I ROMAN ARITHMETICAL PRACTICE
We know very little about the practice of everyday arithmetic in the Roman world. We have no surviving text describing everyday computational processes. The archaeological evidence is extremely limited.Footnote 1 We are primarily reliant on inferences drawn from incidental literary allusions, and reconstructions of likely procedures based on figures in business and other records which lack the underlying computations. We know still less about the handling of fractions in such computations, although some conjectures have been made. For example, a passage from Horace tells us that schoolboys were taught how, by long calculations, to divide into a hundred parts.Footnote 2 Maher and Makowski have suggested a possible formula for those calculations by considering an example from literature.Footnote 3 Columella, writing on investment in viticulture, provides us with a pair of annual interest figures, the second being the interest on the total after the first has been added to the principal.Footnote 4 This pair of figures is useful for computational hypothesis-testing since the first corresponds precisely with that obtained by decimal calculation, while the second appears to be slightly rounded. A hypothesis that explains both should, in principle, carry enhanced plausibility.Footnote 5
The Tabula Alimentaria of Veleia (TAV), an inscription that sets out in detail the financial arrangements for two Trajanic alimentary schemes, provides an opportunity to examine not just two, but dozens of comparable calculations, if it can be shown that those calculations were computed on a like basis. This article argues that this is the case, and considers the implications for our understanding of the Roman arithmetical practices of the time.
II THE IMPERIAL ALIMENTARY SCHEMES AND THE TABULA ALIMENTARIA OF VELEIA
In the late first or early second century a.d., imperial schemes providing cash payments to support the upbringing of children (alimenta) began to appear in Italian towns.Footnote 6 The contemporary evidence for the alimenta is almost entirely epigraphic, comprising some sixty or seventy inscriptions whose findspots are widely distributed across Italy.Footnote 7 Most of the inscriptions are dedicatory, but two large bronze tablets provide significant detail about the financial structure of the alimenta: one is from Ligures Baebiani, some 20 km north of Benevento;Footnote 8 the other, longer inscription is from Veleia, situated about 30 km south of Piacenza in northern Italy.Footnote 9 In each case, the imperial endowment was provided through loans of varying amounts made to local landowners and secured by pledges of specified property. The landowners paid monthly interest to the schemes, thereby ensuring a regular supply of funds for alimenta payments.
The TAV, currently displayed in the Museo Archeologico in Parma, Italy, is approximately 1.37 m in height and 2.85 m wide. The inscription comprises a full-width praescriptio of three lines, followed by 671 lines of text in seven columns. The style of engraving is freehand. It describes two successive alimentary schemes endowed by the emperor Trajan, which provided cash support to the families of 300 children at Veleia.Footnote 10 The two schemes have been dated, based on evidence within the text, to c. a.d. 101 and 106/114 respectively.Footnote 11 This arithmetical analysis is concerned with the later, much larger scheme.Footnote 12 Documented first in the TAV, it occupies six and a half of the seven columns, and accounts for forty-six of the fifty-one loans recorded.Footnote 13 The praescriptio sets out its purpose and basic parameters:
Property pledge in consideration of HS 1,044,000 so that by the indulgentia of the greatest and best princeps, the Emperor Caesar Nerva Trajan Augustus Germanicus Dacicus, male and female children may receive alimenta:
Legitimate boys – 245 in number – HS 16 (per month) making HS 47,040 (annually)
Legitimate girls – 34 in number – HS 12 (per month) making HS 4,896 (annually)
Illegitimate boy – HS 144 (annually)
Illegitimate girl – HS 120 (annually) Footnote 14
Total HS 52,200 (per annum), which is equal to interest at 5 per cent on the above capital sum.Footnote 15
Each of the forty-six landowners’ pledges follows a similar, though not identical, formulation. First, the landowner's name is given. In the ten cases where a single property is involved, that property is identified by name and location, and by adjacent properties or features. The landowner declares an aggregate value for the property after, in some cases, deduction of an unspecified amount for land tax (vectigal). The landowner pledges (obligare) the property and agrees to receive a loan in a stated amount from the emperor. Transactions involving more than one property take a slightly different form, in which first a landowner makes a declaration of aggregate property value, after deduction, if any, for vectigal and properties pledged to earlier alimentary schemes, and commits to receive an overall loan amount. Then individual properties or groups of properties (numbering twenty-six in the largest case, Loan 13) are identified, and a value for each property or group declared, with a portion of the aggregate loan being allocated to each declaration. Thus Loan 11 (col. II, 18–26) takes the following form:
L. Sulpicius Verus has declared praedia rustica (to the value of) HS 71,522. He commits to receive a loan of HS 5,714 and pledges the fundus Lubuatini Obsidianus Arrianus, which is in Veleia, Valerian district, bounded by the public land of Lucensius, and by (the property of) Minicius Verus and Vicrius Sabinus and L Atilius; and the fundus Marianus in Veleia, Salvian district, bounded by (the property of) Granius Piscus and Tarsinius and Sulpicius Bacchus; which two properties combined he has declared at HS 28,600 for HS 2,214; also the fundi Luciliani Didiani, which are in Veleia, Valerian district, bounded by (the properties of) the Lucenses and Valeria Polla, which he has declared at HS 40,000 for HS 3,500.
The Loan 11 sub-loan amounts, HS 2,214 and HS 3,500, sum to the stated overall loan, HS 5,714. This is true for all but one of the thirty-six multi-property loans, the exception being Loan 31, which will be considered shortly. By contrast, the property sub-declarations, HS 28,600 and HS 40,000, sum to HS 68,600, whereas the overall declaration is HS 71,522. Across the multi-property loans, some overall declarations are more than the sum of the parts, some are equal, and some are less. Further, the ratio of sub-loan amount to sub-declaration value appears to follow no discernible pattern. Although this investigation is concerned with the relationship between the overall declarations and the overall loans, these observations are relevant to that analysis.
III THE ANALYSIS OF FéLIX DE PACHTERE AND A REVISED FORMULA
In the early twentieth century, French scholar Félix de Pachtere made a strong case for the existence of a consistent arithmetical relationship between each landowner's overall property declaration and the overall loan received. He determined that loan amounts were calculated using property declarations rounded down to the nearest HS 1,000, and concluded that a common multiplier of 8.05 per cent was applied to each rounded-down declaration to arrive at the related loan amount. He applied this insight to identify around a dozen emendations to the inscribed figures, nearly all of which can be attributed to simple inscribing errors. De Pachtere's emendations are shown in Table 1.
Summary of de Pachtere's emendations. Roman numerals for sub-HS 1,000 elements of the figures are omitted except where relevant to the emendation. In Loans 14 and 29, marked with an asterisk, the emendation applies to the loan figure; all others apply to the property declaration (de Pachtere Reference de Pachtere1920: 98–100).
De Pachtere came very close to providing a full explanation of the arithmetical procedure applied to derive overall loan amounts from overall property declarations. However, an 8.05 per cent multiplier works precisely for only four of the loans, and de Pachtere observes:
Il est remarquable que, sauf exception, la différence soit croissante des plus petites aux plus fortes obligations. Il y a ici et là des interruptions et des régressions dont on pourra apprécier la valeur en consultant la colonne 6 de la table qui suit. Mais, s'il est impossible d'expliquer ces anomalies, elles sont si peu importantes qu'on peut négliger d'en tenir compte.Footnote 16
It is notable that, discounting exceptions, the deviation increases from the smaller to the larger pledges. There are, here and there, some interruptions and regressions, the value of which can be seen by consulting column 6 of the following table. But, if these anomalies are impossible to explain, they are so insignificant that one need not take them into account.
De Pachtere's column 6 is reproduced in column H of Table 2. His claim that the anomalies are insignificant has not been universally accepted. Duncan-Jones says of de Pachtere's analysis:
… it is not clear that his more drastic substitutions are necessarily correct. There is too much residual error in the [TAV] to make it easy to say that the loans were always accurately related to the declared valuation.Footnote 17
De Pachtere's formula and the new formula compared. De Pachtere's emendations are marked with an asterisk (HS).
(a) Based on inscribed declaration. For the reasons set out below, de Pachtere's emendation is rejected.
As will be seen, the residual error lies not in the TAV, but in de Pachtere's arithmetical proposition.
A Revised Arithmetical Formula
De Pachtere inferred from the simple relationships 4,025 to 50,000 (Loans 8, 23 and 29) and 8,050 to 100,000 (Loan 39) that the underlying multiplier must be 8.05 per cent.Footnote 18 But this yields inaccurate and rather puzzling results for the other loans. The problem is resolved through minor adjustments to de Pachtere's formula: the overall declaration, rounded down to the nearest HS 1,000 as in de Pachtere's formula, is multiplied not by 8.05 per cent but 8 
$^{\rm 7}\!/\!_{144}$ per cent; and the non-integer remainder for the resulting loan amount is ignored (so that, for example, the Loan 4 calculated figure HS 12,233.89 is rounded down to HS 12,233 and not up to the nearer HS 12,234).Footnote 19
The results of the revised formula are compared with those of de Pachtere's formula in columns H and I of Table 2. After de Pachtere's emendations, the revised formula yields precisely the inscribed loan figure for thirty-five of the forty-six loans. Of the remainder, the HS 1 deviation in Loan 19 is unexplained.Footnote 20 All others but Loan 13 can be attributed to rounding to HS 5 or 10. We will return to Loan 13 and the rounded figures shortly.
Properties of the Revised Formula
Loan amounts derived from property declarations rounded down to the nearest HS 1,000 vary not continuously but stepwise, by increments of HS 80 or HS 81. A property declaration of HS 69,260 (Loan 35) yields a loan of HS 5,553 (69,000 × 8 
$^{\rm 7}\!/\!_{144}$ per cent). Declarations of HS 71,522 and HS 71,256 (Loans 11 and 40) each yield the same HS 5,714 loan (71,000 × 8 
$^{\rm 7}\!/\!_{144}$ per cent). It is straightforward to construct a table showing the loan amount associated with any property declaration, the relevant extract from which is shown in Table 3(a).
Extracts from table of loan amounts as a function of property declarations (all figures in HS)
Since only every 80th or 81st integer appears in the loan column, the fact that thirty-five of the forty-six inscribed loan figures appear in the table cannot be a result of chance. We can be very confident that these inscribed figures are derived from the revised formula. We can also infer with high confidence that the rounded loan amounts, which, with the exception of Loan 44, are within three sesterces of a figure appearing in the loan table, are derived from the revised formula and then rounded. This is a probabilistic inference that does not depend on individual property declarations or de Pachtere's emendations. Rather, the validity of the loan figures being independently established, the formula can be applied to those figures in order to validate all of de Pachtere's emendations to property declarations except for the Loan 13 declaration, where the loan amount neither appears in the loan table nor can be attributed to rounding.Footnote 21
Possible Objections: Loans 13 and 31
The anomalous result for Loan 13 raises, prima facie, an obvious objection to a claim that the revised formula fully explains the relationship between loan amounts and property declarations. De Pachtere's emendation to the Loan 13 property declaration (for which a simple explanatory inscribing error is not easily identifiable) does not resolve the discrepancy. Indeed, since the inscribed loan amount HS 94,765 does not appear in the loan table (see Table 3(b)), no such emendation can provide a resolution. The recorded loan amount cannot be the result of a simple inscribing error since the Loan 13 sub-loans also sum to HS 94,765. It must be inferred that the loan figure is correctly recorded, and thus deviates from the revised formula. Nonetheless, an explanation consistent with the revised formula may be conjectured, rooted in the limitations of Roman fractional arithmetic. It seems that the administrators who performed the TAV calculations were operating to a precision of no better than 
$^{\rm 1}\!/\!_{144}$ per cent.Footnote 22 At this level of precision they would not have been able to generate a loan total of exactly HS 1,044,000.Footnote 23 The identified arithmetical procedures positively require an adjustment to at least one of the loan amounts if the desired HS 1,044,000 in aggregate is to be achieved. The solution the administrators may have adopted was simply to make an adjustment to Loan 13, the largest property pledge except for Loan 43, the latter being a special case as it is a loan not to an individual but to a neighbouring civic community, the coloni Lucenses.Footnote 24 (It is unclear why the loans sum to HS 1,044,010 and not the exact figure HS 1,044,000.) If this conjecture is correct, the Loan 13 figures as inscribed are not inconsistent with the revised formula, and de Pachtere's unsuccessful attempt at resolution by emendation can be rejected as unnecessary.
An objection might also be raised in relation to Loan 31, where a simple inscribing error cannot easily be identified. Further, uniquely in the TAV, the sum of the eighteen sub-loans (HS 95,910) differs from the overall loan amount (HS 91,110). The editor of CIL 11 (1888), Eugen Bormann, writing before de Pachtere and unsuspecting of any relationship between overall property declarations and overall loans, posits two emendations to reconcile the HS 4,800 discrepancy: the overall loan, inscribed (col. V, 58) HS 
$\overline {{\rm LXXXXI}} $ CX (91,110), should read HS 
$\overline {{\rm LXXXXI}} $ C∞X (91,910); and sub-loan 16 should read not the inscribed (col. V, 91) HS 
$\overline {{\rm XXII}} $ (22,000) but HS 
$\overline {{\rm XIIX}} $ (18,000).Footnote 25 But since the inscribed overall loan figure appears in the loan table whereas neither Bormann's emendation nor the sub-loan total are present (see Table 3(c)), the inscribed figure should be strongly preferred.
An alternative reconciliation which has since, it seems, been overlooked, is found in the eighteenth-century editions of Maffei and de Masdeu.Footnote 26 They accept the inscribed total HS 91,110. They read sub-loan 16 not as HS 
$\overline {{\rm XXII}} $ (22,000) but HS 
$\overline {{\rm XVII}} $ (17,000), thereby reducing the discrepancy between the total of sub-loans and the overall loan from HS 4,800 to HS 200. An inspection of the TAV supports their reading. There is a missing stroke in the second character of the sub-loan 16 figure (one of several missing strokes in the TAV), so that what we see is something like HS 
$\overline {{\rm X}\backslash {\rm II}} $. The characteristic form of the letter and numeral V in the surrounding text has a vertical or near-vertical right-hand stroke. A restoration of the missing stroke to render a V is no less plausible than restoration of an X.Footnote 27 It is then not difficult to conjecture simple inscribing errors that could account for the missing HS 200. De Masdeu proposed an emendation of sub-loan 14 (Col. V, 86) from HS ∞CC (1,200) to HS ∞CD (1,400), but a missing CC from one of the other sub-loans is an alternative possibility. Which is correct is not material to the analysis and probably unknowable. This reconciliation, comprising a credible reading of an incomplete figure and a single emendation, not only retains the overall loan figure of HS 91,110, but is simpler than the two emendations of Bormann. It must be accepted. De Pachtere's emendation to the Loan 31 property declaration must also be retained, since it brings the declaration into line with the loan value according to the revised formula.
IV SOME CONJECTURES ON THE ARITHMETICAL OPERATIONS
Some three or four decades after the Veleian alimentary schemes were set up, the jurist Lucius Volusius Maecianus wrote a short treatise to the future emperor Marcus Aurelius on the subdivision of the as (the unit of Roman currency, but also carrying the meaning ‘the whole’, hence heres ex asse, heir to the whole estate). This, Maecianus says, is necessary for inheritance arrangements and many other purposes.Footnote 28 He explains the Roman fractional system, giving names and signs for duodecimal fractions down to the scriptulum, the 288th part of the whole. His treatise affirms the primacy of the duodecimal system in Roman fractional calculations of the time, and ties this directly to the dividing up of money and property. Maecianus explains at length the relationships between different combinations of fractions but gives no information as to how to carry out division of property in practice. The TAV allows us to test conjectures on how such calculations might have been performed.
The value 8 
$^{\rm 7}\!/\!_{144}$ would conventionally at the time have been expressed as a sum of duodecimal unitary fractions, that is 8 
$^{\rm 1}\!/\!_{24}\,\,^{\rm 1}\!/\!_{144}$ or possibly 8 
$^{\rm 1}\!/\!_{36}\,\,^{\rm 1}\!/\!_{48}$.Footnote 29 From this it might be suspected that the computations would proceed in the same way, with each element calculated separately and the results aggregated. But this does not consistently yield the inscribed result, assuming remainders are ignored at each stage. For example, the computation for Loan 25 would be as follows:
The same HS 16,901 results from a similar computation using 8 
$^{\rm 1}\!/\!_{36}\,\,^{\rm 1}\!/\!_{48}$ per cent. Further, calculating the fractional component as 
$^{\rm 7}\!/\!_{144}$ by first establishing the 144th part (HS 14, as above) then multiplying by 7 (HS 98) yields, after adding the 8 per cent figure of HS 16,800, a total of HS 16,898. Only calculating the fractional component by first multiplying by 7 (2,100 × 7 = 14,700) and then dividing by 144 (to give HS 102, remainder 12) yields the inscribed HS 16,902. This suggests that the accountant not only used non-unitary fractions, but also multiplied by the numerator as the first step, rather than first finding the 144th part.
No procedure has been identified that would generate exactly the figures that appear to be rounded. But if we allow that the accountant might have made time-saving approximations, an alternative conjecture arises. The roundings are hard to explain in terms of a sum-of-the-parts method. For example, Loan 44 (calculated loan HS 18,994) shows both the largest rounding by some distance (HS 6) and the ‘roundest’ result (HS 19,000). But the 8 per cent and 
$^{\rm 7}\!/\!_{144}$ per cent elements of Loan 44 are HS 18,880 and HS 114 respectively. There is no obvious reason to round up either figure, and rounding after adding them together yields negligible economy of computational effort.
The rounding is more readily explained by a calculation that does not involve a summation of parts. This would be the case if the accountant had conceived the loan multiplier entirely in terms of 144th parts. Expressed in this way, 8 
$^{\rm 7}\!/\!_{144}$ represents 1,159 144th parts. The necessary computation would be to multiply the hundredth part of each rounded property declaration by 1,159 and divide the result by 144. This gives the same result as multiplying by 8 and by 
$^{\rm 7}\!/\!_{144}$ and adding the results. We do not know how the Romans would have effected such a division, but one possible method involves repeated subtraction of the divisor from the dividend, using calculi and a counting-table, subtracting at each iteration the largest factor-ten multiple (or intermediate quinary) of the divisor d that is smaller than the remainder.Footnote 31 Each subtraction generates one Roman numeral of the quotient. The application of this method to Loan 44 is illustrated in Table 4. After five subtractions (10,000d, 5,000d and 1,000d three times), the quotient totals 
$\overline {{\rm XVIII}} $ (18,000) and the remainder on the counting-board stands at 
$\overline {{\rm CXXXXIII}} $ CCXXXX (143,240).Footnote 32 A further fourteen subtractions are required to arrive at 
$\overline {{\rm XVIII}} $ DCCCCLXXXXIIII (18,994). But perhaps an experienced accountant, seeing a remainder after the fifth subtraction that was close to, but less than, 1,000d (
$\overline {{\rm CXXXXIIII}} $ or 144,000), would have known that finishing the computation would be a lengthy exercise for little additional accuracy, even if he could not know the exact result or number of further subtractions required.Footnote 33 In this knowledge, he might instead have simply treated the remainder as 1,000d, adding Ī to the quotient for a rounded answer of 
$\overline {{\rm XVIIII}} $ (19,000). He might have applied a similar principle, with smaller roundings, to the other rounded figures.
Conjectural calculation process for Loan 44 (all figures HS)
Final quotient 
$\overline {{\rm XVIII}} $ DCCCCLXXXXIIII (18,994)
The time-saving incentive is not the same where the answer slightly exceeds a round number. This might explain why all but one of the roundings is upward. It does not fully explain why some numbers are rounded and other similar numbers are not, unless this is simply a matter of variations in diligence on the part of the accountant(s).Footnote 34
The plausibility of this conjecture depends in large measure on whether, for the Roman accountant, multiplying the hundredth part by 1,159 then dividing by 144 would, for conceptual or practical reasons, have been preferred to (a) multiplying the hundredth part by eight and recording the result; (b) multiplying the hundredth part by seven and dividing the result by 144; and (c) adding the two. It is not obvious that this would be the case. Nonetheless, the TAV dataset, and the arithmetical conjectures it permits, provide a useful addition to the sparse evidence we have for the arithmetical processes employed at Rome in the second century a.d.
