FOUR MEDIEVAL MANUSCRIPTS WITH MATHEMATICAL GAMES.

In the course of our research for the Index of Middle English Prose, we have come across some interesting variants of two types of mathematical game, both representative of a genre of some antiquity and apparently wide distribution. Both types of game are designed to impress and mystify, and they are also of particular interest for the evidence they provide of considerable exploration into numerical relationships. In fact it is clear from the work of others that these are not the only manuscript instances of such amusements, and perhaps this article will help to reveal the whereabouts of ones previously unrecognized.

Two of these games are versions of a puzzle that has come to be known as the Josephus Problem. It seems to date from an incident recounted in Josephus' De bello Judaico,(1) which involves the author himself. At the siege of Jotapata, Josephus finds himself in a position of either surrendering to Vespasian's troops or participating in a mass suicide with some forty Jewish soldiers. Outvoted, Josephus arranges the soldiers in such an order that every nth soldier is to be killed, the final survivor to commit suicide. Without explaining his precise procedure, Josephus relates that he is the survivor and is thus able to surrender and save his skin.

One late version of the Josephus Problem is found in Oxford, Bodleian Library, MS Bodley 496, a fifteenth-century paper manuscript, written in England and containing mostly medieval Latin poetry. On fol. [215.sup.v], immediately following a Latin poem of some ninety lines dealing with a dissolute and drunken village priest (fols [214.sup.v] - 215), there is a versified version of the problem which involves fifteen Christians and fifteen Jews at sea in a boat that threatens to capsize in a storm. In order for any lives to be saved, half of those on the vessel must be thrown overboard; so it is decided that the passengers will be counted off and every tenth person sacrificed. The problem is to arrange those on board so that in the process all the Christians will survive. In other versions of this problem, discussed below, there is a mnemonic line the arrangement of whose vowels indicate to the adept the correct placement of the persons. In Bodley 496 there are mnemonic verses that explicitly state what the arrangement should be. The text of the verses follows, with abbreviations silently expanded, punctuation added, and emendations in pointed brackets:

 Paganos dudum prudencia cristicolarum
 Vicerat. hunc ludum cerne morando parum.
 Cristicole cum Judeis pontum petiere.
 Vnica nauis erat, numero triginta fuere.
 5 Aura flat in mortem. Faciunt igitur cito sortem,
 Jactari decimum cupientes in mar<e>. Ymum
 Surgit arithmeti<c>us, locat omnes mox et iniqu<e>.
 Quisque perit mersus, sed nullus de grege Cristi.
 Disce per hos versus quo debent ordine sisti:

 10 Post duo, post unum, post tres, composito quinque,
 Post binum, binum, post quatuor, vnus et vnus,
 Post tres, post vnum, post binum, binus et vnus.

 Clericus hic digna dicit, sentit quia metra.
 Hinc volo per signa laicis clares<c>ere tetra:
 15 Cultorem fidei dat crux, nota nigra virumque
 Barbaricum. Signum ter quinque recepit vtrumque.

 + + * + + + * * * * * + + * * + + + + * + * * * + * * + + *

 Grat<ia> mille deo cum quilibet eiciatur
 Perfidus a puppe, plebs et sua saluificatur.
 20 Cessat tempestas. Felix fortuna fideli
 Dat portum populo, mansueto flamine veli.(2)
 (How the prudence of Christians recently overcame some pagans; looking at
 this game won't delay you long. Some Christians and some Jews put to sea.
 There was only one boat, and they were thirty in number. The wind was
 blowing them towards death, so they immediately drew lots, wanting every
 tenth person to be thrown into the sea. Then an arithmetician arose and
 quickly set everyone in uneven order. Many drowned, but no Christians did
 so. Learn by these verses in what order they must be placed:
 After 2, after 1, after 3, put 5;
 After 2, 2, after 4, 1 and 1;
 After 5, after 1, after 2, 2 and 1.
 This scholar said the right thing, because he understood these verses. Now
 I want to make things clear for non-scholars by means of these crude signs.
 The crosses signify the Christians, the black marks the Jews. Fifteen signs
 were allotted to each group:

 + + * + + + * * * * * + + * * + + + + * + * * * + * * + + *

 A thousand thanks to God, for every infidel was thrown overboard while the
 Christians and their goods were saved. The storm ended. A happy fortune
 brought the faithful people to port by a gentle breeze in their sail.)

If we admit one or two false quantities, the lines scan as rhyming hexameters. The rhyme scheme, aabb etc., is on two syllables, except for lines 5 and 6; and the couplet arrangement is broken to accommodate a rhyme of line 7 with line 10. In general the meaning is reasonably clear. As a mnemonic device lines 10-12, which will almost scan as rhythmic amphibrachs as well, serve admirably.

There has been considerable attention given to the Josephus Problem by mathematical historians, and a Latin version of it as early as the tenth century has been found.(3) The version in Bodley 496, however, seems to have been previously overlooked. Furthermore, none of the other instances so far found contains any diagram showing the proper arrangement of the two identifiably distinct sets.

The problem is given in a very different manner in Cambridge, Sidney Sussex College, MS 85, where it occurs twice in truncated form on fols 224 and 225. On fol. 224 it is simply two lines, one of Arabic numerals and the other of marks, in this manner:

[ILLUSTRATION OMITTED]

It is clear that each Arabic numeral in this array does no more than indicate the number of marks beneath it. On fol. 225 the same order of marks occurs, this time without the Arabic numerals and arranged in two lines:

[ILLUSTRATION OMITTED]

In his catalogue of the Sidney Sussex manuscripts, M. R. James explains these symbols as the `puzzle of the Christians and Jews, without explanation'.(4) He continues, `This puzzle I may as well explain: the problem is to arrange 15 Christians and 15 Jews in such a way that every 15th, or 12th, or 10th, person (the number varies) shall always be a Jew.' In more general terms, the nature of the problem is to arrange the elements of two sets in such an array that repeated sequential counting from one to n (where n is a constant fixed at the beginning) of the elements in the array, and discarding each time the nth element, will eliminate all members of one set but leave the other set intact.

Another type of mnemonic device offers a quick and easy solution to the problem. James gives two such mnemonic lines, neither of which actually appears in the Sidney Sussex manuscript, nor does James say where he found them:

 (1) Populeam virgam mater regina tenebat.

 (2) Rex Paphi cum gente bona dat signa serena.

If the vowels a, e, i, o, u are given numerical values one to five, then line (2) will give the arrangement shown above and in the Bodley manuscript. Line (1) will produce the correct arrangement if every ninth person is eliminated. A French mnemonic for the elimination of every tenth member is:

 Mort, tu ne faliras pas
 En me livrant le trepas.

And in English:

 From numbers' aid and art
 Never will fame depart.

And in German:

 Gott schlug den Mann in Amalek
 Den Israel bezwang.(5)

Possibly the most extensive collection of mnemonics for solutions to this problem is found in a work of Nicolo Tartaglia.(6) Employing the same assignment of numerals to vowels as given above, Tartaglia lists three or four mnemonic lines for each case where n, the number of the person to be eliminated, is any number from three to twelve inclusive. One of those he gives for the case n = 10 is `Rex anglicus certe bona flamina dederat.' The sixteenth-century physician Hieronimus Castilioneus Cardanus also mentions the problem in one of his works, but gives only a brief description of it and mentions no mnemonics for solving it.(7)

On fols 224 and 225 of the Sidney Sussex manuscript there are other, apparently unrelated, lines, but James seems not to have thought them worth quoting. In order to indicate the context in which the scribe evidently thought the mathematical game should occur, we give them here (abbreviations silently expanded):

 fol. 224:

 [the two lines quoted above]

 Dicit quidam sanctus

 Virginitas est pax carnis . silencium curarum . carcer libidinum .
 [illegible] secundum . preco fame . jubilus consciencie . porcio nature
 Angelice . decor membrorum vite ornamentum . principatus virtutum . morum
 fundamentum . christi cubiculum / virginitas(8)

 Augustinus

 Quatuor sunt genera sompniorum | augustinus

 Oraculum . visio . sompnium . et ffantasma(9) [The remainder of the leaf
 is blank.]

 fol. 225

 felix qui rerum poterit cognoscere causas ptholomeus(10)

 suum consilium ne committas ei qui secretum proprium non celauit(11)
 [the two lines of marks given above] ait scriptura

 ante mortem ne laudes hominem quemquam(12)

 [The remainder of the leaf is blank, except for some notes of various
 amounts of money and some blind tooling of miscellaneous designs.]

The other two manuscripts we have studied deal with a quite different arithmetical amusement. A fairly complete description of it is given in Oxford, Bodleian Library, MS Ashmole 360, fol. 110.(13) Black dates the manuscript to the fifteenth century, and the bulk of it probably was written then; but the puzzle, on the last folio of this section of the manuscript, seems to us to be in an early sixteenth-century hand. This is the text (abbreviations silently expanded, punctuation added):

 Salve stella maris mater firma dei rite
 Salve stella maris lapsis via venia rite
 Tak pe nowmer of xxiiij cardes and gyf one card to one man, two cardes to
 anoder, and iij cardes to the thyrd man. Than lay downe iii thynges on the
 table and commaund to deyll the resydew of pe cardes so pat he pat hase
 your pryncipall thyng hase as many moo cardes delyverd as he had befor; and
 he pat hase your second principall thyng twyse as many moo cardes delyuerd
 as he had afor; and he pat hase your leyst principall thyng has iiij tymes
 as many cardes delyverd as he had afor. Then aske apon hym pat deylles how
 many cardes is left vndelt. And if he say one, go to pe fyrst word of your
 verse and lok apon your wooles; for `a' betokyns one carde, `e' two cardes,
 `j' thre cardes. And so remembryng whomn ye gafe your cardes to, ye shall
 knaw whoo has euerythyng.

In the manuscript arabic numerals occur above the vowels in the first line of the Latin verse. In agreement with the text, a 1 is written above a, 2 above e, and 3 above i. Though the directions are incomplete, it would seem that after the first six cards are given to three individuals, an assistant then distributes, out of sight of the master, the objects or `principal things' and the other cards as prescribed by the text. The assistant then informs the master as to the number of cards remaining, whereupon the master announces who has received which of the objects.

In our analysis of this trick we will use a bottle of wine (B) as the first principal thing, a half-bottle (H) for the second, and a glass of wine (G) for the third. We give in the table below the names Rob, Sam, and Tom to the individuals who initially received one, two, and three cards respectively. There are in all six ways in which the three objects (B, H, G) can be distributed to these three persons. Each of these assignments is shown in the table, as well as the number of cards then given to each person, the total number distributed (recalling that six cards were initially distributed in each case), and finally the remainder (R) of the pack of twenty-four.

Rob (1) Sam (2) Tom (3) Total R

 B H G

 1 4 12 17 1

 H B G

 2 2 12 16 2

 B G H

 1 8 6 15 3

 H G B

 2 8 3 13 5

 G B H

 4 2 6 12 6

 G H B

 4 4 3 11 7

It will be noted that the first line of the Latin couplet at the beginning of the text contains seven words, each having two of the vowels a, e, i. The first three words and the last three in that line display all six ways of arranging two of the vowels from the set {a, e, i}, and each of those arrangements of vowels corresponds to a distribution of the three principal things.

As an example of the way in which the procedure would operate, let us suppose that after distributing the cards and the objects, there were five cards remaining. Informed of this, the master of ceremonies would then find the fifth word in the first line of the Latin verse. As its vowels are i and a, corresponding to the numerals three and one, he would announce that Tom, who had initially received three cards, was the one with the bottle of wine, B; that Rob, who had one card initially, had the half-bottle, H; leaving Sam with a glass of wine, G.

It should be noted that in the table above there is no case in which the remainder, R, equals four. Consequently, the fourth word is never used, and so the fact that the fourth word in each line in the Latin couplet duplicates the arrangement of vowels found in another word in that same line, `mater' with `salve' and `lapsis' with `maris,' does not interfere with the solution.

Although this trick appears not to have gained as much attention from historians of mathematics as the Josephus Problem, it seems to be of some antiquity and doubtless is considerably older than our manuscript. It appears in numerous collections of arithmetic amusements, as well as in early printed general works on arithmetic. Tartaglia describes it in his Prima parte del general trattato, fol. [263.sup.r-v], and it is included in one of the earliest collections devoted solely to mathematical recreations, Claude Gaspard de Bachet's Problemes plaisans et delectables, qui se font par les nombres.(14) The presentation of the trick given by Bachet, and by the other writers we have studied, differs in two ways from that found in the Ashmole manuscript. First, in place of the manuscript's Latin couplet Bachet and others give a mnemonic string of words and numerals as follows:

 1 2 3 5 6 7

Par fer, Cesar, iadis, devint, si grand, Prince.

This will certainly work as well as the Latin couplet, but it would appear more direct and less designed to conceal than that in the manuscript.

What is more important is that the position of the vowels in the words beneath five and six of the above string is not the same as that of the vowels in firma and dei, the fifth and sixth words of the Latin couplet. Both are actually correct within the context of the solutions as they are presented. Bachet and others employ the vowels a, e, i to correspond to the three principal things, where the Ashmole manuscript does otherwise: its statement that `"a" betokyns one carde, "e" two cardes, "j" thre cardes' clearly shows that the vowels are to correspond to the three individuals receiving that number of cards rather than to the objects. In four of the six possible assignments it makes no difference which view is taken regarding the correspondence. For example, in the case where R = 2, we see in the table above that just as the first object, B, is given to Sam, who had two cards, likewise Rob, with one card, receives the second object H. Consequently, it makes no difference in that case whether we regard a, e as indicating the first and second object or the persons with one and two cards. It is only in case of remainders of five and six that a difference arises, and so only in those cases does the order of vowels need to differ. The Ashmole manuscript seems to be unique in setting the correspondence of the three vowels with the three persons, and hence unique in the mnemonic given to aid in performing the trick.

What might be termed a fragment of the trick is found in Cambridge, Gonville and Caius College, MS 176/97, p. 12.(15) Although no directions for performing the trick are given, the notes seem to relate to the same, or very nearly the same, amusement as that in Ashmole 360. In addition to being incomplete, the fragment contains errors that make any interpretation uncertain. The full text is as follows:

 Duke Noble D A
Salue Erle III half-noble E E
 Baron xxd B I

 Erle 3 Noble 1 E
Stella Duke 1 half 2 A
 Baron 2 xxd 3 I

 Duke half-noble A
Maris Baron II Noble E
 Erle xxd I

The three Latin words in the leftmost column, being identical to the first three words in the Ashmole mnemonic, strongly suggest that the other columns are intended to display three of the six possible ways of assigning to three individuals, a duke, an earl, and a baron, three amounts of money: a noble, half-noble, and twenty pence. Errors, however, lead to some problems in interpretation. First, to agree with the customary ranking of peers the numerical order should be Duke 1, Earl 2, and Baron 3, rather than what is given in the manuscript.

It is unclear whether the three vowels, A, E, I, are meant to correspond with the persons, as in the Ashmole manuscript, or with the objects distributed, as in all the other accounts we have found of the trick. In either case there are inconsistencies. If the vowels correspond to the peers, A with Duke, E with Earl, and I with Baron, then the listings opposite the words Salue and Stella are correct. With respect to Maris, its corresponding vowel order, AIE, should appear in the right-hand column, rather than AEI, which is in the manuscript and duplicates the one assigned to Salue. The list of peers opposite Maris is in the appropriate order, but the money corresponding to the Duke and Baron should be reversed so that it is in decreasing value as in the two preceding cases.

On the other hand, if the vowels are intended to correspond to the coins rather than to the peers, then I is in each case correctly aligned with twenty pence, but A and E are inconsistent in their correspondence to the other two coins. On balance it would seem likely that the correspondence between the vowels and the peers was the one intended, but due to the errors in the text the evidence is not conclusive.

This trick is also found in Nicolas Chuquet's Triparty en la science des nombres.(16) Though the treatise, written in 1484, was not published until the nineteenth century, it is now regarded as an outstanding work and the earliest Renaissance algebra. Chuquet gives no mnemonics to assist in obtaining the solution, but he does have a table showing the six possible remainders and the assignment of objects to players that each indicates. In so doing he makes letters correspond with objects and the order he gives of results is the same as that found in all the presentations we have seen, except that of the Ashmole manuscript and possibly the Gonville and Caius fragment. This trick and variants of it are found in many printed works of the seventeenth century, but aside from Chuquet's Triparty we know of only the two manuscripts we have discussed.(17)

L. M. ELDREDGE KARI ANNE RAND SCHMIDT M. B. SMITH

University of Ottawa University of Oslo Oxford

NOTES

We should like to thank David Howlett, Tony Hunt, and George Keiser for helpful suggestions. We are also grateful to the anonymous readers for Medium/AEvum. We are, of course, solely responsible for any errors that may remain.

(1) Flavius Josephus, De bello Judaico: der judische Krieg, 3 vols in 4, ed. Otto Michel and Otto Bauernfeind (Darmstadt, 1959-69), I, book iii, [paragraph] 361-90, pp. 370-4.

(2) Textual notes: 6. MS maris. 7. MS arithmetius, iniquus. 14. MS clarestere, 18. MS grates.

(3) For a comprehensive study of the Josephus Problem, see W. Ahrens, `Das "Josephsspiel", ein arithmetisches Kunststuck: Geschichte und Literatur', Archiv fur Kulturgeschichte, 11 (1913), 129-51.

(4) M. R. James, A Descriptive Catalogue of the Manuscripts in the Library of Sidney Sussex College, Cambridge (Cambridge, 1895), pp. 68-71.

(5) The French, English, and German mnemonics are given by W. Ahrens, Mathematische Unterhaltungen und Spiele (Leipzig, 1901), pp. 286-301. Ahrens gives many other mnemonics for other numbers of people and other elimination numbers.

(6) Nicolo Tartaglia, La prima parte del general trattato di numeri et misure (Venice, 1656), book xv1, no. 195, fol. [263.sup.r-v].

(7) Hieronimus Castilioneus Cardanus, Practica mathematice (Milan, 1539), cap. 61, `De extraordinariis et ludis,' [paragraph] 18, sig. [Tiiii.sup.v].

(8) `A certain saint says: Virginity is peace of the flesh, silence of cares, prison of the libido, second --, the herald of rumour, the joyous cry of the conscience, a portion of angelic nature, elegance of the members, an ornament of life, the origin of virtue, the foundation of morals, the resting place of Christ. / Virginity.' The identity of `quidam sanctus' remains unknown, though the sentiment is common enough. For example, in his Regula pastoralis (PL LXXVII 26B), Gregory the Great uses the expression `decor membrorum' with reference to chastity, and his phrasing is repeated by Taio Caesaraugustinus in his Expositio veteris ac novae testamentarum (PL LXXX 838C), Symphorius Amalarius in his Forma institutionis canonicorum (PL CV 862C), Hrabanus Maurus in his Expositiones in Leviticum (PL CVIII 486D), Anon. (possibly Hugh of St Victor?) in Posteriores exceptiones (PL CLXXV 667D), Gratianus in his Concordantia discordantum canonum (PL CLXXXVII 254C). `Christi cubiculum' is an expression used by Petrus Cellensis, Sermo xii, `In nativitate domini' (PL CCII 673A). A model for the rhetoric might have been something like Jerome's `nocivum genus femina, janua diaboli, via iniquitatis, scorpionis percussio', in his Regula monachorum iii, `De castitate' (PL XXX 328C).

(9) `Augustine / There are four types of dream: | Augustine / The oracle, the vision, the dream and the fantasm.' Augustine appears to say nothing of this sort. The vocabulary is derived from Macrobius' commentary on Cicero's `Dream of Scipio' (Macrobii Ambrosii Theodosii Commentariorum in Somnium Scipionis libri duo, ed. Luigi Scarpa (Padua, 1981), 1.iii. 2; p. 82), where he identifies five types of dream: somnium, visio, oraculum, insomnium, and fantasma. The insomnium has been omitted from this list. Isidore of Seville, Sententiarum libri IV, IV.xiii, `quot sint genera somniorum' (PL LXXXIII 1163), discusses the sources of dreams, but his vocabulary is not derived from Macrobius.

(10) `Happy the man who can distinguish the causes of things.' This is not Ptholomeus but Virgil, Georgics ii.490, `felix qui potuit cecum cognoscere causas'.

(11) `Don't give your advice to anyone who cannot keep his own secrets.' No direct source has been found for this, which seems to have been misquoted. It should surely read `tuum' instead of `suum'. Hans Walther, Proverbia sententiaeque latinitatis medii aevi, Carmina medii aevi posterioris latina (Gottingen, 1966), 3158, lists a proverb of similar sentiment but quite different words.

(12) `Scripture says: / Don't praise any man before his death.' Ecclesiasticus xi.30.

(13) Text printed in LME, The Index of Middle English Prose Handlist IX: Manuscripts Containing Middle English Prose in the Ashmole Collection, Bodleian Library, Oxford (Cambridge, 1992), pp. 15-16. Manuscript described in W. H. Black, A Descriptive ... Catalogue of the Manuscripts Bequeathed unto the University of Oxford by Elias Ashmole ... (Oxford, 1845), cols 271-6.

(14) 2nd edn (Lyons, 1624), pp. 187-93.

(15) Manuscript described in M. R. James, A Descriptive Catalogue of the Manuscripts in the Library of Gonville and Caius College, 2 vols (Cambridge, 1907-8), I, 201-3. James's transcription errors are here silently corrected. The text will appear in KARS, The Index of Middle English Prose Handlist [number to be assigned]: Manuscripts Containing Middle English Prose in the Library of Gonville and Caius College, Cambridge (Cambridge, forthcoming).

(16) Ed. Aristide Matte, in Bolletino di bibliografia e distoria delle scienze, 14 (1881), 413-60.

(17) For this and similar tricks, see: Tartaglia, La prima parte del general trattato di numeri et misure (Venice, 1656), book xvi, nos 195 and 196, fol. [263.sup.r-v]; Henry Van Etten, Mathematical Recreations or a Collection of Sundrie Excellent Problems out of Ancient and Moderne Philosophers both Useful and Recreative (London 1633), problem VIII, pp. 19-21; Pierre Forcadel, Le Troysieme Livre de l'arithmitique (Paris, 1557), fols 89-[92.sup.v]; R. P. Adalbert Tylkowski, Arithmetica curiosa (Cracow, 1668), pp. 89-90.