Вот ряд римских математиков и геометров (некоторые под вопросом о происхождении, в связи с расплывчатостью дат жизни):
Albinus (Encyclo.) (ca 320 – 345 CE?)
Latin encyclopedist, wrote on music (Cassiodorus, Inst. 2.10), geometry, and dialectic
(Boethius, Inst. Mus. 1.12, 26), all lost. Perhaps identifiable with one of the men
named Ceionius Rufinus Albinus, and/or Albinus the poet of De Metris and Res Romanae
(FLP 425–426); cf. Macrobius, Sat. 1.24.19.
PLRE 1 (1971) 33–34 (#4,5), 37–38 (#14,15); OCD3 50, R.A. Kaster; BNP 1 (2002) 431 (#2),
L. Zanoncelli.
Ammo¯nios of Alexandria, son of Hermeias (ca 470 – after 517 CE)
Born ca 440 CE; Neo-Platonist philosopher from Alexandria, Proklos student.
Damaskios, Simplicius, Asklepios of Tralleis, and Ioannes Philoponos attended
his lectures. Preserved are these commentaries: In Porphyrii Isagogen, In Aristotelis Categorias,
In Aristotelis De Interpretatione, In Aristotelis Analytica Priora (CAG 4.3–6). Askle¯pios’ Comm. In
Metaph. and Philoponos’ In Anal. (I and II), De Gen. and De An. derive heavily from his
lectures. Damaskios describes Ammo¯nios (Vita Is. 79) as philopono¯tatos, an expert in Aristotle, geometry, and astronomy, and critical of Proklean metaphysics. He argued that god(s)
know all of time, but that such knowledge does not constrain future events: they have
knowledge of future contingents but not as future (Tempelis; cf. Iamblikhos’ suggestion
that divine knowledge is definite but is about indefinites). Ammo¯nios observed planetary
occultations or near-conjunctions (with his brother Heliodoros and his uncle), and Arcturus’
longitude (with Simplicius), the latter to check Ptolemy’s value of the precession of
the equinoxes (which he erroneously confirmed); his work on the use of the astrolabe has
been rediscovered and published.
KP 1.306, H. Dörrie; DSB 1.137, Ph. Merlan; Neugebauer (1975) 1031–1041; Ch. Soliotis,
“Unpublished Greek texts on the use and construction of the Astrolabe,” Praktika te¯s Akade¯mias
Athe¯no¯n 61 (1986) 423–454; E. Tempelis, “Iamblichus and the School of Ammonius, Son of Hermias,
on Divine Omniscience,” SyllClass 8 (1997) 207–217; ECP 25–26, H.J. Blumenthal; REP
1.208–210, Chr. Wildberg; Athanassiadi (1999); BNP 1 (2002) 590–591 (#12), P. Hadot.
Anatolios of Laodikeia (250 – 282 CE)
Christian polymath born in Alexandria, succeeded Eusebios as bishop of Laodikeia (268
CE). Because of his high reputation for learning (in arithmetic, geometry, astronomy, dialectics,
physics, and rhetoric), Alexandrians asked him to establish a school of Peripatetic
philosophy (Eus., EH 7.32.6–21). As testimony to Anatolios’ intellectual scope, Eusebios
quotes (rather, misquotes: McCarthy 126–139) the second part of his De ratione Paschali – his
only attested work of substantial length – a defense of an original method of computing the
date of Easter based on a 19-year lunar cycle and drawing from various sources, Greek
(Ptolemy), Christian (mainly Origen), and Jewish (McCarthy 114–125). An hagiographic
anecdote, underscoring Anatolios’ attitude and virtues during the Roman siege of Alexandrian
Pyrucheum in 264, suggests his political influence.
At least three other Anatolioi might be identifiable with our bishop: (1) most plausibly, the
Neo-Platonist philosopher of Alexandria mentioned in Eunapios as Iamblikhos' master
and Porphurios renowned contemporary, probably the author of the short neo-
Pythagorean tract On the Decade and the numbers within it included in the Theologumena Arithmeticae, perhaps part of the Arithmetical introductions mentioned by Eusebios (EH
7.32.20); (2) the source of an appendix to Heron’s Definitiones (4.160.8–166.22 H.), proposing
answers to general questions about the nature and parts of mathematics; (3) one “very
learned Anatolios” mentioned in a letter of Michael Psellos discussing a nomenclature of
“powers of the unknown” similar to, but more general than Diopantos', adding names
for the 5th and 7th powers and proceeding to the 10th power instead of the 7th (Diophantos
2.37–39 Tannery). Tannery’s conjectural identification of the latter with the bishop is weak.
Ed.: D.P. McCarthy, The ante-Nicene Christian Pasch ‘De ratione paschali’ (2003).
DPA 1 (1989) 179–183, R. Goulet.
Andro¯n of Rome (ca 120 – 170 CE)
Marcus Aurelius’ childhood geometry and music teacher, afterwards honored by the
emperor (SHA 2.2).
RE 1.2 (1894) 2159 (#10), P. von Rhoden; Netz (1997) #98.
Balbus (102 – 106 CE?)
Nothing is known of this man’s life. If the beginning of his treatise does refer to Trajan’s
expedition to Dacia, it can be dated to between 102 and 106. The Expositio et ratio omnium
formarum has come to us in mutilated form; this is why, contrary to what its title promises,
it does not deal with all figures. This handbook of geometry was written by an agrimensor:
the author therefore always keeps in mind the relationship between geometry and a surveyor’s
work. Beside definitions concerning the categories of Roman land management
(rigor, extremitas, decumanus, cardo, ager arcifinius), the geometrical definitions (point, line, parallel,
area, and so on) to be found in the extant part of the Expositio testify that Euklid’s Elements
had already been translated into Latin, at least Books 1 to 3, when Balbus wrote, therefore a
long time before such translations as attested by Martianus Capella or attributed to
Boethius. The fortunes of this handbook, to be found in many MSS and still used in
the medieval Demonstratio artis geometricae edited by Lachmann, are precisely due to its offering
only definitions together with very elaborate figure classifications. In this respect it is
comparable with the Greek Definitiones attributed to Heron of Alexandria, actually
apocryphal but whose substance may date back to him.
K. Lachmann, Die Schriften der Römischen Feldmesser, v.1 (1848); French translation and commentary:
Guillaumin, Balbus (1996) = CAR 3.
Book of Assumptions by Aqa¯ t˙
un (Hekato¯n? Agatho¯n?) (200 – 600 CE?)
The Arabic translation of a Greek treatise containing 43 demonstrated geometrical theorems
pertaining to the geometry of triangles and circles (chords and tangents). This collection
of lemmas drawing from various sources (but with no explicit reference to other works)
is probably of late antique origin. The first half contains 19 propositions also found (with slight
differences) in the MS “the Book by Archimedes on the Elements of Geometry.” The
third proposition recalls prop. 10 of Archime¯de¯s’ liber assumptorum, and some others are
(sometimes strongly) similar to propositions found in Pappos, probably suggesting common
sources rather than derivation. The names “Hekato¯n” or “Agatho¯n” are only possible
guesses, and no Greek geometers with such names are known.
Jones (1986) 2.603–605; Y. Dold-Samplonius, “Some Remarks on the ‘Book of Assumptions by
Aqa¯t˙
un,’ ” JHAS 2.2 (1978) 255–263.
Martianus Minneius Felix Capella of Carthage (ca 430 CE?)
Everything concerning the life of Capella is a matter of conjecture; some date him to the
last decades of the 5th c. He may have been a rhetor, and wrote a curious work entitled De
nuptiis Philologiae et Mercurii. When the god Mercury wished to marry, Jupiter approved the
choice of Philology as bride. The marriage ceremony is the subject of the nine books of
De nuptiis. After the first two books have dealt with the mythos, depicting Philology’s apotheosis,
a necessary condition for her union with a god to be made possible, seven books are
devoted to the three literary disciplines (i.e. Book III: Grammar; Book IV: Dialectic; Book V:
Rhetoric) and to the four mathematical ones (i.e. Book VI: Geometry; Book VII: Arithmetic;
Book VIII: Astronomy; Book IX: Harmony).
Every one of these is personified as a maiden who will attend the new bride. Capella’s
originality lies in his uniting Roman encyclopedism, going back to Varro, with the Neo-
Platonic doctrine that knowledge, mainly mathematical, can save the soul. The technical
contents of the books dealing with the four sciences can be traced back in large part to ancient
sources known to the author through school compendia written during the preceding centuries.
His Geometry is made up of a geography derived mostly from Pliny with borrowings
from Iulius Solinus, and in fine of Euclidian fundamentals, mostly definitions. The book on
Arithmetic starts with an arithmology, goes through elements present in Nikomakhos of Gerasa's Introduction to Arithmetic, and ends with a statement of several Euclid’s arithmetical
propositions: these last are valuable evidence of the translation of part of the Elements into
Latin, even before Boethius. The originality of Capella’s book on astronomy is twofold: first in his two-part plan dealing with cosmography to start with and then with the planets, secondly
in the semi-heliocentric system he describes, where Mercury and Venus revolve round
the Sun, whereas the other planets and the Sun itself turn round the Earth. Book IX largely
copies Aristeides Quintilianus by means of intermediaries unknown to us.
Beside the works of Boethius, Cassiodorus and Isidorus of Hispalis, the De nuptiis
became a basic textbook with Carolingian schools, because it provided a genuine encyclopedia.
Many Carolingian MSS from the 9th c. bear witness to its systematic use in schools,
and the pictures of the seven maidens standing for the seven disciplines are plentiful in
medieval and Renaissance iconography.
Ed.: J. Willis, De nuptiis Mercurii et Philologiae (1983); Book VI: G. Gasparotto, Geometria: De nuptiis
Philologiae et Mercurii, liber kextus (1983); VII: Jean-Yves Guillaumin (2003); VIII: A. Le Boeuffle,
Martianus Capella, Astronomie (1998); IX: L. Cristante, Martiani Capellae De nuptiis Philologiae et Mercurii.
Liber IX (1987).
W.H. Stahl, R. Johnson, E.L. Burge, Martianus Capella and the Seven Liberal Arts (1971); S. Grebe, Martianus
Capella (1999); M. Bovey, Disciplinae cyclicae (2003).
De¯me¯trios (Math.) (250 – 300 CE)
Platonic geometer who discussed the properties of odd and even numbers; teacher of
Porphurios (Proklos, in Plat. Remp. 2.23.14 Kroll), who met at Athens with Cassius Longinus, Porphurios, and others, to celebrate Plato’s birthday (Eusebios, Pr. Ev.
10.3.1). Chronology and similarity of interests suggest, though not definitively, that our
De¯me¯trios may be identifiable with Demetrios (Music).
PLRE 1 (1971) 247.
De¯me¯trios of Alexandria (200 BCE – 100 CE)
Wrote Linear Considerations (Grammikai Epistaseis: the third of the three types of geometrical
problems: plane, solid, linear) on the geometrical curves, which he discovered in attempting
to trisect rectilinear angles by studying the interaction of various surfaces (e.g., plektoids/
spiral curves with other surfaces), and which exhibit “astonishing properties” (n.b., the
quadratrix of Deinostratos). De¯me¯trios discovered many complex solutions including,
perhaps, spirals, quadratrices, conchoids, and cissoids. De¯me¯trios predates Menelaos, who
called one of his, or hilo of Tuana’s, curves “paradoxical”: Pappos, Coll. 4.36 (p. 270 H.)
RE 4.2 (1901) 2849 (#116), Fr. Hultsch.
De¯me¯trios of Lako¯nika (150 – 80 BCE)
Epicurean student of Protarkhos of Bargulia and younger associate of Zenon of Sidon. The Herculaneum papyri contain numerous fragments of De¯me¯trios’ works on
poetry, physics, mathematics, theology, and philology. Some of his views can be
reconstructed from these fragments and later reports of his doctrines. He defended Epicurus’ views on the size of the Sun, the nature of the gods, the infinity of space, the infinite
number of atoms, inference from similarities (Philodemos, On Signs 45–46), the nature of
time (“an accident of accidents”), the Epicurean theory of minima, and, against
Karneades, he defended the existence of proof. He also considered the possibility that
some apparent shortcomings in Epicurean philosophy were caused by scribal errors in
earlier MSS.
Ed.: V. de Falco, Demetrio Lacone (1923).
RE 4.2 (1901) 2842 (#89), H. von Arnim; C. Romeo, “Demetrio Lacone sulla grandezza del sole
(PHerc. 1013),” CrErc 9 (1979) 11–35; Long and Sedley (1987) §7C; OCD3 450, D. Obbink; ECP
178, J.S. Purinton; BNP 4 (2004) 250 (#21), T. Dorandi.
Diodo¯ros (Astron.) (150 BCE – 250 CE)
Geometer and astronomer, possibly identifiable with a commentator on Aratos named
Diodo¯ros, and with a Diodo¯ros of Alexandria who wrote on astronomical topics. The
geometer Diodo¯ros authored a treatise On Analemma concerning the theory underlying sundials.
Pappos wrote a lost commentary on this work, and Proklos speaks of Diodo¯ros as
one of the earlier writers on sundials. Diodo¯ros’ method of determining the cardinal directions
from three measured shadows (a problem equivalent to finding the axis of a hyperbola
given three points on the curve and one on the axis) is reported by Hyginus Gromaticus,
Abu Sa id ad-Darir, and al-Biruni. We also know from Pappos’ reference that On Analemma
contained a construction requiring the trisection of a given angle. According to al-Nairizi,
the commentator on Euclides’s Elements, Diodo¯ros attempted a geometrical demonstration
of Euclid’s fifth postulate, perhaps in a different book.
Diodo¯ros, the commentator, is cited intermittently in the scholia to Aratos’ poem; he
criticized Stoic interpreters such as Krates as well as Hipparkhos, and was in turn
attacked by an otherwise unknown Do¯sitheos. Diodo¯ros of Alexandria figures, together with
Meton, Eudoxos, Hipparkhos, and Lasos, in an anonymous list of astronomers, and he
is cited by Akhilleus and Macrobius on the distinction between mathematical and
physical astronomy, the meaning of the words kosmos and “star” (aste¯r), and the nature of
the Milky Way.
Ed.: D.R. Edwards, Ptolemy’s Περ αναλ μματο – An Annotated Transcription of Moerbeke’s Latin Translation
and of the Surviving Greek Fragments with an English Version and Commentary (1984) 152–182.
Neugebauer (1975) 840–843; NDSB 2.304–305, J.L. Berggren.
Dionusios of Kure¯ne¯ (160 – 110 CE)
A student of Antipatros of Tarsos and of Diogenes of Seleukeia, Dionusios was an
acclaimed Stoic geometer (according to a Herculaneum papyrus, Index Stoicorum, col. 52)
who wrote against Poluainos, and was attacked by Demetrios of Lakonika (P. Herc. 1642, fr.4). He insisted that induction must be based upon what is always and everywhere
observed.
GGP 4.2 (1994) 641–642, P. Steinmetz; BNP 4 (2004) 476 (#10), B. Inwood.
Epaphroditos and Vitruuius Rufus (200 – 300 CE?)
A collection of geometrical problems to be found in Latin gromatic MSS (i.e. collections of
texts about land surveying) has survived with these two otherwise unknown names attached
to it; but Lachmann did not include them in his edition of the corpus. Following the same
order as that in the works attributed to eron of Alexandria (Metrika I, authentic, and
Geo¯metrika, considered apocryphal), whose influence is obvious, the calculations of perimeters
and areas of triangles, of quadrangular figures, regular polygons, and of the circle
and its segments are all dealt with practically, with detailed figures but no attempt at demonstration,
which is a great difference from the Metrika. Surprisingly, the polygonal areas
(pentagon and so on up to dodecagon) are here dealt with arithmetically, not geometrically;
they are looked at in the Pythagorean manner as sums, not products. The origin of these
developments ought to be looked for in Diophantos’ treatise Polygonal Numbers, which
provides evidence for dating. As they show similarities with the Podismus (Lachmann,
pp. 295–301), Epaphroditos’ and Vitruuius’ excerpts may bear some link with the calculation
of triangular, trapezoidal, and pentagonal subseciua (minor areas of a centuriation not
allotted to any owner), such as presented by Iunius Nipsius (Lachmann p. 290).
Ed.: N. Bubnov, Gerberti opera mathematica (1899); CAR 3 (1996).
Hermeias (Math.) (40 – 100 CE)
One of Plutarch's interlocutors in Table Talk 9.3 (738D–739A), a geometer who addressed
why the Greek alphabet contains 24 letters. His solution rested upon perfect numbers
(for which he provided two definitions), squares, and cubes. The number of letters of the
alphabet are 3x8 (the first perfect number with a beginning, middle, and end times the first
cube) or 6x4 (the first perfect number equal to the aliquot sum of its factors times the first
square).
RE 8.1 (1912) 732 (#12), C.R. Tittel.
He¯ro¯n of Alexandria (ca 62 CE)
Date: He¯ro¯n’s dates have been the topic of extended discussion; the only explicit markers
cover a 500-year span: he quotes Archimedes, and is quoted by Pappos. Neugebauer
settled the question, observing that He¯ro¯n, in his Dioptra, described a lunar eclipse visible
in both Rome and Alexandria. The only eclipse fitting He¯ro¯n’s data occurred in 62 CE.
Neugebauer argued that He¯ro¯n was earlier than Ptolemy, as he did not make use of his
results, and one of his devices is described as a “new invention” by Pliny (15.5)...
Hyginus Gromaticus (100 – 300 CE?)
The second of the two writers named Hyginus in the Corpus Agrimensorum Romanorum, and often referred to as “Gromaticus” on the basis of the rather
confused MS headings. He refers to the poet Lucanus, but otherwise makes no datable
references. His approach is partly historical in that he discusses the foundation of colonies,
but he also describes the procedures of land survey in a way that offers guidance to other
surveyors. He is particularly informative on the establishment of limites, the dimensions of
land division units (centuriae), and their proper designation with inscribed stones so that plots
of land could be found easily and without ambiguity. Hyginus describes methods of orientation
and the alignment of limites, using a sundial and the measurement of shadows, and a
more complex method based on solid geometry. He also outlines a method for measuring
parallel lines using similar right-angled triangles. Hyginus sets out the best methods of
land division starting from the principle that the two main limites, aligned north-south and
east-west, intersected in the middle of the settlement and extended through four gates.
Although this could rarely be achieved, surveyors with their professional, scientific approach
worked with the administrative bureaucracy to overcome and exploit physical terrain. In
a way, they represented the power of the Roman state to control natural resources.
Thulin (1913); CAR 4 (1996); Campbell (2000) 134–163.
Hyginus, pseudo, de Metatione Castrorum (ca 200 – 212 CE)
A military geometer of good theoretical training and practical experience (§45, 47), who
wrote probably in the beginning of the 3rd c. CE, but not later than 212 (edict of Caracalla:
Grillone 1987: 407–411). De metatione castrorum is a more suitable title than the commonlyaccepted
de munitionibus castrorum, proposed by a copyist: the author treats fortifications only
briefly at the end (§48–58), where however he expends no small attention on geometrical
matters, coxae and clauiculae (§54–55). Coxae round and thus strengthen the angles of the
castra; clauiculae form a vertical quarter-cylinder, extending from the door’s right jamb until
the point corresponding to the central point of the opening part of the wall reserved to the
door (width = 60 feet: §14,49; Grillone 2000: 378–379). Clauiculae and small fossae (§50: titula)
aim to impede frontal attacks, to defend retreating soldiers, and to allow defenders to hit
assailants everywhere.
The rest of the booklet (§1–47), mutilated at the beginning, addresses only metatio,
i.e., how a camp’s surface is distributed between the units of an army of three legions
in three parts, to the front ( praetentura), in the middle (latera praetorii), to the back (retentura);
cohortes partly are symmetrically disposed along the four sides of the castra (8+8 on left
and right hand [§36], 4+4 to the front and to the back [§44]: Grillone 1984: n. 25), partly
in praetentura and in latera praetorii (4+2; §3,9; Grillone 1984: n. 24). In calculating the
area necessary for any unit, the geometer allots 11/5 foot for each infantryman, and
three feet for each horseman (width fixed at 30 feet for arms, animals . . .; §1). Cohortes
legionariae and other units – auxiliarii and gentes (nationes and symmachares) – differ in that
cohortes legionariae take up quarters according to a fixed plan, also if they have less than
600 soldiers (720×30 feet: §1–2), while other troops have an area corresponding to
the number of soldiers (i.e., cohors peditata: 600 men = 720×30 feet: §27–28), and sometimes
the usable area accords to the circumstances (for gentes, if they are less or more
people: §40).
Ed.: Antonino Grillone, Hygini qui dicitur de metatione castrorum liber (1977); M. Lenoir, Pseudo-Hygin, Des
fortifications du camp (CUF 1979).
Antonino Grillone rev. of Lenoir, in: Gnomon 56 (1984) 15–26; Idem, “Problemi tecnici e datazione del
de metatione castrorum dello ps.-Igino,” Latomus 46 (1987) 399–412; Idem, “Soluzioni tecniche
e linguaggio di un geometra militare del III secolo: lo pseudo-Igino,” in Atti del IV Seminario
Internazionale di studi sulla letteratura scientifica e tecnica greca e latina (Messina 29–31 ottobre 1997)
(2000) 365–395; Idem, “Lessico ed espressioni della gromatica militare dello ps. Igino,” in Atti del Congresso Internazionale “Les vocabulaires techniques des arpenteurs romains,” Besançon (19–21 Septembre 2002)
(2005) 125–136.
Nikomakhos of Gerasa (100 – 150 CE)
Neo-Pythagorean philosopher; the Gerasa whence he came is likely to have been the
one in Palestine. Nikomakhos composed two surviving treatises, Introductio Arithmetica (two
books) on the philosophy of number and number theory, and Harmonicum Enchiridion on the
Pythagorean theory of pitches and tuning systems. In the latter (11) he cites Thrasullos,
while Cassiodorus, Institutiones ( p. 140 Mynors) writes that Apuleius translated the Introductio
Arithmetica into Latin, thus bracketing his date. The contents of two further lost works
are known in great part. Arithme¯tika Theologoumena (“arithmetic subjected to theology”), an
exposition of Pythagorean number symbolism, was one source of the Theologoumena
Arithmeticae, and Pho¯tios also summarized it (Bibl. cod.187). A work on the life of
Pythagoras is cited by Porphurios, Vita Pythagorae (20, 59) and was also exploited without
acknowledgement by Iamblikhos in his De Vita Pythagorica. Nikomakhos himself alludes to
a lost Introduction to Geometry in Introductio Arithmetica 2.6.
The Introductio Arithmetica moves fairly rapidly from discussing the ontology of numbers
to exposing elementary number-theoretic classifications of numbers, e.g. into even and odd,
prime and composite. Other prominent topics are ratio equalities and inequalities and
figurate numbers. The presentation is discursive and eschews proofs. In the Harmonicum
Enchiridion, Nikomakhos presents in a comparably discursive manner a Pythagorean theory
of the celestial and numerical foundations of musical pitch, while tacitly incorporating
elements from Aristoxenos.
DSB 10.112–114, L. Tarán; Barker (1989) 245–269; Dillon (1996) 352–361; DPA 4 (2005) 686–694,
G. Freudenthal.
Niko¯n of Pergamon, Aelius (120 – 150 CE)
Architect and geometer, father of Galen (Souda Gamma-32), who does not name his father
but thanks him for his grounding in mathematics and logic (2.116.22–26, 119.2–9
MMH). Gale¯n’s father is probably the Aelius Niko¯n who erected isopsephic inscriptions
at Pergamon (IGRR 4, #502–506; Schlange-Schöningen). Using π = 22/7, Niko¯n compares
the volumes of a cone, cylinder, and sphere, all with a common given radius (that
radius equal also to the height of the cylinder and cone), and compares the surface areas of
a cube (superposed over a cone), of a cylinder, and of a sphere, likewise with a common
radius, yielding a proportion of 42 : 33 : 22 (#503).
H. Schlange-Schöningen, Die römische Gesellschaft bei Galen: Biographie und Sozialgeschichte (2003) 45–54;
DPA 4 (2005) 696–698, V. Boudon-Millot; BNP 9 (2006) 740 (#4), M. Folkerts.
Panaitios the Younger (135 BCE – 300 CE)
Mathematician and music theorist known only from Porphurios commentary on
Ptolemy’s Harmonics. Porphurios provides no biographical information about Panaitios
except to refer to him as “the Younger” (ho neo¯teros, 65.21 Düring), presumably to distinguish
him from the more famous Stoic philosopher Panaitios of Rhodes. Of his work we
know no more than what is preserved by Porphurios, who quotes briefly from Panaitios’
book On the Ratios and Intervals in Geometry and Music (Peri to¯n kata geo¯metrian kai mousike¯n logo¯n kai
diaste¯mato¯n, 65.21); no other titles survive. A single-sentence paraphrase giving the rationale
for the analysis of musical notes by means of mathematical proportion (88.5–7) appears
to have been drawn from the same work, and three subsequent references (92.20, 92.24,
94.24), almost certainly to the same Panaitios, link him with De¯me¯trios on a point of
scientific vocabulary.
The only substantial quotation is the first (65.26–66.15 according to Düring, but probably
extending at least as far as 67.8; Barker translates the passage to 67.10), an argument
intended to prove that the term “semitone” (he¯mitonion) is an invalid term of reference,
because sense-perception, on the one hand, is not sufficiently accurate to divide musical
intervals exactly in half, and because “canonic theory,” on the other hand, denies that such
a division is mathematically possible in the first place. For this conclusion Panaitios relies on
several premises: (1) that the intervals in music can be shown to correspond to certain
mathematical ratios, which he demonstrates by means of a brief canonic division; (2)
that from this division it is evident that the ratio corresponding to the tone (tonos) is 9:8; (3)
that the 9:8 ratio cannot receive a geometric mean expressible in a ratio of whole numbers
(relying on a proof attributed by Boethius to Arkhutas and spelled out at Euclideas Sectio Canonis prop. 3). From this he concludes that the tone cannot be divided into
two equal intervals (cf. Sect. can. prop. 16), and that the term he¯mitonion is consequently as
much a misuse of language as the term he¯mionos (mule, lit. “half-ass”).
The fragment is also noteworthy for its mention of sympathetic vibration of strings.
The phenomenon was noted by other ancient authors (Adrastos, the Aristotelian Corpus Problems, Aristeides Quintillianus), but Panaitios is the only extant author
to connect it with the discovery of the concord-ratios.
Düring (1932); RE 18.3 (1949) 440–441 (#6), K. Ziegler; Barker (1989); Mathiesen (1999).
Pandrosion, and anonymous students (ca 285 – 320 CE)
The female teacher of mathematics to whom Pappos addresses the tract forming the first
part of what later became the third book of the Mathematical Collection (1.30–131 Hultsch), a
long and skillful response to a challenge set to him by (at least) three of Pandrosion’s
students, seeking his opinion about some geometrical constructions (30.17–22). The first
one (1.32), a clever (though erroneous) construction, perhaps derived from Eratosthenes's,
mesolabe¯ (Knorr 1989: 63–69) and was meant to find two geometrical means between two
given lines. The second one (68.17–25), seemingly not fully understood by Pappos himself, is also an elegant solution to the problem of finding in the same figure the geometrical,
harmonic and arithmetical means between two given lines. The last one (114.14–20) is a
paradoxical theorem akin to Erukinos’ paradoxes, as Pappos remarked. Despite Pappos’
(calculated) claim that his mathematical knowledge is superior to Pandrosion’s and her
students’, the tone and content of his response, as well as the level of their achievements,
show their mathematical competence. A feminine name, perhaps of Athenian origin,
the diminutive of Pandrosos, Kekrops’ dewy daughter, Pandrosion was never common
(LGPN ).
Cuomo (2000) 127–128, 170; Jones (1986) 4, n.8; Alain Bernard, “Sophistic aspects of Pappus’s
Collection,” AHES 57 (2003) 93–150.
Pappos of Alexandria (ca 285 – 320 CE)
Influential polymath, astronomically dated to 320 CE; although a marginal note places
him under Diocletian. He wrote on theoretical and computational astronomy, classical
geometry, practical arithmetic, geography, and perhaps astrology. Many of his works are
known only through later quotations (in Proklos, Marinos, Eutokios or scholia to
Ptolemy’s Almagest), heavily interpolated commentaries (e.g., on Euclid’s Elements, Book
10 = IE ), and the collection of originally separate treatises later known as the Mathematical
Collection (MC), probably compiled after the 6th c. CE (Decorps 47–51) and much interpolated.
His geography (kho¯rographia oikoumenike¯ ) is known through an Armenian translation
(see Jones 3–15), and Books 5 and 6 of his commentary on the Almagest are extant
(IA). Pappos’ scientific contribution consists not in any substantial innovation but in the way
he used, organized and compared an impressive mass of scientific texts. He claims originality
usually only for variations on traditional inventions, according to his own values (see his
revealing criticism of Apollonios’ alleged attitude toward Euclid, MC 7, pp. 119.16–
120.12 Jones). It is therefore necessary to outline Pappos’ social and intellectual context as
well as the scope of his sources to assess his key interests and contributions.
Biography and Intellectual Context: Pappos’ tracts addressed various audiences,
interested either in philosophy (MC 5; Cuomo 57–90), mechanics or architecture (MC 8;
Cuomo 91–103), astronomy (MC 6, IA), geometry (MC 3, 4, 5, 7). In MC 3, Pappos
addresses his competitor Pandrosion, her students and some of his own friends (including
Hierios “the philosopher,” perhaps among Iamblikhos’ followers, Jones 5) and thereby
tries to attract new students by displaying his mathematical knowledge and skill, plausibly
implying he worked as a private teacher. He also cleverly shows Pandrosion’s students how
to improve their own propositions and consequently their geometrical knowledge and skill,
especially in analysis. In general Pappos seems to situate himself as a professional mathematician
or at best as a teacher of liberal arts, seemingly confirmed by the scope and variety
of his interests, his care for learning and his Atticist language.
Pappos’ Sources: In computational and theoretical astronomy, Pappos utilized Ptolemy’s
Almagest, Geography, Planispherium, the lost Meteo¯roskopeion, the Handy Tables, repeatedly alluded
to in IA, and introductory works belonging to the corpus of “little astronomy,” some of
which are criticized or amplified in MC 6. Repeated allusion to the Handy Tables, lost works
on the interpretation of dreams as well as the building of a hydroscope suggest ( plausibly
but conjecturally) astrological interests.
In arithmetic and logistic, Pappos paid interest to practical calculations (IA, MC 3 and 8)
and Apollo¯nios’ system of notation for large numbers (MC 2). Neo-Pythagorean hints
contained in MC 3 and IE are probably interpolated: Pappos’ approach to mesotai seems
predominantly geometrical, in the tradition of Theaitetos and Eratosthenes.
In practical and theoretical mechanics, Pappos heavily uses Heron (esp. in MC 8; see
also IA and MC 3); Archimedes, Karpos, Philon Buzantion and Ptolemy are also
mentioned.
In geometry, Pappos relies on a considerable number of works, many known only
through his allusions in MC 7 (also MC 3, 4 and 5), several from Euclid and Apollo¯nios.
Pappos includes them in the “field of analysis” described as “useful material” for the
invention of geometrical problems ( Jones 66–70).
Key Interests and Contribution to Ancient Science: In modern commentaries,
Pappos is sometimes characterized as only interested in geometry (e.g. Heath 2.358). But in
late antiquity, Pappos was mainly known for IA (to which MC also refers) and anthologies
of geometrical and mechanical problems, well reflecting the content and structure of many
of his works (mainly MC 3, 4, 7 and 8). His exposition is often structured by problems or
series of problems, for which he provides various approaches: fully articulated demonstrations,
analyses that open to new problems, mechanical devices, missing lemmas, or calculations.
In some cases unsolved problems or questions are presented with a variety of solutions
(e.g. the treatment of the “squaring” curve in MC 4). In one famous case, the generalization
of Euclid’s problem of three and four lines, Pappos proposes a generalization of the problem
itself (MC 7, pp. 120–123 Jones). This taste for the variety of problems or solutions
is related to Pappos’ interest in problem-solving and mathematical heuristic and to his
endeavor to provide his students treasuries of solutions as a resource for their own efforts as well
as guidance through the use of these works.
This core feature of Pappos’ work is related to his other characteristics: his emphasis on
the dichotomy between invention and demonstration (MC 3.30 Hultsch and 7.1, p. 83
Jones); his tendency to classify and generalize problems according to kinship, either formally
or according to the solutions used; his mixture of mechanics and geometry (MC 8);
his mixture of calculation and geometrical reasoning, akin to the techniques used in the
Almagest; his complex use of the tradition, simultaneously reverential and critical (Cuomo
186–199).
Heath (1921); RE 18 (1949) 1084–1106, K. Ziegler; Jones (1986); Cuomo (2000); Decorps-Foulquier
(2000); Alain Bernard, “Sophistic aspects of Pappus’s Collection,” AHES 57 (2003) 93–150.
Alain Bernard
Papyrus Ayer (70 – 140 CE)
This papyrus contains fragments of a work on mensuration, giving computations of areas
of irregular quadrilaterals, performed by slicing them into triangles (often right) and regular
quadrilaterals, whose more-easily computed areas are summed. The text uses aroura as
an abstract areal unit (not in its Hellenistic sense of a concrete unit of land area); contrast
Heron of Alexandria, who uses monas for abstract units (Metr. 1.1, linear and areal; even
volumetric: e.g., Metr. 2.11). Likewise, the papyrus considers a parallelogram to be any
quadrilateral with at least one pair of sides parallel; whereas He¯ro¯n follows Euclid in requiring
both pairs parallel. Other terminology (koruphe¯ for the “upper side” of a quadrilateral)
and the procedure are similar to He¯ro¯n, Metr. A similar document is P. Cornell inv. 69
(Buelow-Jacobson and Taisbak).
E.J. Goodspeed, “The Ayer Papyrus: A Mathematical Fragment,” AJPhilol 19 (1898) 25–39; A.
Buelow-Jacobson and Ch. M. Taisbak, “P. Cornell inv. 69: Fragment of a Handbook in Geometry,”
in A. Piltz et al., edd., For particular reasons: studies in honour of Jerker Blomqvist (2003) 54–70.
Papyrus Geneva inv. 259 (100 – 200 CE?)
This papyrus contains three problems of increasing complexity, each seeking to solve an
integer right triangle (of sides 3, 4, 5) with two givens: (a) the hypotenuse plus one side, (
the sum of the hypotenuse and one side plus the other side, and © the hypotenuse plus the
sum of the two sides. The method is algebraic, making use of the “Pythagorean” formula.
J. Rudhardt, “Trois problèmes de géometrie, conservés par un papyrus genevois,” MusHelv 35 (1978)
233–240, pl. 7.
Paterios (250 – 400 CE)
In his explanations of the “geometrical number” governing human generations in Plato’s
Republic (546b3–c7), Proklos exposes two geometrical methods to find the numbers 27,
36, 48 and 64, in continuous proportion with epitrite ratio (4:3). The second method (In Resp. 2.40.25–42.10 Kroll) uses elementary constructions within the right triangle with
sides 3,4,5 as well as basic calculations on fractions and is attributed to a certain Paterios,
probably the exegete of Plato’s Phaedo whom Proklos approvingly mentions in explaining
the myth of Er (2.134.10). Damaskios uses Paterios’ exegesis of the Phaedo to solve a
difficulty raised by Harpokration (In Phaed. p. 137 Westerink, on Phaedo 68c1–3). Paterios
is thus either a middle- or Neo-Platonist who commented on Plato between Harpokratio¯n
and Proklos.
RE 18.4 (1949) 2562–2563, R. Beutler.
Peitho¯n (of Antinoeia?) (330 – 390 CE?)
Geometer, contemporary with Serenus of Antinoeia, who preserves Peitho¯n’s definition
of parallel lines (p. 96, ed. Heiberg).
RE S.7 (1940) 836 (#6), M. Kraus.
Proklos of Lukia, diadokhos (ca 430 – 485 CE)
Life: Proklos’ life is amply recorded in the edifying encomium by his student Marinos of Neapolis, composed the year after his mentor’s death (Vita Procli = VP), and in
Damaskios’ Vita Isidori. Born in Constantinople in 412 to a Lukian noble family, Proklos
began his studies with a grammatikos in Lukian Xanthos and continued them in
Alexandria. His father Patricius, a high-ranking advocate, had practiced in the capital and
wanted his son to learn Roman Law. Proklos also began studying rhetoric under the Sophist
Leo¯nas, whom Proklos accompanied on an embassy to Constantinople (VP 8–9). Returning
to Alexandria, Proklos studied philosophy (especially Aristotle’s) and mathematics under
Heron (Math). Ca 430, Proklos went to Athens to study with Plutarkhus of Athens
and Syrianus, the latter connected with Athenian Sophists, especially Lakhare¯s and his
student Nikolaos who welcomed Proklos on his arrival (VP 10–11). Among his fellow students
were Domninos of Larissa and Hermeias. After Syrianus died, ca 437, Proklos
became his “successor” (diadokhos), and, for the rest of his life, lived and taught in Athens,
except during one year when Christian threats forced him into exile in Lydia (VP 15).
Several of his students would become influential in government and/or philosophy, including
Ammonios of Alexandria and his brother Heliodoros, Marinos and Isido¯ros of
Alexandria (who successively succeeded Proklos). His students also included high-ranking
notables of the late empire, many of them Christians (see Saffrey and Westerink, edd.,
Theologie Platonicienne: Proclus I. XLIX - LIV). Proklos himself was an influential political figure,
extending patronage to many contemporaries (VP 16–17). He was also a devoted pagan
who scrupulously observed traditional rites (VP 18–19).
Nature of his scientific works and activities: Marinos describes Proklos as a hard
worker, devoting time to courses, lectures and discussions which he subsequently recorded in
commentaries (VP 22). Some of his extant works greatly influenced philosophy, theology,
and science. Of particular significance are his commentaries on Plato’s Timaeus (IT ) and
Republic (IR), on Euclides’s first book of the Elements (IE), his Outline of astronomical hypotheses
(or Hupotuposis), and his Elements of Physics.
Proklos’ written works derive from a method of reading and discussion that could be
qualified as mystagogical, eclectic, conciliatory, and agonistic. (a) Mystagogical: Proklos remained
faithful to Syrianus’ idea that preparatory readings (like the study of Aristotle) should
lead one to Plato’s mystagogy (VP 13), i.e., to the idea that Plato’s dialogues (especially
Timaeus and Parmenides) were designed to lead their readers to higher hypostases. Proklos
thus considered it his duty to imitate Plato by providing his own “guidance” into Neo-
Platonist metaphysics, e.g., a teaching both inspired and inspiring. In particular, he
considered reading Euclid, Plato’s Timaeus or Ptolemy as steps along the same path. In
general, he considered commentary in itself as a kind of religious performance, akin to
prayer and theurgy. (
Eclectic, since Proklos chose from among his extensive literary knowledge everything relevant to attain “true reality” ( pragmata) and Platonic theology. The
literature upon which Proklos relied included important mathematical works like Euclid’s
Elements, Heron's or Pappos’ commentaries thereon, Neo-Pythagorean arithmetic
works, Geminus’ encyclopedia of mathematical science, and astronomical works (especially
Ptolemy’s Almagest and Hypotheseis). © Conciliatory, since Proklos also tried to build a harmony
(sumpho¯nia) between different kinds of reasoning or theories – e.g., Euclid’s proofs and
Aristotle’s theory of demonstration (revised by Syrianus). (d) Agonistic: the interpretation of
texts was discussed in a closed circle, some of them sometimes raising valid objections
(e.g. IE 29–30). This, in turn, was consistent with Proklos’ view that teaching should awaken
the souls of his listeners, controlled by, and directed toward, higher levels of cognition.
Main scientific works and influence: IT, Proklos’ favorite work (VP 38), is an ambitious
attempt to reconcile Plato’s dialogue with Aristotelian physics and cosmology, which
Proklos substantially criticized and modified. Likewise, his Hupotuposis attempts to criticize
Ptolemy’s cosmology by emphasizing the artificiality of his hypotheses as compared with
the simplicity and the independence from human needs characterizing natural processes,
ideas also explored in IR (2.213–236 Kroll). The 13th dissertation of IR also includes a
long discussion, in which Proklos discusses various issues pertaining to astrology or
Neo-Pythagorean arithmetic. He addresses in particular side and diagonal numbers and
confronts the Neo-Pythagorean procedure with a geometrical proof drawn from Euclid.
Proklos’ IE contains an original theory of mathematical activity and invention, derived from
Syrianus’ own projectionist theories about the activity of the soul (IE 49–57). In its first
Prologue, Proklos also developed Iamblikhos’ earlier idea of “general mathematics” (hole¯
mathe¯matike¯) by expressing it according the late Neo-Platonist metaphysics (IE 5–10).
Proklos’ Elements of Physics, as well as his Elements of Theology, show his eagerness to adapt the
Euclidean paradigm of demonstration to other subjects, such as Aristotelian physics and
Neo-Platonist theology. Proklos’ immediate influence is seen in the interest that some of
his pupils took in ancient science, particularly Ammo¯nios and Marinos.
DSB 11.160–162, G.R. Morrow; A.-Ph. Segonds, “Proclus: astronomie et philosophie,” in J. Pepin and
H.D. Saffrey, edd., Proclus, lecteur et interprète des anciens (1987) 319–334; O’Meara (1989) 142–208;
L. Siorvanes, Proclus, Neo-Platonic Philosophy and Science (1996); ECP 452–454, D.J. O’Meara;
H.D. Saffrey and A.-Ph. Segonds, Marinus: Proclus ou sur le bonheur (CUF 2001).
Ptolemy (“Claudius Ptolemaeus,” 127 – after 146 CE) ...
Serenus of Antinoeia (200 – 230 CE?)
Wrote various geometrical treatises, The section of the cylinder (SCy), The section of the cone
(SCo), both edited by Heiberg (French translation by Ver Eecke), a lost commentary on
Apollonios’ Ko¯nika to which SCy 17 alludes, and geometrical le¯mmata from which a small
addendum to Theon of Smurna’s treatise is borrowed (Heiberg XVIII). MSS cite him as a
[Platonist] philosopher, as a geometer, or according to his birthplace, long considered
Antissa (Lesbos), now Antinoeia according to Heiberg’s plausible correction to the corrupt
subscription of MS Vat. gr. 206 (Heiberg XVII, 116, 120). MS Par. gr. 1918 indicates that
“Sirinos the geometer” followed the views of Harpokration of Argos, implying perhaps
that Serenus was temporally close to Harpokratio¯n and (less plausibly) that he was
himself a Platonist (Decorps-Foulquier 2000: 19).
SCy and SCo are both dedicated to a certain Kuros, and the last four propositions of SCy
are presented in defense of his friend Peithon’s views on parallel lines (Heiberg 96). Both
treatises, of respectable length, imitate Apollo¯nios’ Ko¯nika, less by the results on which they
rely (elementary properties of Apollo¯nios’ ellipse for SCy and of right and oblique cones for SCo) than by their structure: a skillful combination of theorems and problems, the first
progressively leading the reader to conceive solutions to the second.
Ed.: J.L. Heiberg, Sereni Ant. Opuscula (1896).
P. Ver Eecke, Le Livre de la section du cylindre, et le livre de la section du cône (1929); Decorps-Foulquier (2000)
33–39.
kextus Empiricus (ca 100 – 200 CE)
The only ancient Greek skeptic of whom complete works survive. Virtually the only thing
known about him is that he was a doctor. Diogenes Laertios 9.116, and others, refer to
him as a member of the Empiric school, as his name suggests. In one puzzling passage
kextus expresses a preference for the Methodic school over the Empiric; but his criticism
may be only of one particular form of Empiricism. kextus belonged to the Pyrrhonist
skeptical tradition, whose method, as he explains it, was as follows.
The skeptic assembles opposing arguments and impressions on any given topic. These
arguments and impressions are found to exhibit isostheneia, “equal strength”; each of them
appears no more or less persuasive than any of the others. Given this situation, the skeptic
suspends judgment. And this suspension in turn is supposed to yield ataraxia, “freedom from
worry.” The Pyrrhonist skeptic does not claim that knowledge of things is impossible; that too
is a topic about which he suspends judgment. Rather, the skeptic refrains from all pretensions
to knowledge – or even to belief – about how things really are. There is considerable dispute
about what falls under the heading of “how things really are.” But it is at least clear that the
findings of natural science are among the matters on which the skeptic suspends judgment.
kextus applied this method, unrestricted as to subject-matter and clearly intended to be
employed globally, to the central topics of ancient physics in Against the Physicists ( part of a comprehensive but incomplete work), and in the complete but more synoptic Outlines of
Pyrrhonism (“PH”). In addition, kextus’ third surviving work, Pros Mathe¯matikous (Against the
Learned ), discusses six specialized fields of study, of which several are scientific: grammar,
rhetoric, arithmetic, geometry, astrology, and music (i.e., musical theory).
OCD3 1398–1399, G. Striker; ECP 488–490, J. Allen; NP 12/2.1104–1106 (#2), M. Frede.
Sporos of Nikaia (200 – 300 CE)
Six fragments and three testimonia, hard to synthesize, have reached us under this name. (F1)
Eutokios paraphrases his solution to the duplication of the cube (In Arch. Circ. dim. 4.57–58
Mugler). (F2) Pappos approvingly reports his criticism to the quadratrix curve (Math. Coll.
1.252–256 Hultsch): its generation requires determining the ratio of the circumference of
a circle to its radius, although it is meant to find it. Attributed to Sporos are (F3–5) three
nominal scholia to Aratos’ Phainomena (Scholia in Aratum Vetera 541.40–46, 881.21–27,
1093.1–8 Martin) giving physical explanations for natural phenomena (end of the visual
ray pointing to the sky, parhelia, comets), and (F6) a short excerpt in one Aratos MS explaining
why Aratos began with boreal constellations, introduced by the mention “Hipparkhou
Sporos.”
Additionally, (T1) Eutokios (In Arch. Circ. dim. 4.162.18–24) mentions that Poros ho Nikaieus
blamed Archimedes for his vague approximation of the circumference of a circle, contrary
to his teacher Philon of Gadara’s more precise estimation, as reported in his Ke¯ria
(Honeycombs). (T2) This work may be the same as the Aristotelika Ke¯ria mentioned by Eutokios
in the same commentary (4.142.21), with no mention of author but as well-known to his
readers (Aulus Gellius, Pr.1.6, signals keria as an example of a curious book-title). ( T3)
Leontios reports that “Sporos the commentator” (3.6) excused Aratos’ lack of precision,
since his work was aimed only at navigators. Modern commentators strongly diverge on the
positive conclusions to be drawn from such weak and disparate bases. ( T3) and (F3–6) show
that Sporos probably commented on Aratos; (F2) and (T1–2) might indicate that he wrote a
compilation entitled Ke¯ria, containing critical discussions of solutions to classical problems
of geometry; (F4) and (T2) might indicate Sporos’ relative obedience to Aristotle.
Martin (1956) 205–209; DSB 12.579–580, M. Szabo; Knorr (1989) 87–93.
M. Terentius Varro of Reate (81 – 27 BCE)
Ancient Rome’s greatest scholar, born 116 BCE, came from the Sabine territory to the
north-east of the city. Varro studied first at Rome under the philologist and antiquarian
L. Aelius Stilo, then at Athens with the Academic philosopher Antiokhos of Askalon.
A partisan of Pompey the Great, he was active in politics, being elected tribune, aedile, and
quaestor, and serving several times as a naval and army commander. He was also a member of a 20-man commission set up by Iulius Caesar in 59 to redistribute public land in
southern Italy. During the civil war he led forces for Pompey in Spain, and saw his property
confiscated when the latter was defeated. He was given a pardon by Caesar, who selected
him to oversee the establishment of Rome’s first public library. After Caesar’s death the
project was abandoned, and Varro’s property was once again targeted for confiscation, its
owner marked out for death; only the protection of powerful friends ensured his survival.
He devoted the rest of his life to scholarship. In his will he requested that he be buried in
accordance with Pythagorean custom.
During his lifetime Varro composed some 620 books under 74 different titles. His efforts
included humorous essays on human nature (Saturae Menippeae, 150 books), dialogues on
philosophical topics (Logistorici, 76 books), a massive encyclopedia of Roman antiquities
(Antiquitates rerum humanarum et diuinarum, 41 books), and his 700 collected portraits of famous
Greeks and Romans (Hebdomades uel de imaginibus, 15 books). From this vast corpus only nine
books under two titles survive intact, the works De Rebus Rusticis, and De Lingua Latina.
Surviving Works: The characteristic features of Varro’s writings are prodigious erudition,
keen interests in terminology, etymology, and numerology, close attention to the organization
of his material, and occasional brilliant insights. These traits are all on display in the
one work of his which has come down to us complete, the De Rebus Rusticis (RR). Divided
into three books, the treatise deals with all the activities of cultivation that might be found
on a typical large Roman estate in the 1st c. BCE. The first book covers the essentials of
farming and the raising of plant crops, the second focuses on animal husbandry, while the
third deals with uillatica pastio, or the raising of specialty products such as birds, bees, rabbits,
and fish. It was apparently published in stages, with an early version of Book 1 appearing
before 55 BCE, Book 2 composed somewhat later, and Book 3 added to complete the trilogy
at the time of final publication in 37 BCE. Much of the material is clearly based on firsthand
observation, but Varro introduces his treatise with a list of 52 authors whose writings
he claims to have read.
The first book opens with an introduction of the dramatis personae (each book is presented
as a dialogue among prominent Roman land-owners), and a lengthy debate as to the subjects
that fall under the purview of agriculture proper. A guide to different types of land and
soil is followed by directions for preparing vineyards and farm-buildings, procuring the right
staff and equipment for the farm, and a discussion of the most profitable crops. Varro then
inserts a farmer’s calendar with tasks arranged according to an eight-fold division of the
solar year, and concludes by describing how to sow, care for, and harvest various field crops.
The book’s advice (18.8) that the farmer combine imitation of his predecessors with systematic
experimentation was to inspire many a later agronomist, and there are several interesting
reports on agricultural technology, e.g. that in Spain farmers used a riding thresher
on which the driver sat while it cleaned the grain (52.1). Nevertheless, the author’s grasp
of agricultural technique is uneven in this book. Varro is particularly weak on grafting,
for which he paraphrases and in many cases misunderstands his source (cf. 40–41, and
Theophrastos, Caus. Pl. 1.6).
The second book, by contrast, stands out for the depth and accuracy of its treatment of
stock-breeding – a fact no doubt connected to Varro’s possession of large-scale sheepranches
in Apulia and mule-farms near his hometown of Reate. The book as a whole is
structured according to a notional matrix: it has 81 subdivisions, to cover nine different
varieties of animal and nine different kinds of animal care in every possible combination –
that is, everything from “sheep, feeding of ” to “dogs, health problems of.” The longest section is devoted to swine (4), and detailed accounts are given of wool-shearing (11.5–12),
the breaking of horses (7.12–14), and the “points” that breeders were to look for in cattle,
horses, and donkeys (5.7–8, 7.5, 9.3–5). His discussions of transhumance (1.17, 2.9), and his
notices regarding the primacy of mules and donkeys as sources of power for ground transport
in ancient Italy (6.5, 8.5), are important texts for our understanding of the ancient
economy. In a surprise attestation of rural literacy, he claims to have copied out by hand
texts on veterinary health from Mago the Carthaginian and assigned them to his headherdsmen
to read (2.20, etc.).
The third book, on uillatica pastio, focuses mainly on the economics of animal breeding for
urban niche-markets, although it also offers observations on the social organization of bees
(16.4–9), patterns of bird migration (5.7) and learned behavior in birds, stags, boars, and
fish (7.7, etc.). Varro exposes the ingenuity that went into the construction of Roman coops
and bird houses, which were built to accommodate the special requirements of each breed
and featured running water as well as elaborate networks of perches and roosts (5.2–6, etc.).
A true tour de force is Varro’s description of the aviary at his villa in Casinum, which included
a stage for ducks to walk across, a lazy-susan bird feeder, and a planetarium (5).
The other Varronian work of which a substantial part survives intact is his De Lingua
Latina (DLL). Published in the late forties BCE, it originally consisted of 25 books: one book
of introduction, followed by six books on etymologies and meanings, six on inflectional
morphology, and 12 dealing with syntax and grammaticality. Only Books 5–10 have survived
complete. The first of these considers etymologies for the names of locations, the
second for names of periods of time, and the third for the words poets use. The lacunose
texts of Books 8–10 document Varro’s major contribution to theoretical linguistics, which
was a coherent account of variation among Latin roots and inflections that harmonized the
contrary principles of anomaly and analogy.
Lost Works: Among Varro’s lost works are many known to have provided impetus to
the study of science at Rome. Perhaps the most important was his Disciplines, an encyclopedia
of the artes liberales – the technical disciplines which it was felt proper for a free man to
pursue – in nine books, one for each art: grammar, rhetoric, dialectic, arithmetic, geometry,
astronomy, music, medicine, and architecture. This classification of sciences exerted a lasting
influence on later scholarship, which dropped the last two fields due to their banausic
character and consolidated the remainder into what would eventually become the trivium
and quadrivium of medieval tradition. Varro’s known contributions to the scientific subjects
of the quadrivium, as well as medicine and geography, can be summarized as follows.
Geometry: Varro traced the origins of geometria to the needs of surveying and to a
primitive interest in the size of the Earth (Cassiodorus Inst. 2.6.1). He translated into
Latin Euclid’s definitions of “plane,” “solid,” “line” and “cube” (ibid., 1.20), revealed that
in a circle the shortest distance from center to circumference is a line (RR 1.51.1), classified
optics as a geometrical subject, and explained the causes of various optical illusions (Gell.
16.18). The elementary character of this material is patent, yet in RR 3.16.5 Varro betrays a
familiarity with isoperimetry problems.
Arithmetic: Varro devoted a good deal of attention to Latin number terminology, seeking
for instance to establish the precise difference in meaning between secundum and secundo
(Gell. 10.1); a subtle examination of the “rule of nines” can be found at DLL 9.86–88.
Beyond that, a deep and abiding interest in arithmology pervades the Varronian corpus. This
comes across in his habit of working elaborate binary and ternary classificatory schemes into
his treatises, as in RR bk. 2, with its 81 subtopics of animal husbandry, or in the lost De Philosophia, which established as a theoretical possibility that 288 distinct schools of philosophy
could exist (Augustine CD 19.1–3). The Tubero de Origine Humana sought to explain
the viability of fetuses after seven months by dividing the 210–day period of gestation into
35–day chunks, then analyzing those sub-periods as compounds of harmonic ratios; a similar
rationale was given for nine-month pregnancies (Censorinus De Die Nat. 9, 11). Finally,
Varro’s Hebdomades included a long catalogue of entities that come in groups of seven (Gellius
3.10), and there apparently existed a catalogue for threes as well (Ausonius 15.pr Green).
Astronomy: The eighth book of Martianus Capella's De Nuptiis Mercurii et Philologiae
is thought to constitute a rough facsimile of the book on astronomy from the Disciplines.
Martianus explains the nature of the poles, polar circles, colures, the ecliptic, the constellations,
sunanatolai (simultaneous constellation risings), and the planets; the dimensions of the
planetary orbits betray their Varronian provenance by their numerological invention
(8.861). Elsewhere Varro observes the distinction between the sideral month (Fauonius Eulogius In Somn. Scip. 17.1) and the synodic month (Gell. 3.10), though in the interest of
numerology he rounds the figures for their respective lengths to 27 days and 28 days.
Music: Varro divided harmonics into three sub-fields dealing respectively with rhythm,
melody, and meter (Gell. 16.18); it was probably in the book on music from the Disciplines
that he described the major modes, from hyperlydian to hypodorian (Cassiod. Inst. 2.5.8). He
records numerous observations about animals attracted to music, such as the pigs he trained
to respond to a horn (RR 2.4.20, etc.), and shows in other texts (cf. Mart. Cap. 9.926–929) a
Pythagorean’s fascination with the inner connection between music and psychology.
Medicine: In his book on medicine Varro traced the science back to practices at the
temple of Askle¯pios at Ko¯s, and noted Hippokrates’ association with that temple; he
seems also to have assembled a large collection of herbal remedies for diseases (Pliny
20.152, 22.114, 141). A number of the logistorici were devoted to medical topics such as
diaetetics and mental health, and in the Tubero de Origine Humana he gave a detailed if
theoretical account of fetal gestation (Censor. De Die Nat. 9, 11). Varro observes in passing
at RR 1.12.2–4 that diseases could be caused by microorganisms, and accordingly recommends
that country-houses built near swamps not have windows facing in the direction of
the prevailing winds.
Geography: Varro’s survey of geography appears to have concentrated on such subjects
as ethnography (Pliny 3.8), the primary exports of different cities (4.62), the lengths of coasts
and rivers (4.78; cf. Gell. 10.7), and the etymologies of place names (3.8); he made particular
use of information Pompey Magnus gathered while on campaign near the Caspian Sea in
the second Mithridatic War (6.38, 51–52). He also wrote a periplous of the Mediterranean
(De Ora Maritima), a treatise on weather signs for use by sailors (Vegetius De Re
Militari 4.41), and one on tides (De Aestuariis) which was probably based on Poseidonius’
pioneering work.
Such fragments represent only the tip of an iceberg. Much of the technical material in
Pliny, Augustine, Martianus Capella and other late writers must also derive from Varro,
although we are no longer in a position to determine precisely its extent. Placed in the broadest
perspective, Varro’s contributions to ancient science were twofold. Like Cicero, he played a
crucial role in standardizing Latin technical terminology and digesting bodies of Hellenistic
science and philosophy with a view towards transmitting them to later generations. In addition,
he performed original work in several fields: first and foremost in the theory of grammar
and linguistics, in the classification of the sciences, and in the development of the Pythagorean
insight that numbers and patterns are crucial to our understanding of the world.
Ed.: D. Flach, Marcus Terentius Varro. Gespräche über die Landwirtschaft 3 vv. (1996–2002).
RE S.6 (1935) 1172–1277, H. Dahlmann; J.E. Skydsgaard, Varro the Scholar (1968); K.D. White, Roman
Farming (1970); W.H. Stahl, Martianus Capella and the Liberal Arts (1971); K.D. White, “Roman Agricultural
Writers I: Varro and his Predecessors,” ANRW 1.4.1 (1973) 439–497; B. Riposati, ed., Atti
del Congresso Internazionale di Studi Varroniani (1976); Rawson (1985); E.B. Cardauns, Marcus Terentius
Varro (2001); OCD3 1582, R.A. Kaster.
Theodosios (of Bithunia) (200 – 50 BCE)
Mathematician, wrote three extant treatises on mathematical astronomy and a lost commentary
on Archimedes’ Method, which establishes the only sure terminus post for his career.
Strabon lists him (together with his unnamed sons) as a noteworthy Bithunian mathematician,
and Vitruvios 9.8.1 as the inventor of a kind of sundial. An entry on Theodosios in
the Souda, Theta-142, which ascribes philosophical and poetic works to him and states that
he came from Tripolis, apparently confuses him with two other homonymous men.
The Spherics, in three books, was much studied in later antiquity (at least from the time of
Pappos, who commented on it in Book 6 of his Collection). It is an elementary work on
spherical geometry, with applications to astronomical problems that though obvious are
never mentioned in the text; Theodosios’ contribution was primarily to edit and organize
material already known in the 3rd c. BCE if not earlier. Underlying the work are the conventional
assumptions of contemporary astronomy, that the Earth and heavens are both
spherical and concentric and that the Earth has a point-like magnitude in relation to the
celestial sphere. The most advanced theorems demonstrate inequalities subsisting among
the arcs of the horizon or the celestial equator corresponding to equal rising arcs of the
ecliptic circle for observers situated either on or away from the terrestrial equator; for
example these theorems allow comparison of the length of time required for successive
signs of the zodiac to cross one’s horizon. The treatise lacks theorems on configurations of
great circle arcs (“Menelaos’ Theorem”) by which Ptolemy derives numerical values for
quantities in spherical astronomy in Almagest Books 1–2.
On Habitations is a collection of 12 theorems concerning risings and settings of stars and
length of daylight for different locations on the Earth; since most of the situations discussed
are either close to the equator or near the poles, the book is clearly an intellectual exercise,
not related to real observing conditions. The two books of On Days and Nights are similarly
impractical, dealing with such questions as criteria for having day and night exactly equal at
an equinox, taking into account the Sun’s small movement along the ecliptic during the
day in question.
O. Schmidt, On the Relation Between Ancient Mathematics and Spherical Astronomy (1943); DSB 13.319–321, I.
Bulmer-Thomas.
THE ENCYCLOPEDIA OF
ANCIENT NATURAL
SCIENTISTS
The Greek tradition and its many heirs
Edited by Paul T. Keyser and Georgia L. Irby-Massie
2008
