‘Problems’ at School: Mathematical Testimonies from the Fayum in the Roman Period

Introduction

The common association between the teaching of numbers and basic arithmetical operations, on the one hand, and linguistic-literary instruction, on the other, is a datum on which interpreters of Greek papyrological sources are essentially agreed (on which further, below). A more controversial or less frequently considered question is whether the course of non-specialised instruction also encompassed procedures of geometrical and metrological calculation in the form of the ‘problem’. Together with tables of arithmetical operations, such procedures are also typical of the papyrological mathematical tradition.^[1]

A connection between the teaching or learning of mathematics and its practical application characterises, in one way or another, the majority of the relevant papyrological sources.

As Giuseppina Azzarello has shown us,^[2] for tables of arithmetical operations, which are by far the best represented category, there is no doubt that this is a type of instrument that was produced and employed in educational contexts. In at least some cases, they were certainly the same ones as linguistic-literary teaching.^[3] It is also very likely that arithmetical tables (at least for some types of operations) were produced and employed as a reference tool for calculations in economic and bureaucratic activities.^[4]

For mathematical and metrological problems,^[5] several assumptions can now be considered generally established and accepted.^[6]

• ‘Standard’ structure and content: Contrary to what was hypothesised in the past,^[7] these are not exercises of particular application (of a previously learnt ‘general rule’) that were solved from time to time in an extemporary way by those to whom, be they students or not, the problem had been presented (as in modern didactic practices). Rather, they constitute the proposition of calculation algorithms (of geometrical and/or arithmetical type) to be learnt. They are presented in the form of procedural instructions – mostly introduced by the formula ὡϲ δεῖ or ὡϲ δεῖ ποιῆϲαι – to solve a particular problem step by step. The problem may be posed in more or less explicit form and sometimes as a question; it is always formulated not in general terms but with specific numerical data; it includes the direct performance and the results of each step. The indication of the final result is sometimes followed by an ἀπόδειξιϲ, which is not a theoretical demonstration but a counterproof/verification of the correctness of the calculation. The data involved are concrete realities and objects with their units of measurement.^[8]

• The character of the concrete situations presented in the problems is fictitious and exemplary, or at any rate simplified: The procedures of calculation very often have a more general applicability than the specific problem. The utility of these problems in some cases seems to lie in the reasoning they presuppose, more than or rather than in the specific problem’s practical application.

• Similarities and inter-relationships between the papyrological tradition of these problems in Greek and other traditions: i. e. their counterparts in Demotic, the older Egyptian and Babylonian mathematical tradition, and the Greek mathematical-metrological tradition known from medieval witnesses, in particular the writings attributed to Hero and Diophantus.

In contrast, questions about the actors, motives and contexts both of production and of use of this category of texts are more controversial and remain open.

For the category of witnesses to problems, the tendency in research has been to privilege the hypothesis of a connection to ‘scholastic’ or ‘instructional’ contexts.^[9] It has also occasionally been argued that the training offered by this type of mathematical tradition overwhelmingly benefitted tradespeople whose activities were envisaged by the problems, such as land surveyors and architects.^[10] In more recent years, however, different hypotheses have been proposed. On the relationship between these mathematical texts and educational contexts, it has been considered more likely that they were produced and used in training courses reserved for professional scribes or secretaries^[11] and that, with the exception of learning numbers and an initial smattering of basic arithmetical operations, the study of mathematics was independent of ‘ordinary’ (so to speak) education, namely linguistic and then linguistic-literary instruction.^[12] On the other hand, it has also been noted that learning the mathematical-metrological procedures of problems was, at least potentially, just as necessary for anyone actively and variously involved in economic activities and administrative tasks.^[13] Regardless of the particular educational path at stake, the ‘scholastic’ nature of some witnesses to the problems tradition has been questioned.^[14] In addition, it has been found likely that problems (and not only those of a strictly metrological nature) – like arithmetical tables, which are associated with them in the tradition (e. g. P.Mich. III 145; P.Cair.Cat. 10758) – could also have been used as models or reference tools for calculations by those who by reason of their profession or social status dealt with counting and measuring.^[15]

Indeed, among the Greek witnesses to this type of text, only a few items have characteristics that positively indicate an educational context.^[16] The oldest of these date between the 1st and 3rd centuries AD, and are of proven or hypothetical provenance from the Arsinoite nome: P.Vindob. inv. G 26011e r (P.Rain.Unterricht [= MPER N.S. XV] 179), 1st c.; P.Vindob. inv. G 26740 (partim P.Rain.Unterricht [= MPER N.S. XV] 178), 1st/2nd c.; T.Tebt. inv. 3033 (SB XXIV 16038), 1st/2nd c.; O.Mich. inv. 9815 (O.Mich. IV 1140 = SB XIV 11527), 2nd/3rd c. Despite being so few in number, they reveal aspects of the teaching of mathematics that are worthy of note.

1 O.Mich. inv. 9815

O.Mich. inv. 9815 (O.Mich. IV 1140 = SB XIV 11527), dated to the 2nd/3rd century AD, is an ostrakon found in 1932 during the excavation campaigns of the University of Michigan (1924–1935) at the site of Karanis. The text is on the convex face, lacking at least the original upper right corner.^[17]

What remains of the first line consists of a notation in at least four symbols, the fourth of which is mutilated on the right-hand side (where the ostrakon is broken). These symbols constitute an abbreviated notation of the proposition of a problem, indicated by the incipit of the next line – ὡϲ δεῖ or ὡϲ δεῖ ποιῆϲαι – which is the standard formula in this category of texts for introducing the solution’s procedure. In what followed the proposition, an untidy and still not entirely secure hand began to record the solution – or at least a part of it – probably taking dictation. Halfway through the third line (to judge from what can be interpreted of it), he/she seems to have stopped. Perhaps his/her attention lapsed and he/she began to doodle, as those familiar with today’s classrooms know that students often tend to do. Regardless of when the drawing was made, it is paralleled by two similar drawings, certainly produced by students: one in a tablet (BM Add. MS 33368) and the other in a papyrus notebook (P.Vindob. inv. G 29274, p. 16 [MPER N.S. IV 24 = SB XXVIII 17151]). Frederic G. Kenyon even suggested that the former could be some kind of a portrait of the teacher, which of course cannot be proven.^[18] Everything about the hand that began to write the solution on the Michigan ostrakon – perhaps different from the one that penned the symbols in the first line, which seems more skilled – suggests it belonged to a student in training: it displays some uncertainty, is not fully fluent in tracing the letters and, above all, reveals difficulties in knowledge and realisation of the symbols. The condition of the text, combined with the low resolution of the only available reproduction (see above), does not permit all the surviving traces to be deciphered clearly. The difficulties of both reading and interpretation are increased by the obvious uncertainties and carelessness of the writer of the text, who also made mistakes in what little has been written of the solution. It is by no means impossible that, through similar distraction, he also skipped one or more passages or individual elements of the ‘model text’. The following is a preliminary hypothesis of reading:

Figure 1:

O.Mich. inv. 9815r; image freely accessible online at the University of Michigan Library website (https://quod.lib.umich.edu/a/apis/x-1074)

[---] (?) (διακόϲιοι)^? (vac.) (πυρόϲ) (τριώβολον) (vac.) (γίνεται^?)[---]

ὡϲ δεῖ̣ ποιεϲ̣αι̣ (lege ποιῆϲαι)^[19] τ̣οῦ ⟨---⟩ (?) ποί̣ε̣[ι (?) ---]

(ἀρτάβαι+γίνεται)^? (δραχμαὶ)^? ρ ἐπὶ̣ τ̣α̣ϲ̣ (?) (vac.) ⟦χο(ίνικεϲ)^? / κριθ(ή)^?⟧

The editor (Youtie 1975) interpreted the first two symbols of the first line as the numeral 200 followed by the symbol for πυρόϲ “wheat” (cf. e. g. BGU I 20; BGU VII 1570) or πυροῦ ἀρτάβη (cf. e. g. P.Cair.Goodsp. 30 [see Goodspeed 1902, 34]). In light of the subsequent occurrence of the less problematic symbol of the triobol, the text seems to probably belong to a well-documented trend in the papyrological tradition of problems related to operations with quantities of wheat in artabas, their submultiples and other units of volume, sometimes also involving the prices of such quantities: cf. e. g. the problems in P.Mich. III 145 fr. 3 col. ii.1–9; and col. vii.7–10; BL Ostr. inv. 43893 (= SB XXVIII 16920); P.Michael. 62 A; MPER N.S. IV 24 (= SB XXVIII 17151) (P.Vindob. inv. G 29274, p. 12).^[20] The reading of the first symbol is far from certain: it resembles other symbols, including the talent (cf. e. g. BGU I 34, early 4th c. AD). However, Youtie’s interpretation is supported by the fact that, of the symbols still visible, this is the only one that could potentially be a number (namely 200, even if written in a form that is hardly exemplary): this type of text requires at least one numerical datum. Still, the number in this case was followed (and not, as is usual, preceded) by an indication of the unit of measurement. Although the quality of the image does not allow one to go beyond speculation, perhaps a solution to this difficulty could be that before the numeral there was originally (at least in the ‘model text’) another symbol or abbreviation, some slight traces of which can perhaps still be discerned at the fracture line in the upper left corner of the ostrakon, which appears to be abraded. For the possible missing sign(s), an abbreviation χο(ίνικεϲ) (cf. e. g. the problem on the artabas of respectively 36 and 40 χοίνικεϲ in P.Mich. III 145 fr. 3 col. vii.7–10)^[21] perhaps deserves consideration, although it is a highly speculative hypothesis. It may be no more than a coincidence, but 200 χοίνικεϲ could correspond in price to the 100 drachmas most likely recorded in l. 3 (see below): one of the unit prices for artabas of wheat attested in the accounts of P.Cair.Goodsp. 30,^[22] likewise from Karanis and dated to the same period as our problem,^[23] is 20 drachmas per artaba (see P.Cair.Goodsp. 30 col. 15). Given that the most frequent ratio of artabas to χοίνικεϲ is 1 artaba = 40 χοίνικεϲ,^[24] a price of 20 drachmas per artaba would mean 1 triobol (i. e. half a drachma) per χοῖνιξ; 200 χοίνικεϲ would thus cost 100 drachmas. Incidentally, it is perhaps even possible that the other unit price attested by the Goodspeed papyrus – 18 drachmas per artaba (P.Cair.Goodsp. 30 col. 13; col. 20) – represents not a fluctuation in prices (which in this case refer to the same month)^[25] but applies the same price of 1 triobol per χοῖνιξ to an artaba of 36 χοίνικεϲ, a less frequent, but nonetheless attested ratio. Beyond this latter speculation, the ratio of 1 triobol per χοῖνιξ (rounded or not) is in line with real prices attested for the same period and the same place as our ostrakon. As Federico Morelli has noted, some relationship between the figures used in the problems and real-world prices seems, in general terms, more likely.^[26] Moving on, it is possible that the final surviving stroke in the first line was the symbol for γίνεται, introducing the result sought by the problem. In that case, the proposition of the problem, essentially reduced to pure data,^[27] could be more or less as follows: [χοίνικεϲ] 200 (of) wheat (at) 1 triobol (each) equals [(how many) drachmas?], or more likely: [(how many) drachmas per artaba?].

Comparison with similar texts indicates that the first symbol of the third line, which is problematic as it stands, may be the result of a confusion or combination of the symbols for artabas (as hypothesised by the editor) and γίνεται, with the oblique bar and two dots ⸓ (used for example in T.Tebt. inv. 3033, which we will discuss shortly). The editor doubtfully identified the subsequent symbol as the numeral 1,000. I think its shape is by far more consistent with that of a symbol for drachmas, followed by what in this context is almost certainly a rho (i. e. the numeral 100).^[28] If so, this was a monetary figure (“drachmas 100”), the occurrence of which could also befit with the almost certain mention of the triobol in the proposition. As for the erasure at the end of the third line, the mark above it suggests that it was an aborted attempt to write an abbreviation: among the possible matches with the shape of this upper mark, the two abbreviations that would most likely fit the context are that for χοίνικεϲ and that for κριθή “barley/wheat” (for both cf. e. g. their realisation in BGU VII 1570). At the beginning of the solution procedure (l. 2) the genitive article τ̣οῦ is probably present, likely to be followed by an omitted numeral or, more probably, by an entire omitted operation.^[29] If so, it could have been the incomplete statement either of a subtraction (in that case with the unconventional, but attested, omission of the preposition ἀπό) or more probably of a division (in the standard form “of tot an n-th part, equals tot” or “from tot take an n-th part, equals tot”). [ in the second line and ἐπὶ̣ in the third suggest that a multiplication followed (cf. Youtie 1975, 281 ad 2–3). Finally, still concerning the beginning of the solution procedure (l. 2), an alternative reading τ̣ὸ ὑπόλ̣[οιπον, “the remainder”, cannot be ruled out (although this alternative is less likely, since the faded traces following the sequence τ̣ουπο seem more compatible with iota, possibly followed by an epsilon, than lambda). In this case, the text recorded on our ostrakon would nonetheless be only a part of a solution procedure that presupposes at least one additional calculation step which has been omitted.

2 T.Tebt. inv. 3033

T.Tebt. inv. 3033 (= SB XXIV 16038) is a wooden tablet (possibly already used to receive a previous text) with arithmetic-metrological problems written on both sides (parallel to the longer edge) by a single hand, datable to the 1st/2nd century AD. It was found at Tebtunis in 1994 by the joint mission of the Institut Français d’Archéologie Orientale (IFAO) and the Istituto di Papirologia dell’Università degli Studi di Milano.^[30] The editors describe the text as the work of a student, with some practice in writing, but one who still made numerous mistakes and struggled or was careless in maintaining an orderly layout. The text of one side (A) is formulated as a problem of calculating interest and capital, which has a specific parallel in two contemporary problems in P.Mich. III 145 fr. 3, col. vii.1–4 and 5–7 (2nd c. AD).^[31] The problem in our tablet is as follows: given an interest rate (τόκοϲ) of 5 drachmas per 1 mna (100 drachmas: thus 5 %) per month, if 4 months have passed and the total amount of (accrued) interest plus starting capital is 1,000 drachmas, know/find the accrued interest (τόκοϲ) and the starting capital (κεφάλαιον). The procedure for solving the problem – introduced by the classic formula ὡϲ δεῖ – is as follows:

1. multiply 5 by 4 (i. e. monthly interest rate times the number of months elapsed) = 20;

2. multiply the result (20) by 10 mnas = 200 drachmas of accrued interest (sic);

3. subtract this from 1,000 (i. e. the total amount of accrued interest plus starting capital) = 800 drachmas of starting capital.

Clearly, the second part of the procedure and the final results are not consistent with the proposition of the problem.^[32] As I will demonstrate elsewhere,^[33] the tradition of this particular type of problem is vexed, and it cannot be ruled out that the errors of the Tebtunis tablet were already present – in whole or in part – in its model. Although the figures differ, the second step of the algorithm matches almost verbatim that of P.Mich. III 145 fr. 3, col. vii.2: multiply the result of the first step (i. e. the total interest rate) by 10 mnas, which correspond to the 1,000 drachmas of the total amount of accrued interest plus starting capital. Both in our tablet and in P.Mich. III 145, this is ‘incorrect’. Nonetheless, the result given in our tablet, namely 800 drachmas of starting capital and 200 drachmas of accrued interest, relative to a total amount of accrued interest plus starting capital of 1,000 drachmas after 4 months at the rate of 5 % per month, is much closer to the correct answer – i. e. 833 1/3 drachmas of starting capital and 166 2/3 drachmas of accrued interest^[34] – than the one (i. e. 100 drachmas for the accrued interest and 900 for the starting capital) that would have been yielded by the application of the calculation steps shown by P.Mich. III 145 fr. 3, col. vii.1–4.

On the other side of the tablet (B) is a set of four problems for converting artabas of the standard δρόμῳ into artabas of four other kinds (χαλκῷ, Ἑρμοῦ, Φιλίππου, δεξίμῳ).^[35] Numerous elements correspond to those in the ‘complete manual’ preserved on P.Lond. II 265r, which, moreover, begins with the section devoted to the artabas δρόμῳ.^[36] The matching elements are: the wording of the propositions (tot artabas δρόμῳ, how many χαλκῷ/Ἑρμοῦ/Φιλίππου/δεξίμῳ?); the repetition of the same number as the starting data of all the problems; the specific calculation algorithms – again introduced by the classic formula ὡϲ δεῖ – and their wording; the conversion rates (tot art. δρόμῳ: + 28 % = tot art. χαλκῷ; + 25 % [= 1/4] = tot art. Ἑρμοῦ; + 16 2/3 % [= 1/6] = tot art. Φιλίππου; + 23 1/3 % = tot art. δεξίμῳ ~ art. Γάλλου [in P.Lond. II 265r] = art. δρόμῳ + 23 2/3 %); the grouping of problems according to the kind of artaba to be converted; and even the layout of the text – with a separate textual section for each problem and the separation of proposition from solution in individual paragraphs.

The principal difference between the problems on our tablet and the London papyrus is the quantity of artabas to be converted: here they are 100, rather than the 625 in P.Lond. II 265r. The round figure greatly simplifies the calculations, particularly when it comes to percentages of the starting figure (as in problems 1 and 4, i. e. ll. 1–3; 7–9):

• Probl. 1 (cf. P.Lond. II 265r, col. 1.1–5): given 100 artabas δρόμῳ, how many χαλκῷ? How it is to be done: multiply 100 by 28, equals 2,800, of which 1/100 equals 28, add to 100, equals 128.

• Probl. 4 (cf. P.Lond. II 265r, col. 2.25–29): given 100 artabas δρόμῳ, how many δεξίμῳ? How it is to be done: multiply 100 by 23 1/3, equals 123 1/3 (lege 2,333 1/3),^[37] of which 1/100 equals 23 1/3, add to 100, equals 123 1/3.

This ‘variant’, i. e. the choice of the figure 100, recurs in P.Vindob. inv. G 26740 (partim P.Rain.Unterricht 178), col. 8, as we shall see below. It appears to be motivated by didactic considerations – in the most general and neutral sense of the term –, namely to make the mechanism of a particular type of procedure understood by adopting its simplest applicable example. Those considerations are demonstrated by the fact that the calculation of the percentage is not omitted despite being pleonastic for computational purposes, but is set out in full and in the same terms as the corresponding problems of P.Lond. II 265r, where, in contrast, it is an operation that is genuinely indispensable to reach the solution according to the algorithm indicated by each of those problems.

A further instance of reducing procedures to their simplest version can be seen in the other two problems (2 [δρόμῳ → Ἑρμοῦ: ll. 3–4] and 3 [δρόμῳ → Φιλίππου: ll. 5–6]). In each of their counterparts in P.Lond. II 265r, two alternative and equivalent algorithms are proposed, the second of which is introduced by ἄλλωϲ:

• conversion δρόμῳ → Ἑρμοῦ (P.Lond. II 265r, col. 2.30–37):

  1. multiply by 25 the 625 artabas δρόμῳ and calculate 1/100 (= divide by 100) of this product (i. e. calculate 25 % of the number of artabas δρόμῳ to be converted) → add the result to 625 (i. e. to the number of artabas δρόμῳ to be converted);

     alternatively (ἄλλωϲ)

  2. (since 25 % = 1/4) calculate 1/4 (= divide by 4) of the 625 artabas δρόμῳ → add the result to 625;

     cf. T.Tebt. inv. 3033 B, probl. 2 = ll. 3–4: given 100 artabas δρόμῳ how many Ἑρμοῦ? How it is to be done: 1/4 of 100, equals 25, add to 100, equals 125.

• conversion δρόμῳ → Φιλίππου (P.Lond. II 265r, col. 2.15–24):

  1. multiply by 16 2/3 the 625 artabas δρόμῳ and calculate 1/100 (= divide by 100) of this product (i. e. calculate 16 2/3 % of the number of artabas δρόμῳ to be converted) → add the result to 625;

     alternatively (ἄλλωϲ)

  2. (since 16 2/3 % = 1/6) calculate 1/6 (= divide by 6) of the 625 artabas δρόμῳ → add the result to 625;

     cf. T.Tebt. inv. 3033 B, probl. 3 = ll. 5–6: given 100 artabas δρόμῳ how many Φιλίππου? How it is to be done: 1/6 of 100, equals 16 2/3, add to 100, equals 116 2/3.

In both cases, the text of our tablet applies only the second method, i. e. only the simplest and quickest method. Doing so further simplifies a calculation procedure already simplified by taking the number 100 as a starting point. It ‘reduces’ the calculation of a percentage from two operations to one: instead of multiplying by the rate and dividing by 100, in that order (cf. T.Tebt. inv. 3033 B, probl. 1 and 4; P.Lond. II 265r) or in the reverse order (cf. below P.Vindob. inv. G 26740 col. 8), it proceeds by way of a ‘unit fraction’/division.

3 P.Vindob. inv. G 26740

As mentioned above, a Vienna papyrus, P.Vindob. inv. G 26740 (partim P.Rain.Unterricht 178), presents a closely analogous set of problems, only the first two of which are preserved.^[38] They are found in the eighth column (ll. 8 ff.; on the preceding texts, including a different set of problems, see further, below) in the same didactically simplified form which begins with 100 artabas δρόμῳ, converting them into χαλκῷ (+28 %) in probl. 1 and into a standard involving the same ratio as attested for the standard Γάλλου [cf. P.Lond. II 265r, col. 2.25–29] (+23 2/3 %) in probl. 2. The set is introduced by the title διάν̣οια [μ]έ̣τρων and presumably continued in at least one further column (now lost), in order to complete the series devoted to the artabas δρόμῳ. Here too the solutions are introduced by the classic formula ὡϲ δεῖ ποιῆϲαι and include the ‘pleonastic’ calculation of the percentage of 100 (as in T.Tebt. inv. 3033 B, probl. 1 and 4). The only difference from the Tebtunis example is the inversion of the order of the two operations: here, one calculates 1/100 of the starting 100 (i. e. to divide 100 by 100) before multiplying the resulting unit by the percentage rate:

• Probl. 1 conversion δρόμῳ [→ χαλκῷ] (cf. P.Lond. II 265r, col. 1.1–5):

  (given) artabas δρόμῳ 100, how many [χαλκῷ?

  How it is to be done: take from [100

  1/100, equals 1, (multiplied) by 28, equals [28, …

  …

• Probl. 2 conversion δρόμῳ [→ Γάλλου (?)] (cf. P.Lond. II 265r, col. 2.25–29):

  (given) artabas δρόμῳ [100, how many Γάλλου? … How it is to be

  done: take from [100 1/100,

  equals [1], (multiplied) by 23 2/3, equals [23 2/3 …

  …

We have thus arrived at the most significant of the witnesses of interest to us: P.Vindob. inv. G 26740 (partim P.Rain.Unterricht 178).^[39] The probable provenance of this papyrus from Soknopaiou Nesos has been argued on several occasions and especially by the study dedicated to it by Mario Capasso.^[40] It is a palimpsest text dated between the 1st and 2nd centuries AD, written by reusing the perfibral face of a fragment of a scroll, which had previously been used first on this same face for a text that is no longer identifiable (possibly in Demotic) and then on the transfibral face for a text in Demotic that has not yet been deciphered. This witness has been extensively studied both in its material and palaeographical aspects as well as in relation to the interpretation of its content,^[41] but – certainly also due to its complex publishing history^[42] – it has not been considered in more general reflections on the relationship between mathematical and literary education.

The surviving text, written by a single hand and certainly mutilated at the end, is arranged in eight columns. The first five each preserve the text of a geometric problem of surface calculation (eds. Bruins/Sijpesteijn/Worp 1974; Harrauer/Sijpesteijn 1985 [= P.Rain.Unterricht 178]) at top, with a corresponding drawing in the lower part of the column. In all five of these problems, the proposition follows the classical formulation “given tot schoinia for the linear measurements of the geometric shape (of the plot of land), how many arouras?” (cf. e. g. P.Berol. inv. 11529v; P.Gent inv. 1v, col. 1 [SB XVI 12680]; P.Chic. 3 [‘P.Ayer’]). The solution procedure is introduced by the standard formula ὡϲ δεῖ ποιῆϲαι. Probl. 1 calculates the area of a μηνίϲκοϲ “crescent” (properly a circular crown sector). Probl. 2, 3 and 4 calculate the area of a circle, with three different algorithms:^[43] A = 1/12 p²; A = 3/4 d²; A = 1/3 (r + d)². Probl. 5. calculates the area of a semicircle. Thereafter, columns 6–7 and the upper part (ll. 1–7) of the eighth column (eds. Sijpesteijn/Worp 1974 [ed.pr. Oellacher 1938]) contain an excerpt of the parting of Hector and Andromache from the sixth book of the Iliad (6.373–410). The famous letter P.Oxy. VI 930 (2nd/3rd c. AD) shows that this was among the selection of Iliadic texts taught by the ‘teacher of letters’.^[44] The Homeric text is then followed – in the same eighth column and isolated by a paragraphos – by the set of conversion problems of the different kinds of artaba that have been discussed above (ll. 8 ff.: eds. Bruins/Sijpesteijn/Worp 1974; Harrauer/Sijpesteijn 1985 [= P.Rain.Unterricht 178]).

There has been agreement on the ‘scholastic’ nature of this miscellaneous sylloge, not only because of the texts it contains, but also for material reasons (the dynamics of particularly intensive reuse) and especially graphic ones. The type of handwriting, its overall confidence in tracing the letters and the relative regularity in the arrangement of the text could be compatible with a teacher’s hand, but the very high frequency of errors (which also recur in the numerical data on the drawings) and the nature of the errors made decisively indicate work copied by a student still in training.^[45] We can also add the uncommon fact that the forms γίνεται/γίνονται are written in full (and consistently with the phonetic error γει-), instead of being rendered with a symbol as would be normal (and as presumably also in the copy’s exemplar). These features, as well as the presence of a dittography error and, above all, the overall organisation of the writing space all suggest that this is not a series of notes taken under dictation but rather that our text is the overall reproduction of a model or models proposed by the teacher,^[46] which was/were already organised in this way. Although it is not a careful production, it reveals a certain amount of planning (see above and cf. also the second paragraphos in column 8, separating the first conversion problem from the second), even if the individual items or columns were most probably realised in distinct moments.^[47] It should also be noted that the five geometric problems share typologies: not only the type or sub-type of geometric shape involved in all five problems (circular crown sector, circle, semicircle), but also the partially analogous solution methods linking the last two on the circle and semicircle, two different shapes which can be subdivided into the same shapes. Such organization is characteristic of structured collections of geometric problems (in forms that may liken them to a kind of manual), to which we will return shortly.

4 P.Vindob. inv. G 26011e r

The last of our witnesses is P.Vindob. inv. G 26011e r (P.Rain.Unterricht 179),^[48] dated to the 1st century AD. It comes last certainly not from any lack of importance, but because the records about its Fayumic provenance, specifically from Soknopaiou Nesos, is somehow problematic. This provenance is indicated without reservation in the editio princeps (see Harrauer/Sijpesteijn 1985 [= P.Rain.Unterricht], 172; see also Lenaerts 2008, 207–208 and Lougovaya 2021, 254) but the provenance of the texts on the verso of the same papyrus is given in the same volume and by the same editors doubtfully as Hermoupolis (see Harrauer/Sijpesteijn 1985, 27–30), while in the study by Hermann Harrauer and Klaas A. Worp (1993) the papyrus is not mentioned, neither among those of certain nor of probable provenance from Soknopaiou Nesos. However, the provenance from Soknopaiou Nesos is the most probable one (cf. also Lenaerts 2008, 207–208): the confusion is most likely due to an error of the same type as that identified by Giuseppina Azzarello (2018, 103 with n. 25; 2022, 226–229) for two papyri published in the same volume, P.Rain.Unterricht 159 and 160, which she recognised as fragments of a single scroll from Soknopaiou Nesos (2nd/3rd c. AD).

The text is an arithmetic problem of ‘fractional detraction’, the correct reading and interpretation of which is due to the very recent re-edition by Julia Lougovaya (2021). The proposition concerns the weight (measured in mnas) of a statue, from which a series of unit fractions or n-th parts of the initial amount is subtracted, the data provided being these unit fractions and the residual weight after deductions: the goal of the problem is to calculate the initial weight of the statue before deductions. The solution procedure is followed by the counterproof of the calculation (ἀπόδειξιϲ). This particular type of problem is well attested in the papyrological tradition: in the almost contemporary P.Gent inv. 1v (col. 1, probl. 1),^[49] 2nd c. AD, probably traceable to the administrative archive of Theadelphia,^[50] and in two late antique papyrus codices with a collection of mathematical materials, possibly traceable to educational contexts, namely the famous ‘Akhmim mathematical papyrus’ (P.Cair.Cat. 10758) problem no. 13 (ed. Baillet 1892), dated to the late 4th/5th century AD,^[51] and P.Math. Cr 14–21 (eds. Bagnall/Jones 2019), dated to the third quarter of the 4th century AD.

The writing, which is confident and with a rather slow ductus, albeit with some cursive tendencies, and the overall orderly arrangement of the text denote an already quite solid competence in writing; together with the type of errors made, this could be compatible with the hand of a ‘second level’ student. Our text, which is very faded and damaged at several points, was written on the recto of a sheet of papyrus, perhaps reusing the surface.^[52] The verso (P.Vindob. inv. G 26011e v [P.Rain.Unterricht 8]) constitutes the second page of what has been called “un «carnet» scolaire” (see Lenaerts 2008). This ‘carnet’ also comprises two other pages, both written across the fibers of two other sheets of papyrus: p. 1 = P.Vindob. inv. G 26011b (P.Rain.Unterricht 7); p. 3 = P.Vindob. inv. G 26011c+d v (P.Rain.Unterricht 10).^[53] The content of this ‘carnet’, which was most probably written by two distinct hands of developing competence,^[54] evidently relates to a context of language instruction at an elementary level. It consists (p. 1) of the writing of the alphabet in regular order, then in reverse and then again in random order, in two so-called χαλινοί.^[55] There follows (still p. 1) the writing of all the numerical symbols up to the fundamental number 6,000,^[56] a mention of drachmas and talents, followed by two lines that are hard to decipher and a syllabary that runs from p. 1 to p. 3. This syllabary is arranged in columns: initially seven columns (in the lower part of p. 1), then six (p. 2 = verso of the problem) and finally five columns (p. 3), depending on the length of the syllables (first of two and then of three and finally of four letters).^[57]

Conclusions

Perhaps one of the most important aspects that emerges from the four specimens we have analysed here is that, in many respects – the type of material, graphic features, the nature and frequency of errors – they do not seem to be distinguishable from texts attributable to language instruction, especially at ‘second level’.^[58] One of them (P.Vindob. inv. G 26740) is especially significant for attesting to the study both of classical geometrical and metrological problems and of Homer (one could almost say organically integrated) in the same context and in the activity of the same student. Another (P.Vindob. inv. G 26011e) provides at least a clue that a typical arithmetical problem of ‘fractional detraction’ passed into a context of (also) elementary language instruction.

For the specimens of thematically structured collections of geometrical problems,^[59] Giorgio Zalateo (1961, 199 ad no. 330) hypothesised a didactic use such as that which can be deduced upstream of the set of typologically homogeneous geometrical problems copied in the first five columns of P.Vindob. inv. G 26740 (followed by the Homeric text and conversion problems). These collections could have been kept and used by those teaching the subject as models either for dictation or for students to copy one or more problems. It now seems clear that this form of use and circulation was not the only one for this type of collection.^[60] Nonetheless, the aforementioned scenario could perhaps be suggested by the roll on which the collection of geometrical problems MPER N.S. I 1^[61] (1st c. AD)^[62] had been inscribed, which Harrauer and Worp (1993, 36) include among those of “sure origin” from Soknopaiou Nesos. In fact, perhaps a small clue that this collection (at least at some time in its ‘use history’) passed through a context of (also) elementary education may lie in the fact that (what appears in all likelihood to be) the still not entirely secure hand of a student^[63] has, so to speak, momentarily reused the roll to write on the verso a table of additions (P.Rain.Unterricht 151, 1st c. AD) in the space corresponding to the eighth column of the recto problem collection.^[64] The table is organised into six columns and its calculations take the classical form:^[65] 1+1=2; 1+2=3; 1+3=4 etc.; 2+2 = 4; 2+3 = 5 etc.; 9 +9 = 18. The calculations continue with tens (10 + 10 = 20; 10 + 20 = 30 etc.; 90 + 90 = 180), hundreds (100 + 100 = 200 etc.), and thousands (1,000 + 1,000 = 2,000 etc.) and conclude with the first myriad (10,000 + 10,000 = 20,000),^[66] although there is a jump at the end of the series for 20 (20 + 90 = 110) to the beginning of the series for 50 (50 + 50 = 100), evidently the result of a leap in copying or dictation.

Just as Raffaella Cribiore^[67] hypothesised that different levels of education in language and literature could be taught by the same teachers and in the same locations, one envisions a similar dynamic for instruction in the different levels of mathematics, in combination with language teaching. Indeed, for the Fayum of the Roman period there is at least one piece of evidence for a relationship between geometrical problems and the school of an elementary teacher of letters^[68] – γραμματοδιδάϲκαλοϲ (or γραμματιϲτήϲ or διδάϲκαλοϲ) – who probably also dealt with the second level of linguistic-literary education, that of the γραμματικόϲ. This is P.Berol. inv. 11708v (= SB III 7268),^[69] a letter roughly contemporary with the examples cited so far, written on the verso of a document from the reign of Trajan. It was sent by a certain Sarapion in the Herakleopolite nome (where he may have moved), to Ptolemaios in the Arsinoite nome, who was a friend and, probably, a former fellow student. The final greeting of the letter (l. 19) shows that Sarapion is a fairly competent writer. He reports that he is struggling with problems of geometry presented to him: ll. 7–8: ἐπεὶ χειμάζομαι προτάϲεϲί τιϲι τιθεμέναιϲ μοι γεομετρικαῖϲ (lege γεωμετρικαῖϲ). He therefore asks his friend, as (he says) he had also asked him in person, to give to the deliverer of this letter the same papyrus that Ptolemaios, on account of their friendship, had previously mentioned – ll. 10–15: διὸ ὥϲπερ καὶ [κατʼ] ὄψιν ϲὲ̣ παρεκάλεϲα καὶ νῦν προϲ[ερω]τ̣ῶ, ὅπωϲ τῷ κομίζοντί ϲοι τὸ ἐπι[ϲτό]λ̣ι̣ο̣ν̣ τοῦτο δοιϲ (lege δῷϲ) τὸν χάρτην αὐτόν, [ὅν]περ μοι ἐκ προα[ι]ρέϲεωϲ φιλικῆϲ [κατ]ή̣γγειλαϲ. With Ptolemaios’ help, Sarapion hopes he will not be a layman ‘of the subject’ (ll. 9–10: ὅπωϲ διὰ τῆϲ ϲῆϲ βοηθείαϲ μὴ ἀμύητοϲ ὦ τῶν πραγμάτων). The location to which the letter is addressed, signficantly, is the γραμματοδιδαϲκαλεῖον of the teacher Melankomas: εἰϲ τὸ Μελανκόμου γραμματοδιδαϲκαλ(εῖον).^[70]

The teaching of mathematics in the form of numbers and basic arithmetical operations evidently occurred alongside elementary language teaching on a ‘common educational path’ and was not necessarily restricted to the training of professional secretaries, but was intended for the same pupils and could ordinarily be led by the same teacher.^[71] The textual evidence examined here also hints that the next level of instruction in literacy in ‘non-specialist’ training paths could include relatively more advanced computational mathematics of metrological and geometrical problems, as well. This applies whether these teachings were imparted by the two distinct figures of the γραμματικόϲ and the γεωμέτρηϲ^[72] – whose ‘equivalence’ is also suggested by the same salary being fixed for the two professions in Diocletian’s edict on prices^[73] – or whether, especially outside the large towns, the same teacher taught both subjects. The information from other sources, although it is meagre or indirect, inclines in this direction too.^[74] For the period in question, an education in this type of computational mathematics was ‘commonly’ recognised as useful for all those who had properties to manage or activities to supervise.^[75] That utility also underlies the circulation (in general, not necessarily limited to strictly technical-professional training contexts) of texts such as those transmitted in the Late Antique codices mentioned above (cf. e. g. P.Cair.Cat. 10758; P.Math.) and it can be considered one of the primary reasons for the particular fortuna of this type of mathematical texts,^[76] which flowed into the medieval tradition of the so-called Rechenbücher^[77] and the corpora attributed to Hero^[78] and Diophantus.^[79] Also pointing in a similar direction, among the papyrological sources, are two codices of waxed tablets from Late Antiquity, T.Varie 71–78 (5th c. AD) and T.Varie 14–21 (6th c. AD),^[80] the former probably from the Oxyrhynchite nome, preserving a series of problems^[81] (in T.Varie 71–77 and T.Varie 20 [but cf. also the land measurements in T.Varie 18–19; 21]).^[82] Both the type of writing material and the contents of these tablet ‘books’ are compatible with professional use (as reference tools) and with educational contexts. In the latter case, the type of writing and the co-presence of ‘model documents’^[83] (see T.Varie 14–15; but cf. also 18–19; 21) suggest a particularly professional kind of education. Nonetheless, as has been observed by Ruth Duttenhöfer (2021, esp. 241–242) and Giuseppina Azzarello (in this volume, pp. 381–382) precisely about a historical-geographical context that everything suggests was the same of these tablets,^[84] the overall analysis of this type of material reveals an absence of rigidly distinct ‘professional’ training paths. On the contrary, the evidence suggests a more common trend of coexistence, in both location and in the activity of a single teacher, of linguistic(-literary) and mathematical education at different levels. It could also include ‘bureaucratic-administrative’ training, but was or could be adapted to the education level and professional ambitions of various kinds of pupils.

The fact that only a small number of witnesses to geometrical and metrological problems can be clearly located in educational contexts, compared to the abundance of ‘consumer scholastic products’ relating to language teaching,^[85] is not necessarily an indication of their alleged ‘exceptionality’.^[86] It seems to be linked, rather, to the very nature of the texts and their specific didactic function. In language teaching, the production of a copy constitutes in itself the exercise through which the student learns, or is an essential component of it, and the more he produces, the more he learns. In contrast, for arithmetical tables^[87] and even more so for mathematical problems, the exercise lies in the memorisation and understanding of the individual operations and procedural algorithms. The student’s copying of the teacher’s models is thus not indispensable, though it may offer the student a guide to help follow the teacher’s explanation, which is itself likely to have supplemented and expanded upon the text of the problems known to us, as it is extremely abbreviated and often almost shorthand.^[88] The student’s copy will have been, above all, a way for fixing and/or archiving the information to be retained, to facilitate the work of memorising it and for future consultation as either a support or alternative to memory, even after the course of instruction had ended.