4. The reckoning of duodecimal fractions
THE RECKONING OF TIME
newborn.26 Let us see to it that these things are avoided, because such observance is futile and alien to our faith.
The Apostle uses the term for this kind of time in a better sense, to suggest the swiftness of the Resurrection, stating, We shall all rise, but we shall not all be changed, in a moment, in the twinkling of an eye, at the last trumpet.27 This deserves our attention, because although computists make a strict distinction [between these terms], many writers indiscriminately call that tiniest interval of time in which the lids of our eyes move when a blow is launched [against them], and which cannot be divided or distributed,28 either a momentum, a punctus or an atom.
4. THE RECKONING OF DUODECIMAL FRACTIONS
To know how to divide duodecimal fractions [unciae] is no unworthy ambition, for one can apply it to the calculation not only of coinage, but of time and of other things. /279/ Because from time to time various historical narratives, and even Holy Scripture itself, will use the names and symbols [of fractions] we have taken the trouble to record them in summary form here. pound or as or assis 12 unciae deunx or iabus 11 unciae decunx or dextans 10 unciae [˙~¨] dodrans or doras 9 unciae [¯~˘] bes or bisse 8 unciae [˜~¯] septunx or septus 7 unciae semis 6 unciae [ˆ~˜] quincunx or cingus 5 unciae triens or treas 4 unciae [ˆ~¯]
26 Cf. Basil, Hexaemeron 6.5, trans. Eustathius (PL 53.925C^927B); Ambrose, Hexaemeron 4.4, ed. C. Shenkl, CSEL 32.1 (1897):120.9^122.10. Since the passage which follows is from Jerome's Letter 119.5 (see below), this is probably Bede's immediate source, though Jones (BOT 333) opts for Ambrose or Basil.
27 I Corinthians 15.51^52; cf. Augustine, Ep. 205, ed. W. Goldbacher, CSEL 57 (1911):355.1^4. This translation may sound strange to English readers familiar with the Authorized Version or the libretto of Handel's Messiah, but it renders Bede's Vulgate:
Omnes quidem resurgemus, sed non omnes immutabimur . . . This was a di⁄cult crux, and Augustine devotes considerable time to its resolution in Ep. 205.
28 Jerome, Ep. 119.5, ed. I. Hilberg, CSEL 55 (1996):450.8^20.
TRANSLATION
quadrans or quadras 3 unciae [ˆ~˘] sextans or sextas 2 unciae [ˆ~¨] sescunx or sescuncia 1ˆ~˜ unciae [ˆ~˚] uncia 24 scripuli semuncia 12 scripuli two sextulae or sesclae, that is, ˆ~¯ of an uncia
8 scripuli sicilicus 6 scripuli [ˆ~˘ unciae] sextula or sescla 4 scripuli ˆ~˜ sextula or sescla 2 scripuli scripulus 6 siliquae29 These names and symbols for weights apply, I say, not only to measuring money, but to quantifying any body or length of time. Hence it is both logical and customary that in reciting their computations, boys often change ''one'' and ''two'' to ''assis'' and ''dipondium''; likewise they say
''tressis'', ''quartus'', ''quinquis'', ''sexis'', ''septis'' and so forth in this fashion, instead of ''three asses'', ''four asses'', and a sequence of however many numbers in the same manner.30
Therefore, should you wish to divide one hour, or a whole day, /281/ or a month, or a year, or any other speci¢c length of time, greater or less, into twelve, that twelfth part is an uncia; the remaining eleven are called deunx [11 12]. If you divide into six, the sixth part is called sestans [ˆ~¨]; the remaining ¢ve are called dextans, or according to others, distas [˙~¨]. If into four, the fourth part is called quadrans [ˆ~˘]; the residual three take the name of dodrans [¯~˘]. Something which perturbs many beginners is solved by the rule of this discipline; for Philip, in his commentary on the blessed
Job, describing how the ocean tide comes in twice each day, adds that it arrives, whether by day or by night, later [than the previous day's tide] by an interval of a dodrans [i.e. 3
4] of an equinoctial hour, with no break in 29 Priscian, De ¢guris numerorum 10^11, ed. Heinrich Keil, Grammatici latini
(Leipzig: Teubner, 1855^1880):3.408^409; cf. 15^16 (410^411); 31 (416). Cf. Victorius of Aquitaine, Calculus, ed. G. Friedlein, Zeitschrift fu« r Mathematik und Physik 7 (1871):42^
97.
30 If this passage re£ects Bede's own experience, and is not copied from some untraced classical source, it suggests two things: ¢rst, that boys in monasteries learned computational tables by rote, and hence in some kind of programmed way; secondly, that numbers were commonly understood, not as abstractions, but as concrete numbers of things.
THE RECKONING OF TIME
continuity.31 Now if you want to divide by three, you will call the third part triens, and the remaining two parts bisse [˜~¯]. If by two, you will call the half semis; and so on for the rest. These things can be both learned and taught more easily through speech than by the pen of a writer. /282/
If your task is to divide miles or stadia or acres or perches, or even cubits, feet or palms, proceed according to the method given above, and likewise for bodies. Thus in Exodus a half-cubit is called semis, for Moses tells that the Ark of the Covenant was two and one-half [duos semis] cubits long and a cubit and a half high,32 and in the Gospel the fourth part of the universe, that is, earth, is symbolically designated by the term quadrans. It is said to the sinner as he is sent to his punishment,
You shall not go hence until you have paid the last farthing [quadrantem],33 that is until you atone for your earthly sins, as St Augustine explains: For the sinner heard this: You are earth and to earth you shall go.34 Now the fourth and last of the distinct divisions of the universe is found to be earth, so that you begin with heaven, count air as coming second, water third, and earth fourth.35 He also alludes to this branch of study in his commentary on the Gospel, where it is written, And it was about the sixth hour of the preparation of the Passover,36 saying, For it was not fully the sixth hour, but about the sixth hour, that is, the ¢fth hour was ¢nished /283/ and the sixth somewhat begun. For these men would never have said ¢ve and a quarter [quadrans] or ¢ve and a third [triens] or ¢ve and a half [semis] or anything of that kind.37
31 Philippus Presbyter, Commentarii in librum Iob 38 (PL 26.752D). Jones argues that this passage is a criticism of Philippus: ''Bede may have believed that if Philippus had known the proper names for fractions he would not have said dodrans, which is forty-¢ve minutes. Bede's calculation for the retardation of the tides is forty-eight minutes'' (BOT
334^335). But Bede is merely stating that the term dodrans is unfamiliar to many readers, not that Philippus is misusing it. In the translation included in his doctoral dissertation,
Jones misconstrues the ¢rst part of this sentence as ''The rules of this discipline are in disorder, because many untrained persons use them inaccurately'' (''Materials'' 160). This is undoubtedly the source of the misunderstanding.
32 Exodus 25.10.
33 Matthew 5.26.
34 Genesis 3.19.
35 Augustine, De sermone Domini in monte 1.11.30, ed. A. Munzenbecker, CCSL 35 (1967):31.664^665, 660^663.
36 John 19.14.
37 Augustine, De consensu Evangelistarum 3.13.41, ed. F Weihrich, CSEL 43 (1904):324.21^325.3.