTVOTVMSICOMPLEATORE MYSASVOQTAECVMQVEPARATSVELEGÉSONARE SCRYPOSISINNEXAMODISPERFECTAÜAMENIS VVLIRESIONAREMEISETTESTISNOTA|TROPAEA DEPICTISSIGNAREMETRISICVMMVNE[RIESACRO JMENTISDEVOTAEPLACARINTFATAPRÜÔOCELLAS 5 10 15 20 25 30 35 Fig. 5 quadrangles, and other polygons, from which one could compose various figures with enough fantasy and perhaps by repeating a few pieces: simile ut dicas ludicro, quod Graeci òotopáytov vocavere. Ossicula ea sunt: ad summam quattuordecim figuras geometricas habent. Sunt enim aequaliter triquetra vel extentis lineis vel <eiusdem> frontis, <vel rectis> angulis vel obliquis: toooker ipsi vel toórAevpa vocant, ôpBoybvia quoque et oxadnva. Harum verticula- 32 “So that you may say it is like the puzzle which the Greeks call stomachia. There you have little pieces of bone, fourteen in number and representing geometrical . For some are equilateral triangles, some with sides of various lengths, some symmetrical, some with right angles, some with oblique: the same people call them isosceles or equal-sided triangles, and also right-angled and scalene. By fitting these pieces together in various ways, pictures of countless objects ate produced: a monstrous elephant, a brutal boar, a goose in flight, and a gladiator in armor, a huntsman crouching down, and a dog barking — even a tower and a tankard and numberless other things of this sort, whose variety depends upon the skill of the player. But while the harmonious arrangement of the skilful is marvelous, the jumble made by the unskilled is grotesque.\Ausonins trans. by Hugh G. Evelyn White [Cambridge, MA: Harvard University ress, 19191). 3 “Thrice drink or else as many times three cups: thus stands the mystic law — whether three draughts thou drinkest or three thrice multipliest, with nine times three uneven form the cube. The same virtue is in three as in thrice three: all things are in terms of these; the first forming of the human shape, the due completion of the act of birth, and the limit which marks man’s extreme span, years nine times nine.” (Ibid.) 34 “Over three kinds ranges the figure of the triangle, equilateral, isosceles, and scalene. Three

(88-90), where, Finally, let us observe the last three lines of the short poem aux does not hesitate in order to exalt the number three, the poet from Borde to hint at the Christian mystery of the Trinity: ter decies ternos habeat deciesque novenos.”° Ter bibe. Tris numerus super omnia, tris deus unus. Hic quoque ne ludus numero transcurrat inerti, anr Nok. Ar the best In conclusion, we have good reason to believe that in antiquity problems of Latin intellectuals did have an interest in arguments and into scientific arithmetic and geometry that had been introduced knowledge by scholars writing in Greek. IN when the same midmost point of four, six, and twice four, is brackete group thrice three be formed, by parts combined make up the perfect number, in such wise that if a which has an odd, an even, and three times three the same may be resolved. Three is the first number and when it is placed under a medial unit: but, as the unit itself divides three, so does it five and seven; forming a continuous cube, by the central point of the full number, it parts in two a series of thirds numbers thrice find a center, separating even and equal groups from the uneven threes: and even d.” (Ibid.). three, or nine times ten!” (Ibid). God! And that this 35 “Thrice drink! The number three is above all, Three Persons and One have verses thrice ten times conceit may not run its course without significance of number, let it