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In spite of the difficulties presented by their cumbersome numeral signs, the Romans were yet able to perform complicated mathematical problems by means of the abacus and by a system of finger-counting. The truth of this statement is generally recognized; but when one turns to the various works on Roman private life for information as to the actual use of the second method he is doomed to disappointment. A quotation from one of the most recent writers is perhaps typical.1 "In arithmetic mental calculation was emphasized, but the pupil was taught to use his fingers in a very elaborate way that is not now thoroughly understood." It is true that in Friedländer and in Marquardt,2 somewhat fuller statements are found, but they are mainly based upon a later Greek writer, Nicholaus of Smyrna, and fail to present illustrations of any concrete problem. The passage which has long been regarded as a locus classicus is Quintilian 1.10.35:3 numerorum quidem notitia non oratori modo sed cuicumque saltem primis litteris erudito necessaria est. In causis vero frequentissime versari solet: in quibus actor, non dico si circa summas trepidat, sed si digitorum saltem incerto aut indecoro gestu a computatione dissentit, iudicatur indoctus.
No suggestion of an actual problem for computation is found until we come to Apuleius, who in defending himself from the charge of magic, is forced to touch upon the delicate question of Prudentilla's age and thus in rather vague language attempts to p26 refute his opponent's calculations: posses videri pro computationis gestu errasse, quos circulare debueris digitos aperuisse.4
The contributions of another African writer to this subject, and indeed to the general topic of mathematical study, have been strangely neglected. I refer to Augustine, the great bishop of Hippo, in whose works, especially his sermons, may be found certain significant passages that throw light upon the method of counting commonly employed in his time. At first thought, a sermon seems a strange field in which to gather material of this nature; yet to one familiar with Augustine's method of biblical exegesis — a method which he learned from Ambrose, who in turn was influenced by Origen and certain schools of Alexandria — it is not a matter of surprise. Briefly stated, this method is that which with almost a caricature of the term he calls the "spiritual" as opposed to the literal interpretation; it is an effort to find in the plain prose of historical narrative with the frequent occurrence of numbers that seemed to him so inconsistent with the dignity of the Holy Scriptures, some hidden mystery of divine revelation, disclosed only to the diligent searcher after the pearl of sacred truth. In this way only had Augustine succeeded in overcoming his distaste for that which had earlier appeared to him trivial or even revolting in some of the Old Testament stories.5 Naturally the result of this perverted ingenuity is too often something puerile, which would appear ludicrous, were it not tragic in its awful waste of intellectual energy. Add to this a real Pythagorean feeling for the mystery of numbers, so characteristic of many early Christian teachers and so prominent in much of the early literature of the church. One passage may be cited in illustration of Augustine's point of view.6
Et horum quidem numerorum causas cur in scripturis sanctis positi sint, potest alius alias indagare vel quibus istae quas ego reddidi praeponendae sint vel aeque probabiles vel ipsis etiam probabiliores: frustra tamen eos esse in scripturis positos et nullas causas esse mysticas cur illic isti numeri commemorentur nemo tam stultus ineptusque contenderit.
Even proper names are given a mystical interpretation in accordance with the numeral value of the letters with which the p27 word is written in its Greek form. To cite one illustration only, Adam is thus made an unconscious prophet of the forty‑six years of the building of the temple.7
Of the passages more properly illustrating the topic of this paper, the first is found in Sermon. clxxv.1. Here by a species of contaminatio, Augustine attempts to find a connection between the "ninety and nine" of the parable, whom he identifies with the Jews, and the goats standing at the left hand on the day of judgment.
Superbiebant Iudaei, extollebant se, alta sapiebant, iustos se putabant et Dominum colligentem peccatores insuper accusabant. Qui ergo superbiebant et alta sapiebant, relicti sunt in montibus, ad nonaginta novem pertinent. Quid est, relicti sunt in montibus? Relicti sunt in timore terreno. Quid est, ad nonaginta novem pertinent? In sinistra sunt, non in dextera. Nonaginta enim et novem in sinistra numerantur: unum adde, ad dexteram transitur.
From the last words, the one conclusion to be drawn is that the members of the congregation addressed by Augustine were perfectly familiar with a method of counting by which the numbers from one to ninety-nine were represented on the left hand, while the sign for one hundred was formed on the right hand. This view is supported by other passages from the sermons which will be mentioned later and particularly by the Tractatus in Iohannis Evangelium cxxii.7: quia in summa centenarii numerus ad dexteram transit.
An interesting parallel is thus presented to the lines of Juvenal Satire X.248, 249 with their reference to the long life of Nestor.
Felix nimirum qui tot per saecula mortem
Distulit atque suos iam dextra computat annos.
A second series of interesting passages from Augustine's sermons is based upon his interpretation of John 21. 1‑14.8 After p28 the resurrection Jesus appeared at the Sea of Tiberias to certain of his disciples, who in the discouragement and despair that followed the Crucifixion had returned to their former occupation of fishermen. After casting their net to the right hand at the command of their Lord, they drew it back to land "full of great fishes one hundred and fifty-three." But why one hundred and fifty-three rather than any other number?9 Here evidently is a sacred mystery for the preacher to interpret to his people and he succeeds in discovering at least two solutions. His favorite explanation, however, is as follows: Seventeen is a significant number, because it is made up of ten, the number of the decalogue (decem propter legem) plus seven, the number of the Spirit (septem propter Spiritum). Now if all the numbers from unity to seventeen in an ascending arithmetical progression are added, the sum is just one hundred and fifty-three. On several occasions Augustine calls on his audience to follow the calculation for themselves and his words offer us some hint of the method employed, although unfortunately only the earlier combinations in the addition are mentioned.
Sermon. ccxlviii.5 and cclxx.7: Apud vos numerate: sic computate. Decem et septem faciunt centum quinquaginta tres: si vero computes ab uno usque ad decem et septem et addas numeros omnes — unum, duo, tria: sicut unum et duo et tria faciunt sex; sex, quatuor et quinque faciunt quindecim: sic pervenis usque ad decem et septem, portans in digitis centum quinquaginta tres.
We are justified, surely, in drawing from this passage the conclusion that in the congregation of this North African church, composed mainly of persons in humble circumstances and apparently with rather slight educational advantages, there were many who were able to perform rapidly on their fingers this calculation, which, it must be admitted, is not one of great simplicity.
In his commentary on the same text, Augustine again alludes to the use of the right hand for numbers above one hundred. Thus in Sermon. ccli.7 after the explanation of the mystical p29 number, this exhortation follows: cum pertinueris ad decem et septem, iam exinde excrescet numerus ad centum quinquaginta tres. Eris ad dexteram coronandus: ne remaneas ad sinistram damnandus.
A question naturally arises as to the identity of the system of computation thus employed in Africa at the close of the fourth century or the beginning of the fifth, with that in vogue in Italy at the time of Quintilian. In this connection it should be remembered that while Augustine was trained in the African cities of Thageste and Carthage, yet the educational system of these provincial schools was practically identical with that of Italy, where indeed he afterward taught, and in all probability had not been greatly changed since the time of the early empire. More positive evidence, however, is afforded by a comparison with the early mediaeval system of counting as it has been handed down to us in the work of the Venerable Bede (672‑735) entitled De loquela per gestum digitorum et temporum ratione.10 This must have been drawn from early sources; at the same time, it is in complete accord with the passages cited from Augustine.
Fortunately we have additional evidence from a contemporary of Augustine in a far‑distant province, Jerome, in letters written from his retreat at Bethlehem. In support of his favorite thesis of the supreme glory of chastity and the unequal merit of the three orders among Christians, the virgins, the widowed, and the married, he finds an analogy in the seed that "brought forth fruit, some an hundred-fold, some sixty-fold, some thirty-fold." His interpretation of this parable is thus developed in Letter xlviii.2.
Centesimus et sexagesimus et tricesimus fructus quamquam de una terra et de una semente nascatur, tamen multum differt in numero. Triginta referuntur ad nuptias quia et ipsa digitorum coniunctio, quasi molli osculo se complexans et foederans, maritum pingit et coniugem. Sexaginta vero ad viduas, eo quod in angustia et tribulatione sint positae. Unde et superiori digito deprimuntur, quia quanto maior difficultas expertae quondam voluptatis illecebris abstinere, tanto maius est praemium. Porro numerus centesimus (quaeso diligenter lector attende) de p30 sinistra transfertur ad dexteram: et iisdem quidem digitis: sed non eadem manu, quibus in laeva, nuptae significantur, et viduae, circulum faciens, exprimit virginitatis coronam.11
Let Bede's description of the representation of these three numbers serve as a commentary: Quum dicis triginta, ungues indices et pollicis blando coniunges amplexu. Quum dicis sexaginta, pollicem (ut supra) curvatum, indice circumflexo diligenter a fronte praecinges. Centum vero in dextra quemadmodum decem in laeva facies. (Quum dicis decem, unguem indicis in medio figes artu pollicis.)12
A comparison of the figurative passage of Jerome with the prosaic instruction of Bede furnishes convincing proof that each was familiar with the same method of the representation of numbers.
To return to Augustine, one somewhat indirect proof of the frequent use of "finger-counting" may be cited. If it be assumed that this was the ordinary method of performing calculations, it is natural to find digiti at times employed in a way that does not seem to admit of a literal translation. Thus in De civitate Dei XVIII.53, in refutation of certain predictions as to the end of the world, it is said: Omnium vero de hac re calculantium digitos resolvit et quiescere iubet ille qui dicit: non est vestrum scire tempora, quae pater posuit in sua potestate. In the Enarratio in Psalmum xxxi.2.16 he thus declaims against the astrologers: dicant illi electos et doctos numeratores siderum: dicant illi sapientes eos qui quasi digerunt in digitis fata hominum.
In conclusion, certain mathematical terms employed in the passages already cited may be briefly noted. Computare seems to have throughout the specific meaning of "count on the fingers" as opposed to the more general numerare.13 This is consistent also with the use of computare or computatio in the quotations from Quintilian, Juvenal, and Apuleius in the early part of this paper.
The relation between seventeen and one hundred and fifty-three is thus expressed in Sermon. ccli.5: A decem et septem p31 nascitur numerus crescens. In other passages crescendo is consistently used of an ascending arithmetical series. Where seventeen is said to be the seed (semen) of one hundred and fifty-three, the expression is probably figurative, suggested by the preceding sentence.14 An unusual phrase for an ascending series occurs in Enarratio in Psalmum cl.1: quod decem et septem in trigonum missis, id est ab uno usque ad decem et septem omnibus computatis, ad eundem numerum pervenitur. The phrase in trigonum mittere is not mentioned either in Harper's Lexicon or in Forcellini, and it was evidently too technical to be readily understood by Augustine's congregation, for it is directly followed by a definition introduced by the explanatory id est. It is perhaps, analogous to the English "pyramid-building."a
The use of ducere is puzzling: sometimes it is a synonym of the more common multiplicare, sometimes it means "to add" — a use not noted in the dictionaries. Thus the first sense is illustrated by Sermon. cclii.8: septuaginta quinque bis ducti faciunt centum quinquaginta; but in Contra Julianum III.11.22: nisi cum amborum anni computati et simul ducti centum transisse docerentur, the reference is clearly to the added ages of the husband and wife. The confusion is probably to be explained by the fact that with the Romans, whether finger-counting or the abacus was the means employed, the process of multiplication was performed by a series of additions, so that naturally the same term might be employed in either case.
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3 The following passages should also be cited to complete the references to this topic from the classical period. Pliny N. H. 34.7.16: praeterea Ianus geminus a Numa rege dicatus qui pacis bellique argumento colitur digitis ita figuratis ut ccclxv dierum nota aut per significationem anni temporis et aevi esse deum indicent. Suetonius, Claudius 21, adeo ut oblatos victoribus aureos prolata sinistra pariter cum vulgo voce digitisque numeraret.
5 Confessiones 6.3.6.
6 De Trinitate 4.6.10.
7 Tractatus in Iohannis Evangelium x.12: Quomodo ergo ibi invenimus et quadrenarium senarium numerum? Ad litteras computant Graeci. Quod nos facimus α litteram, ipsi lingua sua ponunt alpha et vocatur alpha unum. Ubi scribunt δ, vocatur in numeris ipsorum quatuor — m quod nos dicimus et illi dicunt my μ, quadraginta significat. Iam videte istae litterae quem numerum habeant et ibi invenietis quadraginta sex annis aedificatum templum.
8 Sermones ccxlviii.3; ccxlix.3; ccl.3; ccli.5‑7; cclii.8‑11; cclxx.7; Tractatus in Iohannis Evangelium cxxii.8; Enarrationes in Psalmos xlix.9; cl.1.
9 Sermones cclviii.1: Numquam hoc Dominus iuberet nisi aliquid significare vellet, quod nobis nosse expediret. Quid ergo pro magno potuit ad Iesum Christum pertinere, si pisces caperentur aut si non caperentur? Sed illa piscatio nostra erat significatio.
10 E. F. Wüstemann in Jahn's Jahrbücher, Supplementband XV (1849), pp511‑15 gives the text and also very interesting illustrations from the Regensburg manuscript.
11 Cf. Letter cxxiii.9.
12 Op. cit., p513. The Regensburg illustrations explain Jerome's words at a glance. The system of Nicholaus of Smyrna does not seem to be the same.
13 As in Sermon. ccxlviii.5: apud vos numerate, sic computate.
14 Sermon. cclxx.7.
a Our author need not have fished so far. The standard term used by modern mathematicians is triangular numbers, triangular series, probably of Pythagorean origin. The reason for the name is easy to see:
At any rate, the term exactly corresponds to the Latin trigonus (triangle or triangular), as found in Apuleius (Plat. I.7) among others, and also in standard use by ancient astrological writers to denote what modern astrologers now call the trine aspect.
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