Chapter V: Part 5

_S._ Now wyll I proue by an other exa{m}ple, as this: 40 labourers (after 6 d. y^e day for eche man) haue wrought 28 dayes, I wold [*125a] know what theyr wages doth amou{n}t vnto: In this case muste I worke doublely: fyrst I must multyplye the nomber of the labourers by y^e wages of a man for one day, so wyll y^e charge of one daye amount: then secondarely shall I multyply that charge of one daye, by the hole nomber of dayes, {and} so wyll the hole summe appeare: fyrst therefore I shall set the su{m}mes thus.

------+--------------
|
------+--------------
|
------+--------------
|
------+--o-o-o-o-----
o |
--o---+--------------

Where in the fyrste space is the multyplyer (y^t is one dayes wages for one man) {and} in the second space is set the nomber of the worke men to be multyplyed: the{n} saye I, 6 tymes 4 (reckenynge that second lyne as the lyne of vnites) maketh 24, for whiche summe I shulde set 2 counters in the thyrde lyne, and 4 in the seconde, therfore do I set 2 in the thyrde lyne, and let the 4 stand styll in the seconde lyne, thus.[*125b]

-----+--------------
|
-----+--------------
|
-----+--o-o---------
|
-----+--o-o-o-o-----
|
-----+--------------

So apwereth the hole dayes wages to be 240d’. that is 20 s. Then do I multiply agayn the same summe by the no{m}ber of dayes and fyrste I sette the nombers, thus.

---------+--------------
|
---------+--------------
|
---------+--o-o---------
|
--o-o----+--o-o-o-o-----
o |
--o-o-o--+-------------

The{n} bycause there are counters in dyuers lynes, I shall begynne with the hyghest, and take them vp, settynge for them the multyplyer so many tymes, as I toke vp counters, y^t is twyse, then wyll y^e su{m}me stande thus.

-----+--------------
| o
-----+--------------
| o
-----+--o-----------
|
-----+--o-o-o-o-----
|
-----+--------------

Then come I to y^e seconde lyne, and take vp those 4 cou{n}ters, settynge for them the multiplyer foure tymes, so wyll the hole summe appeare thus.[*126a]

-----+----------
| o
-----+--o-------
| o
-----+--o-o-----
|
-----+--o-o-----
|
-----+----------

So is the hole wages of 40 workeme{n}, for 28 dayes (after 6d’. eche daye for a man) 6720d’. that is 560 s. or 28 l’i.

[Headnote: Division on the Counting Board.]

[Sidenote: Diuision.]

_M._ Now if you wold proue Multiplycatio{n}, the surest way is by Dyuision: therfore wyll I ouer passe it tyll I haue taught you y^e arte of Diuision, whiche you shall worke thus. Fyrste sette downe the Diuisor for feare of forgettynge, and then set the nomber that shalbe deuided, at y^e ryghte syde, so farre from the diuisor, that the quotient may be set betwene them: as for exa{m}ple: Yf 225 shepe cost 45 l’i. what dyd euery shepe cost? To knowe this, I shulde diuide the hole summe, that is 45 l’i. by 225, but that can not be, therfore must I fyrste reduce that 45 l’i. into a lesser denomination, as into shyllynges: then I multiply 45 by 20, and it is 900, that summe shall I diuide by the no{m}ber of [*126b] shepe, whiche is 225, these two nombers therfore I sette thus.

-------+-----+--------------
| |
-------+-----+--------------
| | o
--o-o--+-----+--o-o-o-o-----
| |
--o-o--+-----+--------------
o | |
-------+-----+--------------

Then begynne I at the hyghest lyne of the diuident, and seke how often I may haue the diuisor therin, and that maye I do 4 tymes, then say I, 4 tymes 2 are 8, whyche yf I take from 9, there resteth but 1, thus

-------+-----------+--------
| |
-------+-----------+--------
| |
--o-o--+-----------+--o-----
| |
--o-o--+-----------+--------
o | |
-------+--o-o-o-o--+--------

And bycause I founde the diuisor 4 tymes in the diuidente, I haue set (as you se) 4 in the myddle roume, which [*127a] is the place of the quotient: but now must I take the reste of the diuisor as often out of the remayner: therfore come I to the seconde lyne of the diuisor, sayeng 2 foure tymes make 8, take 8 from 10, {and} there resteth 2, thus.

----------+-----------+----------
| |
-||-------+-----------+----------
| |
-----o-o--+-----------+----------
| |
-----o-o--+-----------+--o-o-----
o | |
----------+--o-o-o-o--+----------

Then come I to the lowest nomber, which is 5, and multyply it 4 tymes, so is it 20, that take I from 20, and there remayneth nothynge, so that I se my quotient to be 4, whiche are in valewe shyllynges, for so was the diuident: and therby I knowe, that yf 225 shepe dyd coste 45 l’i. euery shepe coste 4 s.

_S._ This can I do, as you shall perceaue by this exa{m}ple: Yf 160 sowldyars do spende euery moneth 68 l’i. what spendeth eche man? Fyrst [*127b] bycause I can not diuide the 68 by 160, therfore I wyll turne the pou{n}des into pennes by multiplicacio{n}, so shall there be 16320 d’. Nowe muste I diuide this su{m}me by the nomber of sowldyars, therfore I set the{m} i{n} order, thus.

---------+-----+--o---------
| | o
-||------+-----+--o---------
| |
-----o---+-----+--o-o-o-----
o | |
-----o---+-----+--o-o-------
| |
---------+-----+------------

Then begyn I at the hyghest place of the diuidente, sekynge my diuisor there, whiche I fynde ones, Therfore set I 1 in the nether lyne.

_M._ Not in the nether line of the hole summe, but in the nether lyne of that worke, whiche is the thyrde lyne.

_S._ So standeth it with reason.

_M._ Then thus do they stande.[*128a]

---------+-----+------------
| |
-||------+-----+------------
| |
-----o---+--o--+--o-o-o-----
o | |
-----o---+-----+--o-o-------
| |
---------+-----+------------

Then seke I agayne in the reste, how often I may fynde my diuisor, and I se that in the 300 I myghte fynde 100 thre tymes, but then the 60 wyll not be so often founde in 20, therfore I take 2 for my quotient: then take I 100 twyse from 300, and there resteth 100, out of whiche with the 20 (that maketh 120) I may take 60 also twyse, and then standeth the nombers thus,

---------+-------+-----
| |
-||------+-------+-----
| |
-----o---+--o----+-----
o | |
-----o---+-------+-----
| |
---------+--o-o--+-----

[*128b] where I haue sette the quotient 2 in the lowest lyne: So is euery sowldyars portion 102 d’. that is 8 s. 6 d’.

_M._ But yet bycause you shall perceaue iustly the reason of Diuision, it shall be good that you do set your diuisor styll agaynst those nombres fro{m} whiche you do take it: as by this example I wyll declare. Yf y^e purchace of 200 acres of ground dyd coste 290 l’i. what dyd one acre coste? Fyrst wyl I turne the poundes into pennes, so wyll there be 69600 d’· Then in settynge downe these nombers I shall do thus.

---------+-----+--------------
| | o
----o-o--+-----+--o-----------
| | o
-X-------+-----+--o-o-o-o-----
| | o
---------+-----+--o-----------
| |
---------+-----+--------------
| |
---------+-----+--------------

Fyrst set the diuident on the ryghte hande as it oughte, and then [*129a] the diuisor on the lefte hande agaynst those nombers, fro{m} which I entende to take hym fyrst as here you se, wher I haue set the diuisor two lynes hygher the{n} is theyr owne place.

_S._ This is lyke the order of diuision by the penne.

_M._ Truth you say, and nowe must I set y^e quotient of this worke in the thyrde lyne, for that is the lyne of vnities in respecte to the diuisor in this worke. Then I seke howe often the diuisor maye be founde in the diuident, {and} that I fynde 3 tymes, then set I 3 in the thyrde lyne for the quotient, and take awaye that 60000 fro{m} the diuident, and farther I do set the diuisor one line lower, as yow se here.

----------+---------+--------------
| | o
-||--o-o--+---------+--o-o-o-o-----
| | o
----------+--o-o-o--+----o---------
| |
----------+---------+--------------
| |
----------+---------+--------------

[*129b] And then seke I how often the diuisor wyll be taken from the nomber agaynste it, whiche wyll be 4 tymes and 1 remaynynge.

_S._ But what yf it chaunce that when the diuisor is so remoued, it can not be ones taken out of the diuident agaynste it?

_M._ Then must the diuisor be set in an other line lower.

_S._ So was it in diuision by the penne, and therfore was there a cypher set in the quotient: but howe shall that be noted here?

_M._ Here nedeth no token, for the lynes do represente the places: onely loke that you set your quotient in that place which standeth for vnities in respecte of the diuisor: but now to returne to the example, I fynde the diuisor 4 tymes in the diuidente, and 1 remaynynge, for 4 tymes 2 make 8, which I take from 9, and there resteth 1, as this figure sheweth:

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-||--o-o--+-----------+--o------
| | o
----------+--o-o-o----+--o------
| |
----------+--o-o-o-o--+---------
| |
----------+-----------+---------

and in the myddle space for the quotient I set 4 in the seconde lyne, whiche is in this worke the place of vnities.[*130a] Then remoue I y^e diuisor to the next lower line, and seke how often I may haue it in the dyuident, which I may do here 8 tymes iust, and nothynge remayne, as in this fourme,

----------+-----------+-----
| |
-||--o-o--+-----------+-----
| |
----------+--o-o-o----+-----
| |
----------+--o-o-o-o--+-----
| o |
----------+--o-o-o----+-----

where you may se that the hole quotient is 348 d’, that is 29 s. wherby I knowe that so moche coste the purchace of one aker.

_S._ Now resteth the profes of Multiplycatio{n}, and also of Diuisio{n}.

_M._ Ther best profes are eche [*130b] one by the other, for Multyplication is proued by Diuision, and Diuision by Multiplycation, as in the worke by the penne you learned.

_S._ Yf that be all, you shall not nede to repete agayne that, y^t was sufficye{n}tly taughte all redye: and excepte you wyll teache me any other feate, here maye you make an ende of this arte I suppose.

_M._ So wyll I do as touchynge hole nomber, and as for broken nomber, I wyll not trouble your wytte with it, tyll you haue practised this so well, y^t you be full perfecte, so that you nede not to doubte in any poynte that I haue taught you, and thenne maye I boldly enstructe you in y^e arte of fractions or broken no{m}ber, wherin I wyll also showe you the reasons of all that you haue nowe learned. But yet before I make an ende, I wyll showe you the order of co{m}men castyng, wher in are bothe pennes, shyllynges, and poundes, procedynge by no grounded reason, but onely by a receaued [*131a] fourme, and that dyuersly of dyuers men: for marchau{n}tes vse one fourme, and auditors an other:

[Headnote: Merchants’ Casting Counters.]

[Sidenote: Merchants’ casting.]

But fyrste for marchauntes fourme marke this example here,

o o o o o
o
o o o o
o
o o o o o
o
o o o o o

in which I haue expressed this summe 198 l’i.[{2}] 19 s. 11 d’. So that you maye se that the lowest lyne serueth for pe{n}nes, the next aboue for shyllynges, the thyrde for poundes, and the fourth for scores of pou{n}des. And farther you maye se, that the space betwene pennes and shyllynges may receaue but one counter (as all other spaces lyke wayes do) and that one standeth in that place for 6 d’. Lyke wayes betwene the shyllynges {and} the pou{n}des, one cou{n}ter standeth for 10 s. And betwene the poundes and 20 l’i. one counter standeth for 10 pou{n}des. But besyde those you maye see at the left syde of shyllynges, that one counter standeth alone, {and} betokeneth 5 s. [*131b] So agaynste the poundes, that one cou{n}ter standeth for 5 l’i. And agaynst the 20 poundes, the one counter standeth for 5 score pou{n}des, that is 100 l’i. so that euery syde counter is 5 tymes so moch as one of them agaynst whiche he standeth.

[Sidenote: Auditors’ casting.]

Now for the accompt of auditors take this example.

o o o o o o
o o o o o o o o o o o o
o o o o

where I haue expressed y^e same su{m}me 198 l’i. 19 s. 11 d’. But here you se the pe{n}nes stande toward y^e ryght hande, and the other encreasynge orderly towarde the lefte hande. Agayne you maye se, that auditours wyll make 2 lynes (yea and more) for pennes, shyllynges, {and} all other valewes, yf theyr summes extende therto. Also you se, that they set one counter at the ryght ende of eche rowe, whiche so set there standeth for 5 of that roume: and on [*132a] the lefte corner of the rowe it sta{n}deth for 10, of y^e same row. But now yf you wold adde other subtracte after any of both those sortes, yf you marke y^e order of y^t other feate which I taught you, you may easely do the same here without moch teachynge: for in Additio{n} you must fyrst set downe one su{m}me and to the same set the other orderly, and lyke maner yf you haue many: but in Subtraction you must sette downe fyrst the greatest summe, and from it must you abate that other euery denominatio{n} from his dewe place.

_S._ I do not doubte but with a lytell practise I shall attayne these bothe: but how shall I multiply and diuide after these fourmes?

_M._ You can not duely do none of both by these sortes, therfore in suche case, you must resort to your other artes.

_S._ Syr, yet I se not by these sortes how to expresse hu{n}dreddes, yf they excede one hundred, nother yet thousandes.

_M._ They that vse such accomptes that it excede 200 [*132b] in one summe, they sette no 5 at the lefte hande of the scores of poundes, but they set all the hundredes in an other farther rowe {and} 500 at the lefte hand therof, and the thousandes they set in a farther rowe yet, {and} at the lefte syde therof they sette the 5000, and in the space ouer they sette the 10000, and in a hygher rowe 20000, whiche all I haue expressed in this exa{m}ple,

o o o o
o
o o o
o o o o
o o o
o o o o o
o
o o
o
o o o
o o
o

which is 97869 l’i. 12 s. 9 d’ ob. q. for I had not told you before where, nother how you shuld set downe farthynges, which (as you se here) must be set in a voyde space sydelynge beneth the pennes: for q one counter: for ob. 2 counters: for ob. q. 3 counters: {and} more there can not be, for 4 farthynges [*133a] do make 1 d’. which must be set in his dewe place.

[Headnote: Auditors’ Casting Counters.]

And yf you desyre y^e same summe after audytors maner, lo here it is.

o o o o o o
o o o o o o o o o o o o o o o o o o o
o o o o
o

But in this thyng, you shall take this for suffycyent, and the reste you shall obserue as you maye se by the working of eche sorte: for the dyuers wittes of men haue inuented dyuers and sundry wayes almost vnnumerable. But one feate I shall teache you, whiche not only for the straungenes and secretnes is moche pleasaunt, but also for the good co{m}moditie of it ryghte worthy to be well marked. This feate hath ben vsed aboue 2000 yeares at the leaste, and yet was it neuer come{n}ly knowen, especyally in Englysshe it was neuer taughte yet. This is the arte of nombrynge on the hand, with diuers gestures of the fyngers, expressynge any summe conceaued in the [*133b] mynde. And fyrst to begynne, yf you wyll expresse any summe vnder 100, you shall expresse it with your lefte hande: and from 100 vnto 10000, you shall expresse it with your ryght hande, as here orderly by this table folowynge you may perceaue.

+¶ Here foloweth the table
of the arte of the
hande+

The arte of nombrynge by the hande.

[Transcriber’s Note:

Footnote 3 reads:
“Bracket ([) denotes new paragraph in original.”
For this e-text, the brackets have been omitted in favor of restoring
the paragraph breaks. Changes of speaker (M, S) are also marked by
paragraphs, as in the previous selection.

The illustration includes the printed page number 134; there is
therefore no sidenote *134a. The sidenote for “4” is missing.]

[Sidenote: 1]

[*134b] In which as you may se 1 is expressed by y^e lyttle fynger of y^e lefte hande closely and harde croked.

[Sidenote: 2]

[{3}]2 is declared by lyke bowynge of the weddynge fynger (whiche is the nexte to the lyttell fynger) together with the lytell fynger.

[Sidenote: 3]

3 is signified by the myddle fynger bowed in lyke maner, with those other two.

4 is declared by the bowyng of the myddle fynger and the rynge fynger, or weddynge fynger, with the other all stretched forth.

[Sidenote: 5, 6]

5 is represented by the myddle fynger onely bowed.

And 6 by the weddynge fynger only crooked: and this you may marke in these a certayne order. But now 7, 8, and 9, are expressed w{i}t{h} the bowynge of the same fyngers as are 1, 2, and 3, but after an other fourme.

[Sidenote: 7]

For 7 is declared by the bowynge of the lytell fynger, as is 1, saue that for 1 the fynger is clasped in, harde {and} [*135a] rounde, but for to expresse 7, you shall bowe the myddle ioynte of the lytell fynger only, and holde the other ioyntes streyght.

_S._ Yf you wyll geue me leue to expresse it after my rude maner, thus I vnderstand your meanyng: that 1 is expressed by crookynge in the lyttell fynger lyke the head of a bysshoppes bagle: and 7 is declared by the same fynger bowed lyke a gybbet.

_M._ So I perceaue, you vnderstande it.

[Sidenote: 8]

Then to expresse 8, you shall bowe after the same maner both the lyttell fynger and the rynge fynger.

[Sidenote: 9, 10]

And yf you bowe lyke wayes with them the myddle fynger, then doth it betoken 9.

Now to expresse 10, you shall bowe your fore fynger rounde, and set the ende of it on the hyghest ioynte of the thombe.

[Sidenote: 20]

And for to expresse 20, you must set your fyngers streyght, and the ende of your thombe to the partitio{n} of the [*135b] fore moste and myddle fynger.

[Sidenote: 30]

30 is represented by the ioynynge together of y^e headdes of the foremost fynger and the thombe.

[Sidenote: 40]

40 is declared by settynge of the thombe crossewayes on the foremost fynger.

[Sidenote: 50]

50 is signified by ryght stretchyng forth of the fyngers ioyntly, and applyenge of the thombes ende to the partition of the myddle fynger {and} the rynge fynger, or weddynge fynger.

[Sidenote: 60]

60 is formed by bendynge of the thombe croked and crossynge it with the fore fynger.

[Sidenote: 70]

70 is expressed by the bowynge of the foremost fynger, and settynge the ende of the thombe between the 2 foremost or hyghest ioyntes of it.

[Sidenote: 80]

80 is expressed by settynge of the foremost fynger crossewayes on the thombe, so that 80 dyffereth thus fro{m} 40, that for 80 the forefynger is set crosse on the thombe, and for 40 the thombe is set crosse ouer y^e forefinger.

[Sidenote: 90]

[*136a] 90 is signified, by bendynge the fore fynger, and settyng the ende of it in the innermost ioynte of y^e thombe, that is euen at the foote of it. And thus are all the no{m}bers ended vnder 100.

[Sidenote: 11, 12, 13, 21, 22, 23]

_S._ In dede these be all the nombers fro{m} 1 to 10, {and} then all the tenthes within 100, but this teacyed me not how to expresse 11, 12, 13, {et}c. 21, 22, 23, {et}c. and such lyke.

_M._ You can lytell vnderstande, yf you can not do that without teachynge: what is 11? is it not 10 and 1? then expresse 10 as you were taught, and 1 also, and that is 11: and for 12 expresse 10 and 2: for 23 set 20 and 3: and so for 68 you muste make 60 and there to 8: and so of all other sortes.

[Sidenote: 100]

But now yf you wolde represente 100 other any nomber aboue it, you muste do that with the ryghte hande, after this maner. [You must expresse 100 in the ryght hand, with the lytell fynger so bowed as you dyd expresse 1 in the left hand.

[Sidenote: 200]

[*136b] And as you expressed 2 in the lefte hande, the same fasshyon in the ryght hande doth declare 200.

[Sidenote: 300]

The fourme of 3 in the ryght hand standeth for 300.

[Sidenote: 400]

The fourme of 4, for 400.

[Sidenote: 500]

Lykewayes the fourme of 5, for 500.

[Sidenote: 600]

The fourme of 6, for 600. And to be shorte: loke how you did expresse single vnities and tenthes in the lefte hande, so must you expresse vnities {and} tenthes of hundredes, in the ryghte hande.

[Sidenote: 900]

_S._ I vnderstande you thus: that yf I wold represent 900, I must so fourme the fyngers of my ryghte hande, as I shuld do in my left hand to expresse 9,

[Sidenote: 1000]

And as in my lefte hand I expressed 10, so in my ryght hande must I expresse 1000.

And so the fourme of euery tenthe in the lefte hande serueth to expresse lyke no{m}ber of thousa{n}des,

[Sidenote: 4000]

so y^e fourme of 40 standeth for 4000.

[Sidenote: 8000]

The fourme of 80 for 8000.

[Sidenote: 9000]

[*137a]

And the fourme of 90 (whiche is
the greatest) for 9000, and aboue that
I can not expresse any nomber. _M._
No not with one fynger: how be it,
w{i}t{h} dyuers fyngers you maye expresse
9999, and all at one tyme, and that lac
keth but 1 of 10000. So that vnder
10000 you may by your fyngers ex-
presse any summe. And this shal suf-
fyce for Numeration on the fyngers.
And as for Addition, Subtraction,
Multiplicatio{n}, and Diuision (which
yet were neuer taught by any man as
farre as I do knowe) I wyll enstruct
you after the treatyse of fractions.
And now for this tyme fare well,
and loke that you cease not to
practyse that you haue lear
ned. _S._ Syr, with moste
harty mynde I thanke
you, bothe for your
good learnyng, {and}
also your good
cou{ns}el, which
(god wyllyng) I truste to folow.

Finis.

FOOTNOTES (Accomptynge by counters
_and_ The arte of nombrynge by the hande):

[1: 1342 in original.]
[2: 168 in original.]
[3: Bracket ([) denotes new paragraph in original.]

APPENDIX I.

+A Treatise on the Numeration of Algorism.+

[_From a MS. of the 14th Century._]

To alle suche even nombrys the most have cifrys as to ten. twenty. thirtty. an hundred. an thousand and suche other. but ye schal vnderstonde that a cifre tokeneth nothinge but he maketh other the more significatyf that comith after hym. Also ye schal vnderstonde that in nombrys composyt and in alle other nombrys that ben of diverse figurys ye schal begynne in the ritht syde and to rekene backwarde and so he schal be wryte as thus--1000. the sifre in the ritht side was first wryte and yit he tokeneth nothinge to the secunde no the thridde but thei maken that figure of 1 the more signyficatyf that comith after hem by as moche as he born oute of his first place where he schuld yf he stode ther tokene but one. And there he stondith nowe in the ferye place he tokeneth a thousand as by this rewle. In the first place he tokeneth but hymself. In the secunde place he tokeneth ten times hymself. In the thridde place he tokeneth an hundred tymes himself. In the ferye he tokeneth a thousand tymes himself. In the fyftye place he tokeneth ten thousand tymes himself. In the sexte place he tokeneth an hundred thousand tymes hymself. In the seveth place he tokeneth ten hundred thousand tymes hymself, &c. And ye schal vnderstond that this worde nombre is partyd into thre partyes. Somme is callyd nombre of digitys for alle ben digitys that ben withine ten as ix, viii, vii, vi, v, iv, iii, ii, i. Articules ben alle thei that mow be devyded into nombrys of ten as xx, xxx, xl, and suche other. Composittys be alle nombrys that ben componyd of a digyt and of an articule as fourtene fyftene thrittene and suche other. Fourtene is componyd of four that is a digyt and of ten that is an articule. Fyftene is componyd of fyve that is a digyt and of ten that is an articule and so of others . . . . . . But as to this rewle. In the firste place he tokeneth but himself that is to say he tokeneth but that and no more. If that he stonde in the secunde place he tokeneth ten tymes himself as this figure 2 here 21. this is oon and twenty. This figure 2 stondith in the secunde place and therfor he tokeneth ten tymes himself and ten tymes 2 is twenty and so forye of every figure and he stonde after another toward the lest syde he schal tokene ten tymes as moche more as he schuld token and he stode in that place ther that the figure afore him stondeth: lo an example as thus 9634. This figure of foure that hath this schape 4 tokeneth but himself for he stondeth in the first place. The figure of thre that hath this schape 3 tokeneth ten tyme himself for he stondeth in the secunde place and that is thritti. The figure of sexe that hath this schape 6 tokeneth ten tyme more than he schuld and he stode in the place yer the figure of thre stondeth for ther he schuld tokene but sexty. And now he tokeneth ten tymes that is sexe hundrid. The figure of nyne that hath this schape 9 tokeneth ten tymes more than he schulde and he stode in the place ther the figure of 6 stondeth inne for thanne he schuld tokene but nyne hundryd. And in the place that he stondeth inne nowe he tokeneth nine thousand. Alle the hole nombre of these foure figurys. Nine thousand sexe hundrid and foure and thritti.

APPENDIX II.

Carmen de Algorismo.

[_From a B.M. MS., 8 C. iv., with additions from 12 E. 1 & Eg. 2622._]

Hec algorismus ars presens dicitur[{1}]; in qua
Talibus Indorum[{2}] fruimur his quinque figuris.
0. 9. 8. 7. 6. 5. 4. 3. 2. 1.
Prima significat unum: duo vero secunda:
Tercia significat tria: sic procede sinistre 4
Donec ad extremam venies, qua cifra vocatur;
[{3}][Que nil significat; dat significare sequenti.]
Quelibet illarum si primo limite ponas,
Simpliciter se significat: si vero secundo, 8
Se decies: sursum procedas multiplicando.[{4}]
[Namque figura sequens quevis signat decies plus,
Ipsa locata loco quam significet pereunte: 12
Nam precedentes plus ultima significabit.]
[{5}]Post predicta scias quod tres breuiter numerorum
Distincte species sunt; nam quidam digiti sunt;
Articuli quidam; quidam quoque compositi sunt. 16
[Sunt digiti numeri qui citra denarium sunt;
Articuli decupli degitorum; compositi sunt
Illi qui constant ex articulis digitisque.]
Ergo, proposito numero tibi scribere, primo 20
Respicias quis sit numerus; quia si digitus sit,
[{5}][Una figura satis sibi; sed si compositus sit,]
Primo scribe loco digitum post articulum fac
Articulus si sit, cifram post articulum sit, 24
[Articulum vero reliquenti in scribe figure.]
Quolibet in numero, si par sit prima figura,
Par erit et totum, quicquid sibi continetur;
Impar si fuerit, totum sibi fiet et impar. 28
Septem[{6}] sunt partes, non plures, istius artis;
Addere, subtrahere, duplare, dimidiare;
Sexta est diuidere, set quinta est multiplicare;
Radicem extrahere pars septima dicitur esse. 32
Subtrahis aut addis a dextris vel mediabis;
A leua dupla, diuide, multiplicaque;
Extrahe radicem semper sub parte sinistra.

[Sidenote: Addition.]

Addere si numero numerum vis, ordine tali 36
Incipe; scribe duas primo series numerorum
Prima sub prima recte ponendo figuram,
Et sic de reliquis facias, si sint tibi plures.
Inde duas adde primas hac condicione; 40
Si digitus crescat ex addicione priorum,
Primo scribe loco digitum, quicunque sit ille;
Si sit compositus, in limite scribe sequenti
Articulum, primo digitum; quia sic iubet ordo. 44
Articulus si sit, in primo limite cifram,
Articulum vero reliquis inscribe figuris;
Vel per se scribas si nulla figura sequatur.
Si tibi cifra superueniens occurrerit, illam 48
Deme suppositam; post illic scribe figuram:
Postea procedas reliquas addendo figuras.

[Sidenote: Subtraction.]

A numero numerum si sit tibi demere cura,
Scribe figurarum series, vt in addicione; 52
Maiori numero numerum suppone minorem,
Siue pari numero supponatur numerus par.
Postea si possis a prima subtrahe primam,
Scribens quod remanet, cifram si nil remanebit. 56
Set si non possis a prima demere primam;
Procedens, vnum de limite deme sequenti;
Et demptum pro denario reputabis ab illo,
Subtrahe totaliter numerum quem proposuisti. 60
Quo facto, scribe supra quicquit remanebit,
Facque novenarios de cifris, cum remanebis,
Occurrant si forte cifre, dum demseris vnum;
Postea procedas reliquas demendo figuras. 64

[Sidenote: Proof.]

[{7}][Si subtracio sit bene facta probare valebis,
Quas subtraxisti primas addendo figuras.
Nam, subtractio si bene sit, primas retinebis,
Et subtractio facta tibi probat additionem.] 68

[Sidenote: Duplation.]

Si vis duplare numerum, sic incipe; solam
Scribe figurarum seriem, quamcumque voles que
Postea procedas primam duplando figuram;
Inde quod excrescet, scribens, vbi iusserit ordo, 72
Juxta precepta que dantur in addicione.
Nam si sit digitus, in primo limite scribe;
Articulus si sit, in primo limite cifram,
Articulum vero reliquis inscribe figuris; 76
Vel per se scribas, si nulla figura sequatur:
Compositus si sit, in limite scribe sequenti
Articulum primo, digitum; quia sic jubet ordo:
Et sic de reliquis facias, si sint tibi plures. 80
[{8}][Si super extremam nota sit, monadem dat eidem,
Quod tibi contingit, si primo dimidiabis.]

[Sidenote: Mediation.]

Incipe sic, si vis aliquem numerum mediare:
Scribe figurarum seriem solam, velud ante; 84
Postea procedens medias, et prima figura
Si par aut impar videas; quia si fuerit par,
Dimidiabis eam, scribens quicquit remanebit;
Impar si fuerit, vnum demas, mediare, 88
Nonne presumas, sed quod superest mediabis;
Inde super tractum, fac demptum quod notat unum;
Si monos, dele; sit ibi cifra post nota supra.
Postea procedas hac condicione secunda:[{9}] 92
Impar[{10}] si fuerit hic vnum deme priori,
Inscribens quinque, nam denos significabit
Monos prædictam: si vero secunda dat vnam,
Illa deleta, scribatur cifra; priori 96
Tradendo quinque pro denario mediato;
Nec cifra scribatur, nisi inde figura sequatur:
Postea procedas reliquas mediando figuras,
Quin supra docui, si sint tibi mille figure. 100
[{11}][Si mediatio sit bene facta probare valebis,
Duplando numerum quem primo dimidiasti.]
Si super extremam nota sit monades dat eidem
Quod contingat cum primo dimiabis
Atque figura prior nuper fuerit mediando.]

[Sidenote: Multiplication.]

Si tu per numerum numerum vis multiplicare,
Scribe duas, quascunque volis, series numerorum; 104
Ordo tamen seruetur vt vltima multiplicandi
Ponatur super anteriorem multiplicantis;
[{12}][A leua relique sint scripte multiplicantes.]
In digitum cures digitum si ducere, major 108
Per quantes distat a denis respice, debes
Namque suo decuplo tociens delere minorem;
Sicque tibi numerus veniens exinde patebit.
Postea procedas postremam multiplicando, 112
Juste multiplicans per cunctas inferiores,
Condicione tamen tali; quod multiplicantis
Scribas in capite, quicquid processerit inde;
Set postquam fuerit hec multiplicata, figure 116
Anteriorentur seriei multiplicantis;
Et sic multiplica, velut istam multiplicasti,
Qui sequitur numerum scriptum quicunque figuris.
Set cum multiplicas, primo sic est operandum, 120
Si dabit articulum tibi multiplicacio solum;
Proposita cifra, summam transferre memento.
Sin autem digitus excrescerit articulusque,
Articulus supraposito digito salit ultra; 124
Si digitus tamen, ponas illum super ipsam,
Subdita multiplicans hanc que super incidit illi
Delet eam penitus, scribens quod provenit inde;
Sed si multiplices illam posite super ipsam, 128
Adiungens numerum quem prebet ductus earum;
Si supraimpositam cifra debet multiplicare,
Prorsus eam delet, scribi que loco cifra debet,
[{12}][Si cifra multiplicat aliam positam super ipsam, 132
Sitque locus supra vacuus super hanc cifra fiet;]
Si supra fuerit cifra semper pretereunda est;
Si dubites, an sit bene multiplicando secunda,
Diuide totalem numerum per multiplicantem, 136
Et reddet numerus emergens inde priorem.

[Sidenote: Mental Multiplication.]

[{13}][Per numerum si vis numerum quoque multiplicare
Tantum per normas subtiles absque figuris
Has normas poteris per versus scire sequentes. 140
Si tu per digitum digitum quilibet multiplicabis
Regula precedens dat qualiter est operandum
Articulum si per reliquum vis multiplicare
In proprium digitum debebit uterque resolvi 144
Articulus digitos post per se multiplicantes
Ex digitis quociens teneret multiplicatum
Articuli faciunt tot centum multiplicati.
Articulum digito si multiplicamus oportet 148
Articulum digitum sumi quo multiplicare
Debemus reliquum quod multiplicaris ab illis
Per reliquo decuplum sic omne latere nequibit
In numerum mixtum digitum si ducere cures 152
Articulus mixti sumatur deinde resolvas
In digitum post hec fac ita de digitis nec
Articulusque docet excrescens in detinendo
In digitum mixti post ducas multiplicantem 156
De digitis ut norma docet sit juncta secundo
Multiplica summam et postea summa patebit
Junctus in articulum purum articulumque
[{14}][Articulum purum comittes articulum que] 160
Mixti pro digitis post fiat et articulus vt
Norma jubet retinendo quod egreditur ab illis
Articuli digitum post in digitum mixti duc
Regula de digitis ut percipit articulusque 164
Ex quibus excrescens summe tu junge priori
Sic manifesta cito fiet tibi summa petita.
Compositum numerum mixto sic multiplicabis
Vndecies tredecem sic est ex hiis operandum 168
In reliquum primum demum duc post in eundem
Unum post deinde duc in tercia deinde per unum
Multiplices tercia demum tunc omnia multiplicata
In summa duces quam que fuerit te dices 172
Hic ut hic mixtus intentus est operandum
Multiplicandorum de normis sufficiunt hec.]

[Sidenote: Division.]

Si vis dividere numerum, sic incipe primo;
Scribe duas, quascunque voles, series numerorum; 176
Majori numero numerum suppone minorem,
[{15}][Nam docet ut major teneat bis terve minorem;]
Et sub supprima supprimam pone figuram,
Sic reliquis reliquas a dextra parte locabis; 180
Postea de prima primam sub parte sinistra
Subtrahe, si possis, quociens potes adminus istud,
Scribens quod remanet sub tali conditione;
Ut totiens demas demendas a remanente, 184
Que serie recte ponentur in anteriori,
Unica si, tantum sit ibi decet operari;
Set si non possis a prima demere primam,
Procedas, et eam numero suppone sequenti; 188
Hanc uno retrahendo gradu quo comites retrahantur,
Et, quotiens poteris, ab eadem deme priorem,
Ut totiens demas demendas a remanenti,
Nec plus quam novies quicquam tibi demere debes, 192
Nascitur hinc numerus quociens supraque sequentem
Hunc primo scribas, retrahas exinde figuras,
Dum fuerit major supra positus inferiori,
Et rursum fiat divisio more priori; 196
Et numerum quotiens supra scribas pereunti,
Si fiat saliens retrahendo, cifra locetur,
Et pereat numero quotiens, proponas eidem
Cifram, ne numerum pereat vis, dum locus illic 200
Restat, et expletis divisio non valet ultra:
Dum fuerit numerus numerorum inferiore seorsum
Illum servabis; hinc multiplicando probabis,

[Sidenote: Proof.]

Si bene fecisti, divisor multiplicetur 204
Per numerum quotiens; cum multiplicaveris, adde
Totali summæ, quod servatum fuit ante,
Reddeturque tibi numerus quem proposuisti;
Et si nil remanet, hunc multiplicando reddet, 208

[Sidenote: Square Numbers.]

Cum ducis numerum per se, qui provenit inde
Sit tibi quadratus, ductus radix erit hujus,
Nec numeros omnes quadratos dicere debes,
Est autem omnis numerus radix alicujus. 212
Quando voles numeri radicem querere, scribi
Debet; inde notes si sit locus ulterius impar,
Estque figura loco talis scribenda sub illo,
Que, per se dicta, numerum tibi destruat illum, 216
Vel quantum poterit ex inde delebis eandem;
Vel retrahendo duples retrahens duplando sub ista
Que primo sequitur, duplicatur per duplacationem,
Post per se minuens pro posse quod est minuendum. 220
[{16}]Post his propones digitum, qui, more priori
Per precedentes, post per se multiplicatus,
Destruat in quantum poterit numerum remanentem,
Et sic procedens retrahens duplando figuram, 224
Preponendo novam donec totum peragatur,
Subdupla propriis servare docetque duplatis;
Si det compositum numerum duplacio, debet
Inscribi digitus a parte dextra parte propinqua, 228
Articulusque loco quo non duplicata resessit;
Si dabit articulum, sit cifra loco pereunte
Articulusque locum tenet unum, de duplicata resessit;
Si donet digitum, sub prima pone sequente, 232
Si supraposita fuerit duplicata figura
Major proponi debet tantummodo cifra,
Has retrahens solito propones more figuram,
Usque sub extrema ita fac retrahendo figuras, 236
Si totum deles numerum quem proposuisti,
Quadratus fuerit, de dupla quod duplicasti,
Sicque tibi radix illius certa patebit,
Si de duplatis fit juncta supprima figura; 240
Radicem per se multiplices habeasque
Primo propositum, bene te fecisse probasti;
Non est quadratus, si quis restat, sed habentur
Radix quadrati qui stat major sub eadem; 244
Vel quicquid remanet tabula servare memento;
Hoc casu radix per se quoque multiplicetur,
Vel sic quadratus sub primo major habetur,
Hinc addas remanens, et prius debes haberi; 248
Si locus extremus fuerit par, scribe figuram
Sub pereunte loco per quam debes operari,
Que quantum poterit supprimas destruat ambas,
Vel penitus legem teneas operando priorem, 252
Si suppositum digitus suo fine repertus,
Omnino delet illic scribi cifra debet,
A leva si qua sit ei sociata figura;
Si cifre remanent in fine pares decet harum 256
Radices, numero mediam proponere partem,
Tali quesita radix patet arte reperta.
Per numerum recte si nosti multiplicare
Ejus quadratum, numerus qui pervenit inde 260
Dicetur cubicus; primus radix erit ejus;
Nec numeros omnes cubicatos dicere debes,
Est autem omnis numerus radix alicujus;

[Sidenote: Cube Root.]

Si curas cubici radicem quærere, primo 264
Inscriptum numerum distinguere per loca debes;
Que tibi mille notant a mille notante suprema
Initiam, summa operandi parte sinistra,
Illic sub scribas digitum, qui multiplicatus 268
In semet cubice suprapositum sibi perdat,
Et si quid fuerit adjunctum parte sinistra
Si non omnino, quantum poteris minuendo,
Hinc triplans retrahe saltum, faciendo sub illa 272
Que manet a digito deleto terna, figuram
Illi propones quo sub triplo asocietur,
Ut cum subtriplo per eam tripla multiplicatur;
Hinc per eam solam productum multiplicabis, 276
Postea totalem numerum, qui provenit inde
A suprapositis respectu tolle triplate
Addita supprimo cubice tunc multiplicetur,
Respectu cujus, numerus qui progredietur 280
Ex cubito ductu, supra omnes adimetur;
Tunc ipsam delens triples saltum faciendo,
Semper sub ternas, retrahens alias triplicatas
Ex hinc triplatis aliam propone figuram, 284
Que per triplatas ducatur more priori;
Primo sub triplis sibi junctis, postea per se,
In numerum ducta, productum de triplicatis:
Utque prius dixi numerus qui provenit inde 288
A suprapositis has respiciendo trahatur,
Huic cubice ductum sub primo multiplicabis,
Respectumque sui, removebis de remanenti,
Et sic procedas retrahendo triplando figuram. 292
Et proponendo nonam, donec totum peragatur,
Subtripla sub propriis servare decet triplicatis;
Si nil in fine remanet, numerus datus ante
Est cubicus; cubicam radicem sub tripla prebent, 296
Cum digito juncto quem supprimo posuisti,
Hec cubice ducta, numerum reddant tibi primum.
Si quid erit remanens non est cubicus, sed habetur
Major sub primo qui stat radix cubicam, 300
Servari debet quicquid radice remansit,
Extracto numero, decet hec addi cubicato.
Quo facto, numerus reddi debet tibi primus.
Nam debes per se radicem multiplicare 304
Ex hinc in numerum duces, qui provenit inde
Sub primo cubicus major sic invenietur;
Illi jungatur remanens, et primus habetur,
Si per triplatum numerum nequeas operari; 308
Cifram propones, nil vero per hanc operare
Set retrahens illam cum saltu deinde triplata,
Propones illi digitum sub lege priori,
Cumque cifram retrahas saliendo, non triplicabis, 312
Namque nihil cifre triplacio dicitur esse;
At tu cum cifram protraxeris aut triplicata,
Hanc cum subtriplo semper servare memento:
Si det compositum, digiti triplacio debet 316
Illius scribi, digitus saliendo sub ipsam;
Digito deleto, que terna dicitur esse;
Jungitur articulus cum triplata pereunte,
Set facit hunc scribi per se triplacio prima, 320
Que si det digitum per se scribi facit illum;
Consumpto numero, si sole fuit tibi cifre
Triplato, propone cifram saltum faciendo,
Cumque cifram retrahe triplam, scribendo figuram, 324
Preponas cifre, sic procedens operare,
Si tres vel duo serie in sint, pone sub yma,
A dextris digitum servando prius documentum.
Si sit continua progressio terminus nuper 328
Per majus medium totalem multiplicato;
Si par, per medium tunc multiplicato sequentem.
Set si continua non sit progressio finis:
Impar, tunc majus medium si multiplicabis, 332
Si par per medium sibi multiplicato propinquum. 333

FOOTNOTES (Appendix II, Carmen de Algorismo):

[1: “Hec præsens ars dicitur algorismus ab Algore rege ejus
inventore, vel dicitur ab _algos_ quod est ars, et _rodos_ quod est
numerus; quæ est ars numerorum vel numerandi, ad quam artem bene
sciendum inveniebantur apud Indos bis quinque (id est decem)
figuræ.” --_Comment. Thomæ de Novo-Mercatu._ MS. Bib. Reg. Mus.
Brit. 12 E. 1.]

[2: “Hæ necessariæ figuræ sunt Indorum characteros.” _MS. de
numeratione._ Bib. Sloan. Mus. Brit. 513, fol. 58. “Cum vidissem
Yndos constituisse IX literas in universo numero suo propter
dispositionem suam quam posuerunt, volui patefacere de opere quod
sit per eas aliquidque esset levius discentibus, si Deus voluerit.
Si autem Indi hoc voluerunt et intentio illorum nihil novem literis
fuit, causa que mihi potuit. Deus direxit me ad hoc. Si vero alia
dicam preter eam quam ego exposui, hoc fecerunt per hoc quod ego
exposui, eadem tam certissime et absque ulla dubitatione poterit
inveniri. Levitasque patebit aspicientibus et discentibus.” MS.
U.L.C., Ii. vi. 5, f. 102.]

[3: From Eg. 2622.]

[4: 8 C. iv. inserts
Nullum cipa significat: dat significare sequenti.]

[5: From 12 E. 1.]

[6:
En argorisme devon prendre
Vii especes . . . .
Adision subtracion
Doubloison mediacion
Monteploie et division
Et de radix eustracion
A chez vii especes savoir
Doit chascun en memoire avoir
Letres qui figures sont dites
Et qui excellens sont ecrites. --MS. _Seld. Arch._ B. 26.]

[7: From 12 E. 1.]

[8: From 12 E. 1.]

[9: 8 C. iv. inserts
Atque figura prior nuper fuerit mediando.]

[10: _I.e._ figura secundo loco posita.]

[11: So 12 E. 1; 8 C. iv. inserts--

[12: 12 E. 1 inserts.]

[13: 12 E. 1 inserts to l. 174.]

[14: 12 E. 1 omits, Eg. 2622 inserts.]

[15: 12 E. 1 inserts.]

[16: 8 C. iv. inserts--
Hinc illam dele duplans sub ei psalliendo
Que sequitur retrahens quicquid fuerit duplicatum.]

INDEX OF TECHNICAL TERMS[1*]

[Footnote 1*: This Index has been kindly prepared by Professor
J. B. Dale, of King’s College, University of London, and the
best thanks of the Society are due to him for his valuable
contribution.]

[Transcriber’s Note:
The Technical Terms and Glossary (following) refer to page and line
numbers in the printed book. Information in [[double brackets]] has
been added by the transcriber to aid in text searching.]

+algorisme+, 33/12; +algorym+, +augrym+, 3/3; the art of computing,
using the so-called Arabic numerals.
The word in its various forms is derived from the Arabic
_al-Khowarazmi_ (i.e. the native of Khwarazm (Khiva)). This was the
surname of Ja’far Mohammad ben Musa, who wrote a treatise early in
the 9th century (see p. xiv).
The form _algorithm_ is also found, being suggested by a supposed
derivation from the Greek ἀριθμός (number).

+antery+, 24/11; to move figures to the right of the position in
which they are first written. This operation is performed repeatedly
upon the multiplier in multiplication, and upon certain figures
which arise in the process of root extraction.

+anterioracioun+, 50/5; the operation of moving figures to the
right. [[written anteriorac{i}o{u}n or anterioracio{u}n]]

+article+, 34/23; +articul+, 5/31; +articuls+, 9/36, 29/7,8;
a number divisible by ten without remainder. [[also articull{e}]]

+cast+, 8/12; to add one number to another.
‘Addition is a _casting_ together of two numbers into one number,’
8/10.

+cifre+, 4/1; the name of the figure 0. The word is derived from the
Arabic _sifr_ = empty, nothing. Hence _zero_.
A cipher is the symbol of the absence of number or of zero quantity.
It may be used alone or in conjunction with digits or other ciphers,
and in the latter case, according to the position which it occupies
relative to the other figures, indicates the absence of units, or
tens, or hundreds, etc. The great superiority of the Arabic to all
other systems of notation resides in the employment of this symbol.
When the cipher is not used, the place value of digits has to be
indicated by writing them in assigned rows or columns. Ciphers,
however, may be interpolated amongst the significant figures used,
and as they sufficiently indicate the positions of the empty rows or
columns, the latter need not be indicated in any other way. The
practical performance of calculations is thus enormously facilitated
(see p. xvi).

+componede+, 33/24; +composyt+, 5/35; with reference to numbers, one
compounded of a multiple of ten and a digit.
[[written componed{e}]]

+conuertide+ = conversely, 46/29, 47/9.
[[written co{n}u{er}tid{e} or {con}u{er}tid{e}]]

+cubicede+, 50/13; +to be c.+, to have its cube root found.
[[written cubiced{e}]]

+cubike nombre+, 47/8; a number formed by multiplying a given number
twice by itself, _e.g._ 27 = 3 × 3 × 3. Now called simply a cube.
[[written cubik{e} ...]]

+decuple+, 22/12; the product of a number by ten. Tenfold.

+departys+ = divides, 5/29. [[written dep{ar}tys]]

+digit+, 5/30; +digitalle+, 33/24; a number less than ten,
represented by one of the nine Arabic numerals.
[[written digitall{e}]]

+dimydicion+, 7/23; the operation of dividing a number by two.
Halving. [[written dimydicioñ]]

+duccioun+, multiplication, 43/9. [[written duccio{u}n]]

+duplacion+, 7/23, 14/15; the operation of multiplying a number by
two. Doubling.
[[written duplacioñ or duplacioɳ with fancy “n”]]

+i-mediet+ = halved, 19/23.

+intercise+ = broken, 46/2; intercise Progression is the name given
to either of the Progressions 1, 3, 5, 7, etc.; 2, 4, 6, 8, etc.,
in which the common difference is 2. [[written int{er}cise]]

+lede into+, multiply by, 47/18.
[[words always separated, as “lede ... into”]]

+lyneal nombre+, 46/14; a number such as that which expresses the
measure of the length of a line, and therefore is not _necessarily_
the product of two or more numbers (_vide_ Superficial, Solid). This
appears to be the meaning of the phrase as used in _The Art of
Nombryng_. It is possible that the numbers so designated are the
prime numbers, that is, numbers not divisible by any other number
except themselves and unity, but it is not clear that this
limitation is intended.

+mediacioun+, 16/36, 38/16; dividing by two (see also +dimydicion+).
[[written mediacioɳ with fancy “n”, generally without “u”]]

+medlede nombre+, 34/1; a number formed of a multiple of ten and a
digit (_vide_ componede, composyt). [[written medled{e} ...]]

+medye+, 17/8, to halve; +mediete+, halved, 17/30; +ymedit+, 20/9.

+naturelle progressioun+, 45/22; the series of numbers 1, 2, 3, etc.
[[written naturell{e} p{ro}gressio{u}n]]

+produccioun+, multiplication, 50/11. [[written produccio{u}n]]

+quadrat nombre+, 46/12; a number formed by multiplying a given
number by itself, _e.g._ 9 = 3 × 3, a square.

+rote+, 7/25; +roote+, 47/11; root. The roots of squares and cubes
are the numbers from which the squares and cubes are derived by
multiplication into themselves.

+significatyf+, significant, 5/14; The significant figures of a
number are, strictly speaking, those other than zero, _e.g._ in 3 6
5 0 4 0 0, the significant figures are 3, 6, 5, 4. Modern usage,
however, regards all figures between the two extreme significant
figures as significant, even when some are zero. Thus, in the above
example, 3 6 5 0 4 are considered significant.

+solide nombre+, 46/37; a number which is the product of three other
numbers, _e.g._ 66 = 11 × 2 × 3. [[usually written solid{e}]]

+superficial nombre+, 46/18; a number which is the product of two
other numbers, _e.g._ 6 = 2 × 3.
[[written sup{er}ficial or sup{er}ficiall{e}]]

+ternary+, consisting of three digits, 51/7.
[[written t{er}nary]]

+vnder double+, a digit which has been doubled, 48/3.

+vnder-trebille+, a digit which has been trebled, 49/28;
+vnder-triplat+, 49/39.
[[written vnder-trebill{e}, vnder-t{r}iplat]]

+w+, a symbol used to denote half a unit, 17/33
[[shown in e-text as superscript ʷ]]

GLOSSARY

[Transcriber’s Note:

Words whose first appearance is earlier than the page cited in the
Glossary are identified in double-bracketed notes. To aid in text
searching, words written with internal {italics} are also noted,
and context is given for common words.]

+ablacioun+, taking away, 36/21 [[written ablacio{u}n]]
+addyst+, haddest, 10/37
+agregacioun+, addition, 45/22. (First example in N.E.D., 1547.)
[[written ag{r}egacio{u}n]]
+a-ȝenenes+, against, 23/10
+allgate+, always, 8/39
+als+, as, 22/24
+and+, if, 29/8;
+&+, 4/27;
+& yf+, 20/7
+a-nendes+, towards, 23/15
+aproprede+, appropriated, 34/27 [[written ap{ro}pred{e}]]
+apwereth+, appears, 61/8
+a-risyȝt+, arises, 14/24
+a-rowe+, in a row, 29/10
+arsemetrike+, arithmetic, 33/1 [[written arsemetrik{e}]]
+ayene+, again, 45/15

+bagle+, crozier, 67/12
+bordure+ = ordure, row, 43/30 [[written bordur{e}]]
+borro+, _inf._ borrow, 11/38;
_imp. s._ +borowe+, 12/20;
_pp._ +borwed+, 12/15;
+borred+, 12/19
+boue+, above, 42/34

+caputule+, chapter, 7/26 [[written caputul{e}]]
+certayn+, assuredly, 18/34 [[written c{er}tayɳ]]
+clepede+, called, 47/7 [[written cleped{e}]]
+competently+, conveniently, 35/8
+compt+, count, 47/29
+contynes+, contains, 21/12; [[written {con}tynes]]
_pp._ +contenythe+, 38/39 [[written co{n}tenyth{e}]]
+craft+, art, 3/4

+distingue+, divide, 51/5

+egalle+, equal, 45/21 [[written egall{e}]]
+excep+, except, 5/16]
+exclusede+, excluded, 34/37 [[written exclused{e}]]
+excressent+, resulting, 35/16 [[written exc{re}ssent]]
+exeant+, resulting, 43/26
+expone+, expound, 3/23

+ferye+ = ferþe, fourth, 70/12
+figure+ = figures, 5/1 [[written fig{ure}]]
+for-by+, past, 12/11
+fors; no f.+, no matter, 22/24
+forseth+, matters, 53/30
+forye+ = forþe, forth, 71/8]
+fyftye+ = fyftþe, fifth, 70/16

+grewe+, Greek, 33/13

+haluendel+, half, 16/16;
+haldel+, 19/4;
_pl._ +haluedels+, 16/16
+hayst+, hast, 17/3, 32
+hast+, haste, 22/25 [[in “haue hast to”]]
+heer+, higher, 9/35
+here+, their, 7/26 [[in “in her{e} caputul{e}”]]
+here-a-fore+, heretofore, 13/7 [[written her{e}-a-for{e}]]
+heyth+, was called, 3/5
+hole+, whole, 4/39;
+holle+, 17/1;
+hoole+, of three dimensions, 46/15
+holdyþe+, holds good, 30/5
+how be it that+, although, 44/4

+lede+ = lete, let, 8/37
+lene+, lend, 12/39
+lest+, least, 43/27 [[in “at the lest”]]
+lest+ = left, 71/9 [[in “the lest syde”]]
+leue+, leave, 6/5;
_pr. 3 s._ +leues+, remains, 11/19; [[first in 10/40]]
+leus+, 11/28;
_pp._ +laft+, left, 19/24
+lewder+, more ignorant, 3/3 [[written lewd{er}]]
+lust+, desirest to, 45/13
+lyȝt+, easy, 15/31
+lymytes+, limits, 34/18;
+lynes+, 34/12;
+lynees+, 34/17;
Lat. limes, _pl._ limites.

+maystery+, achievement; [[written mayst{er}y]]
+no m.+, no achievement, i.e. easy, 19/10
+me+, _indef. pron._ one, 42/1 [[first in 34/16]]
+mo+, more, 9/16
+moder+ = more (Lat. majorem), 43/22
+most+, must, 30/3 [[first in 3/12 and many more]]
+multipliede+, +to be m.+ = multiplying, 40/9
+mynvtes+, the sixty parts into which a unit is divided, 38/25
[[written mynvt{es}]]
+myse-wroȝt+, mis-wrought, 14/11

+nether+, nor, 34/25 [[in “It was, nether is”]]
+nex+, next, 19/9
+noȝt+, nought, 5/7 [[first in 4/8]]
+note+, not, 30/5

+oo+, one, 42/20; +o+, 42/21 [[first in 34/27; 33/22]]
+omest+, uppermost, higher, 35/26;
+omyst+, 35/28
+omwhile+, sometimes, 45/31 [[first in 39/17]]
+on+, one, 8/29 [[in “on vnder an-oþ{er}”]]
+opyne+, plain, 47/8 [[written opyn{e}]]
+or+, before, 13/25 [[in “or þou be-gan”]]
+or+ = þe oþ{er}, the other, 28/34 [[in “or by-twene”]]
+ordure+, order, 34/9;
row, 43/1 [[word form is “order”]]
+other+, or, 33/13, 43/26;
[[in “art other craft” on 33/13, “other how oft” on 43/26;
note also “one other other” on 35/24]]
+other . . . or+, either . . . or, 38/37
[[in “other it is even or od{e}” on 38/37;
there are earlier occurrences]]
+ouerer+, upper, 42/15 [[written ou{er}er]]
+ouer-hippede+, passed over, 43/19 [[written ou{er}-hipped{e}]]

+recte+, directly, 27/20 [[in “stondes not recte”;
also on 26/31 in “recte ou{er} his hede”]]
+remayner+, remainder, 56/28
+representithe+, represented, 39/14 [[written rep{re}sentith{e}]]
+resteth+, remains, 63/29 [[first in 57/29 and others]]
+rewarde+, regard, 48/6 [[written reward{e}]]
+rew+, row, 4/8
+rewle+, row, 4/20, 7/12;
[[in “place of þe rewle”, “þe rewle of fig{ure}s”]]
+rewele+, 4/18;
+rewles+, rules, 5/33

+s.+ = scilicet, 3/8 [[in “s. Algorism{us}”]]
+sentens+, meaning, 14/29
+signifye(tyf)+, 5/13. The last three letters are added above the
line, evidently because of the word ‘significatyf’ in l. 14.
But the ‘Solucio,’ which contained the word, has been omitted.
+sithen+, since, 33/8
+some+, sum, result, 40/17, 32
[[first in 36/21 in “me may see a some”, then in “the same some”
and “to some of”]]
+sowne+, pronounce, 6/29
+singillatim+, singly, 7/25
+spices+, species, kinds, 34/4 [[first in 5/34 and others]]
+spyl+, waste, 14/26
+styde+, stead, 18/20
+subtrahe+, subtract, 48/12;
_pp._ +subtrayd+, 13/21
+sythes+, times, 21/16

+taȝt+, taught, 16/36
+take+, _pp._ taken;
+t. fro+, starting from, 45/22 [[in “fro oone or tweyn{e} take”]]
+taward+, toward, 23/34
+thouȝt+, though, 5/20
+trebille+, multiply by three, 49/26 [[written trebill{e}]]
+twene+, two, 8/11 [[first in 4/23]]
+þow+, though, 25/15 [[in “þow þ{o}u take”]]
+þowȝt+, thought;
+be þ.+, mentally, 28/4
+þus+ = þis, this, 20/33 [[in “þus nombur 214”]]

+vny+, unite, 45/10

+wel+, wilt, 14/31 [[in “If þ{o}u wel”]]
+wete+, wit, 15/16;
+wyte+, know, 8/38;
_pr. 2 s._ +wost+, 12/38
+wex+, become, 50/18
+where+, whether, 29/12
[[written wher{e} in “wher{e} in þe secunde, or”]]
+wher-thurghe+, whence, 49/15 [[written Wher-thurgh{e}]]
+worch+, work, 8/19; [[first in 7/35]]
+wrich+, 8/35;
+wyrch+, 6/19;
_imp. s._ +worch+, 15/9; [[first in 9/6]]
_pp._ +y-wroth+, 13/24
+write+, written, 29/19;
[[first in 6/37 in “hast write”, “be write”]]
+y-write+, 16/1
+wryrchynge+ = wyrchynge, working, 30/4 [[written wryrchyng{e}]]
+w^t+, with, 55/8

+y-broth+, brought, 21/18
+ychon+, each one, 29/10 [[written ychoɳ]]
+ydo+, done, added, 9/6
[[first in 8/37 in “haue ydo”; 9/6 in “ydo all to-ged{er}”]]
+ylke+, same, 5/12
+y-lyech+, alike, 22/23
+y-myȝt+, been able, 12/2
+y-nowȝt+, enough, 15/31;
+ynovȝt+, 18/34
+yove+, given, 45/33
+y^t+, that, 52/8
+y-write+, _v._ +write.+
+y-wroth+, _v._ +worch.+

* * * * *
* * * *
* * * * *

MARGINAL NOTES:

+Headnotes+ have been moved to the beginning of the appropriate paragraph. Headnotes were omitted from the two Appendixes, as sidenotes give the same information.

+Line Numbers+ are cited in the Index and Glossary. They have been omitted from the e-text except in the one verse selection (App. II, _Carmen de Algorismo_). Instead, the Index and Glossary include supplemental information to help locate each word.

+Numbered Notes+:

Numbered sidenotes show page or leaf numbers from the original MSS.
In the e-text, the page number is shown as [*123b] inline; mid-word
page breaks are marked with a supplemental asterisk [*]. Numbers are
not used.