Textual Sources for Roman Mathematics

Ambrose, De Tobia 7
Reference to finger counting

Apuleius, Apologia 2
Reference to finger counting

Augustine, Against the Academics 3.11
illustration of immutability

Augustine, The city of God 11.3
meaning of the number 6

Augustine, Confessions 10.19
could not rid his memory of mathematical notions because of their abstractness

Augustine, On the quantity of the Soul 7.11
on the definition of a figure (some space shut in by some boundaries)

Augustine, On the quantity of the Soul 3ff.
on whether number, measure and weight pre-existed creation

Augustine, The Genesis Interpreted literally 4.7
on whether number, measure and weight pre-existed creation;

Augustine, The Genesis Interpreted literally 4.14 4.14
on the perfection of the number 6

Augustine, On the Immortality of the Soul 1.5
use of numbers to illustrate theological concepts

Augustine, On the Immortality of the Soul 1.6
illustration of immutability (examples with geometry)

Augustine, On free will 2.20.
use of numbers to illustrate theological concepts

Augustine, On Music, book 6
on the distinction between different types of numbers and their relative order from the pov of spiritual excellence.

Augustine, On the Trinity 4.17 4.17
On numerology

Augustine, De Civitate Dei 23.53
Reference to finger counting

Ausonius, Epigrammata 7
Reference to finger counting

Balbus, expositio et ratio omnium formarum
Geometrical / metrological treatise addressed to Celsus

Catullus, Carmina 5.7–13
possible reference to finger counting, and a gestus impudicus.

Cicero, Tusculan Disputations 1.2.5
claim that Romans restricted mathematics to measuring and reckoning

Cicero, Tusculan Disputations 5.23
History of mathematicians; tomb of Archimedes

Columella, on Agriculture 5.1.4-12
Detailed exposition of Roman land measurement, deriving units from the pes (foot) to passus, actus, and iugerum, and setting out the area of a iugerum (28,800 sq. feet) together with its fractional subdivisions (scripulum, uncia, sextans, etc.). The material is presented systematically.

Columella, De re rustica 5.1.1-3.9
Series of geometrical problem-texts concerining fields, preserved both in the MS tradition of Columella and in the Corpus Agrimensorum Romanorum

Epaphroditus and Vitruivus Rufus, fragments
Treatise(s?) that form the basis of the mathematical texts edited in the AgRoMa project.

Epaphroditus and Vitruivus Rufus, Epaphroditus and Vitruvius Rufus auctus [= Bubnov’s Varro F2]
Treatise containing geometrical (and other) problem-texts. To be edited as part of the AgRoMa project.

Epiphanius Latinus, De mensuris et ponderibus

Festus, de uerborum significatione 1.Pref
Festus compares his summary style to that of calculatores, who write large sums with compact notation.

Horace, Satires 1.6.71–75
Schoolboys with tabula and coin reckoning

Isidore, Etymologiae 15.15
Sets out the Roman system of agrarian measures, with description of regional variations

Isidore, Etymologiae 15.16
Description of measures of roads and distances

Isidore, Etymologiae 16.25
Description of Roman units of weights

Isidore, Etymologiae 16.26
Description of Roman capacity measures

Isidore, Etymologiae 16.27
Description of the symbols and abbreviations used for weights and measures

Jerome, Adversus Iovinianum 1
Refernce to finger counting

Juvenal, Satires 10.249–251
Reference to finger counting (the right hand was used to count hundreds and thousands)

Juvenal, Satires 9.39–44
Counting with calculus and tabula; financial arithmetic

L. Volusius Maecianus, assis distributio
Treatise on numerical divisions, weights and measures

Macrobius, Saturnalia 1.9
Reference to finger counting

Marcus Iunius Nipsus, fluminis uariatio
Treatise (uncertain date) on the geometry of surveying. Includes a section on estimating the width of a river

Martianus Capella, 6
Possible reference to abacus (check exact lines)

Martianus Capella, 7
Possible reference to abacus (check exact lines)

Martianus Capella, De Nuptiis Philologiae et Mercurii 2
Reference to finger counting

Martianus Capella, De Nuptiis Philologiae et Mercurii 7
Reference to finger counting

Ovid, Fasti 3.121–123
Reference to finger counting

Ovid, Epistulae ex Ponto 2.3.17
Reference to finger counting

Persius, Satires 1
Reference to abacus and counting in dust (pulvis)

Peter Chrysologus, Sermo 108
Reference to finger counting

Plautus, Miles Gloriosus 204–212
Reference to finger counting

Pliny the Elder, Naturalis Historia 34.16
Reference to finger counting

Pliny the Younger, Letters 10.17a–17b
Correspondence with Trajan requesting surveyors to sort out a city's expenditure

Pliny the Younger, Epistulae 2.2
Reference to finger counting

Priscian, De figuris numerorum quos antiquissimi habent codices
Short treatise explaining the signs and abbreviations used in Latin manuscripts to represent numbers and weights

Ps-Balbus, De asse minutisque eius portiunculis

Quintilian, Institutio Oratoria 1
Reference to finger counting

Quintilian, Institutio Oratoria 1.10.3
Importance for an orator of having reasonably good mathematical skills

Quintilian, Institutio Oratoria 1.10.39-41
Examples from geometry and arithmetic to illustrate how plausible but false arguments can mislead: e.g. the mistaken assumption that equal perimeters imply equal areas; the historical error of estimating the size of islands by the length of their coastal voyage.

Remmius F(l)avi(a)nus, , carmen de ponderibus
Latin didactic poem (208 hexameter verses), once ascribed to Priscian but now attributed to a Remmius Favinus (or Flavinus). It surveys Greek and Roman systems of weights and measures, and includes technical descriptions of a hydrometer for measuring specific gravity (ll. 103–121), and a hydrostatic balance for testing alloys (ll. 125–162) (cf. OCD entry)

Seneca, Epistulae Morales ad Lucilium 88
Reference to finger counting

Servius, Commentary on Virgil, Georgics 2.18
Defines ‘calculus’ as a small stone used for counting

Sidonius Apollinaris, Epistulae 9.9
Reference to finger counting

Suetonius, Divus Claudius 21
Reference to finger counting

Tertullian, Apologeticus 47
Pythagoreans think god is composed of numbers (as opposed to Epicureans who think he is composed of atoms)

Varro, de lingua latina 5.169-74.
Description of the decimal and fractional systems of Roman coinage

Varro, de lingua latina 5.34-5
Description of Roman land measures (actus, iugerum, centuria)

Varro, on Agriculture 1.18

Varro, on Agriculture 1.10.
Description of Roman land measures, comparing regional units (iugum, versus, iugerum), defining subdivisions like the scripulum (1/288 of a iugerum), and larger units such as the haeredium, centuria, and saltus

Vergil, Eclogues 3.4
Reference to astronomer Conon, useof compass/radius

Vitruvius, De architectura 1.1.3-4
Architect must be know geometry and optics, be skilled in figures. Explanation of why these are important, and mention of arithmetic for measurement aid and for estimation of costs.

Vitruvius, De architectura 1.1.16-8
Architects’ knowledge of all of the various subdisciplines of mathematics must be sufficient; those who know all the liberal arts in detail ‘go beyond what is required of architects can be called mathematicians’.

Vitruvius, De architectura 1.2.3
Proportion is the regular agreement of parts of a building: height to width to length to whole.

Vitruvius, De architectura 1.5.5
Towers (in a city) should be round or polygonal; square is a bad form.

Vitruvius, De architectura 1.6.5-16
Geometric construction for city orientation (wind rose).

Vitruvius, De architectura 3.1.1-2
The design of temples should reflect the ideal proportions of a human figure.

Vitruvius, De architectura 3.1.5
The measures used in buildings and all other works are derived from the members of the human body.

Vitruvius, De architectura 3.1.6-8
The mathematicians consider the number six perfect, but ‘our ancestors’ prefer the number ten.

Vitruvius, De architectura 3.3.7-11
Visual perception in column production

Vitruvius, De architectura 3.5.5-15
Mathematical constructions for column capitals; proportions for elements of temple construction.

Vitruvius, De architectura 4.Pref.3-4
The importance of the cube

Vitruvius, De architectura 4.6
Geometrical construction of the form of a Roman theatre.

Vitruvius, De architectura 4.7
Geometrical description of the form of Greek theatres

Vitruvius, De architectura 6.2.1
Dimensions of buildings are found by calculation

Vitruvius, De architectura 6.8.5-7
To remedy wall extrusion, the thickness of the wall must be in proportion, and buttresses carried up against it: geometrical construction of buttresses.

Vitruvius, De architectura 8.6.4
Pipes should be proportioned to the quantity of the water (with extensive numerical examples)

Vitruvius, De architectura 9.Pref
Importance of Pythagoras, Democritus, Plato, Aristotle, ‘and other sages.’’ Includes description of doubling the square and of ‘Pythagoras’ theorem’ about right-angled triangles.

Vitruvius, De architectura 9.7
Construction of the analemma.

Vitruvius, De architectura 10.Pref.2
Architects would be more careful in their calculations of expenses if they feared punishment.

Vitruvius, De architectura 10.6
Description of the machine for raising water on the principle of the screw.

Vitruvius, De architectura 10.11
Description of the construction of ballistae, with reference to ‘geometers and arithmeticians’.

Censorinus, de die natali 2
Eratosthenes calculated the size of the world 'by geometrical calculations'

Seneca, Letters to Lucilius 88.10-13
Seneca expresses a preference for arithmetic and 'practical' mathematics over geometry; he also makes a reference to counting on the fingers

Anonymous, de iugeris metiundis
Treatise containing geometrical problems linked to surveying (measuring the number of iugera in fields of various shapes. To be edited in the AgRoMa project.

Anonymous, liber podismus
Treatise containing geometrical problem-texts to be edited as part of the AgRoMa project

Sextus Empiricus, Against the Arithmeticians,
Text criticising notions such as number, unit or addition and subtraction.

,

Sextus Empiricus, Against the Geometers
Text criticising geometers, and particularly their use of hypotheses and definitions

Sextus Empiricus, Against the Logicians 105-6
Life judges everything by standards which are based on number: units of measurement are all based on number.

Vegetius, On military matters 2.19
desirability of arithmetical competence for soldiers and potential recruits

Vegetius, On military matters 3.15
desirability of arithmetical competence for soldiers and potential recruits

Vettius Valens, Anthology 1.8, 1.9, 1.15
instructions given to find the position of the moon and five planets geometrically and astronomically

Cato, On agriculture 2.2, 2.5, 5.4.