+John of Holywood’s De Arte Numerandi.+
[_Ashmole MS. 396, fol. 48._]
+Boys seying in the begynnyng of his Arsemetrik{e}:--All{e}
[*Fol. 48.] thynges that ben{e} fro the first begynnyng of thynges
have p{ro}ceded{e}, and come forth{e}, And by reso{u}n of nombre
ben formed{e}; And in wise as they ben{e}, So oweth{e} they to be
knowen{e}; wherfor in vniu{er}sall{e} knowlechyng of thynges the
Art of nombrynge is best, and most operatyf{e}.+
[Sidenote: The name of the art. Derivation of Algorism. Another.
Another. Kinds of numbers. The 9 rules of the Art.]
Therfore sithen the science of the whiche at this tyme we intenden{e} to
write of standith{e} all{e} and about nombre: ffirst we most se, what is
the p{ro}pre name therof{e}, and fro whens the name come: Afterward{e}
what is nombre, And how manye spices of nombre ther ben. The name is
cleped{e} Algorisme, had{e} out of Algor{e}, other of Algos, in grewe,
That is clepid{e} in englissh{e} art other craft, And of Rithm{us} that
is called{e} nombre. So algorisme is cleped{e} the art of nombryng,
other it is had of{e} en or in, and gogos that is introduccio{u}n, and
Rithm{us} nombre, that is to say Interduccio{u}n of nombre. And thirdly
it is had{e} of the name of a kyng that is cleped{e} Algo and Rythm{us};
So called{e} Algorism{us}. Sothely .2. maner{e} of nombres ben
notified{e}; Formall{e},[{1}] as nombr{e} i{s} vnitees gadred{e}
to-gedres; Materiall{e},[{2}] as nombr{e} is a colleccio{u}n of vnitees.
Other nombr{e} is a multitude had{e} out of vnitees, vnitee is that
thynge wher-by eu{er}y thynge is called{e} oone, other o thynge. Of
nombres, that one is cleped{e} digitall{e}, that other{e} Article,
Another a nombre componed{e} oþ{er} myxt. Another digitall{e} is a
nombre w{i}t{h}-in .10.; Article is þ{a}t nombre that may be dyvyded{e}
in .10. p{ar}ties egally, And that there leve no residue; Componed{e} or
medled{e} is that nombre that is come of a digite and of an article. And
vndrestand{e} wele that all{e} nombres betwix .2. articles next is a
nombr{e} componed{e}. Of this art ben{e} .9. spices, that is forto sey,
num{er}acio{u}n, addicio{u}n, Subtraccio{u}n, Mediac{i}o{u}n,
Duplacio{u}n, Multipliacio{u}n, Dyvysio{u}n, Progressio{u}n, And of
Rootes the extraccio{u}n, and that may be had{e} in .2. maners, that is
to sey in nombres quadrat, and in cubic{es}: Amonge the which{e}, ffirst
of Num{er}acio{u}n, and aft{er}ward{e} of þe oþ{er}s by ordure,
y entende to write.
[Headnote: Chapter I. Numeration.]
[*Fol. 48b]
+For-soth{e} num{er}acio{u}n is of eu{er}y numbre by
competent figures an artificiall{e} rep{re}sentacio{u}n.+
[Sidenote: Figures, differences, places, and limits. The 9 figures.
The cipher. The numeration of digits, of articles, of composites.
The value due to position. Numbers are written from right to left.]
Sothly figure, difference, places, and lynes supposen o thyng other the
same, But they ben sette here for dyue{r}s resons. ffigure is cleped{e}
for p{ro}traccio{u}n of figuracio{u}n; Difference is called{e} for
therby is shewed{e} eu{er}y figure, how it hath{e} difference fro the
figures before them: place by cause of space, where-in me writeth{e}:
lynees, for that is ordeyned{e} for the p{re}sentacio{u}n of eu{er}y
figure. And vnderstonde that ther ben .9. lymytes of figures that
rep{re}senten the .9. digit{es} that ben these. 0. 9. 8. 7. 6. 5. 4. 3.
2. 1. The .10. is cleped{e} theta, or a cercle, other a cifre, other a
figure of nought for nought it signyfieth{e}. Nathelesse she holdyng
that place giveth{e} others for to signyfie; for with{e}-out cifre or
cifres a pure article may not be writte. And sithen that by these .9.
figures significatif{es} Ioyned{e} w{i}t{h} cifre or w{i}t{h} cifres
all{e} nombres ben and may be rep{re}sented{e}, It was, nether is,
no nede to fynde any more figures. And note wele that eu{er}y digite
shall{e} be writte w{i}t{h} oo figure allone to it ap{ro}pred{e}. And
all{e} articles by a cifre, ffor eu{er}y article is named{e} for oone of
the digitis as .10. of 1.. 20. of. 2. and so of the others, &c. And
all{e} nombres digitall{e} owen to be sette in the first difference:
All{e} articles in the seconde. Also all{e} nombres fro .10. til an
.100. [which] is excluded{e}, with .2. figures mvst be writte; And yf it
be an article, by a cifre first put, and the figure y-writte toward{e}
the lift hond{e}, that signifieth{e} the digit of the which{e} the
article is named{e}; And yf it be a nombre componed{e}, ffirst write the
digit that is a part of that componed{e}, and write to the lift side the
article as it is seid{e} be-fore. All{e} nombre that is fro an
hundred{e} tille a thousand{e} exclused{e}, owith{e} to be writ by .3.
figures; and all{e} nombre that is fro a thousand{e} til .x. Mł. mvst be
writ by .4. figures; And so forthe. And vnderstond{e} wele that eu{er}y
figure sette in the first place signyfieth{e} his digit; In the
second{e} place .10. tymes his digit; In the .3. place an hundred{e} so
moche; In the .4. place a thousand{e} so moche; In the .5. place .x.
thousand{e} so moch{e}; In the .6. place an hundred{e} thousand{e} so
moch{e}; In the .7. place a thousand{e} thousand{e}. And so infynytly
mvltiplying by [*Fol. 49.] these .3. 10, 100, 1000. And vnderstand{e}
wele that competently me may sette vpon figure in the place of a
thousand{e}, a prik{e} to shewe how many thousand{e} the last figure
shall{e} rep{re}sent. We writen{e} in this art to the lift side-ward{e},
as arabien{e} writen{e}, that weren fynders of this science, other{e}
for this reso{u}n, that for to kepe a custumable ordr{e} in redyng,
Sette we all{e}-wey the more nombre before.
[Headnote: Chapter II. Addition.]
[Sidenote: Definition. How the numbers should be written. The method
of working. Begin at the right. The Sum is a digit, or an article,
or a composite.]
Addicio{u}n is of nombre other of nombres vnto nombre or to nombres
aggregacio{u}n, that me may see that that is come therof as
exc{re}ssent. In addicio{u}n, 2. ordres of figures and .2. nombres ben
necessary, that is to sey, a nombre to be added{e} and the nombre wherto
the addic{i}oun shold{e} be made to. The nombre to be added{e} is that
þat shold{e} be added{e} therto, and shall{e} be vnderwriten; the nombre
vnto the which{e} addicio{u}n shall{e} be made to is that nombre that
resceyueth{e} the addicion of þat other, and shall{e} be writen above;
and it is convenient that the lesse nombre be vnderwrit, and the more
added{e}, than the contrary. But whether it happ{e} one other other, the
same comyth{e} of, Therfor, yf þow wilt adde nombre to nombre, write the
nombre wherto the addicio{u}n shall{e} be made in the omest ordre by his
differences, so that the first of the lower ordre be vndre the first of
the omyst ordre, and so of others. That done, adde the first of the
lower ordre to the first of the omyst ordre. And of such{e} addicio{u}n,
other þ{er}e grow{i}t{h} therof a digit, An article, other a
composed{e}. If it be digit{us}, In the place of the omyst shalt thow
write the digit excrescyng, as thus:--
+----------------------------+---+
|The resultant | 2 |
+----------------------------+---+
|To whom it shal be added{e} | 1 |
+----------------------------+---+
|The nombre to be added{e} | 1 |
+----------------------------+---+
If the article; in the place of the omyst put a-way by a cifre writte,
and the digit transferred{e}, of þe which{e} the article toke his name,
toward{e} the lift side, and be it added{e} to the next figure folowyng,
yf ther be any figure folowyng; or no, and yf it be not, leve it [in
the] void{e}, as thus:--
+---------------------------------+----+
| The resultant | 10 |
+---------------------------------+----+
| To whom it shall{e} be added{e} | 7 |
+---------------------------------+----+
| The nombre to be added{e} | 3 |
+---------------------------------+----+
+----------------------+---+---+---+---+---+
| Resultans | 2 | 7 | 8 | 2 | 7 |
+----------------------+---+---+---+---+---+
| Cui d{ebet} addi | 1 | 0 | 0 | 8 | 4 |
+----------------------+---+---+---+---+---+
| Num{erus} addend{us} | 1 | 7 | 7 | 4 | 3 |
+----------------------+---+---+---+---+---+
And yf it happe that the figure folowyng wherto the addicio{u}n shall{e}
be made by [the cifre of] an article, it sette a-side; In his place
write the [*Fol. 49b] [digit of the] Article as thus:--
+---------------------------------+----+
| The resultant | 17 |
+---------------------------------+----+
| To whom it shall{e} be added{e} | 10 |
+---------------------------------+----+
| The nombre to be added{e} | 7 |
+---------------------------------+----+
And yf it happe that a figure of .9. by the figure that me mvst adde
[one] to, In the place of that 9. put a cifre {and} write þe article
toward{e} þe lift hond{e} as bifore, and thus:--
+---------------------------------+----+
| The resultant | 10 |
+---------------------------------+----+
| To whom it shall{e} be added{e} | 9 |
+---------------------------------+----+
| The nombre to be added{e} | 1 |
+---------------------------------+----+
And yf[{3}] [therefrom grow a] nombre componed,[{4}] [in the place of
the nombre] put a-way[{5}][let] the digit [be][{6}]writ þ{a}t is part of
þ{a}t co{m}posid{e}, and þan put to þe lift side the article as before,
and þus:--
+---------------------------------+----+
| The resultant | 12 |
+---------------------------------+----+
| To whom it shall{e} be added{e} | 8 |
+---------------------------------+----+
| The nombre to be added{e} | 4 |
+---------------------------------+----+
This done, adde the seconde to the second{e}, and write above oþ{er} as
before.
[Sidenote: The translator’s note.]
Note wele þ{a}t in addic{i}ons and in all{e} spices folowyng, whan he
seith{e} one the other shall{e} be writen aboue, and me most vse eu{er}
figure, as that eu{er}y figure were sette by half{e}, and by
hym-self{e}.
[Headnote: Chapter III. Subtraction.]
[Sidenote: Definition of Subtraction. How it may be done. What is
required. Write the greater number above. Subtract the first figure
if possible. If it is not possible ‘borrow ten,’ and then subtract.]
Subtraccio{u}n is of .2. p{ro}posed{e} nombres, the fyndyng of the
excesse of the more to the lasse: Other subtraccio{u}n is ablacio{u}n of
o nombre fro a-nother, that me may see a some left. The lasse of the
more, or even of even, may be w{i}t{h}draw; The more fro the lesse may
neu{er} be. And sothly that nombre is more that hath{e} more figures, So
that the last be signyficatife{s}: And yf ther ben as many in that one
as in that other, me most deme it by the last, other by the next last.
More-ou{er} in w{i}t{h}-drawyng .2. nombres ben necessary; A nombre to
be w{i}t{h}draw, And a nombre that me shall{e} w{i}t{h}-draw of. The
nombre to be w{i}t{h}-draw shall{e} be writ in the lower ordre by his
differences; The nombre fro the which{e} me shall{e} with{e}-draw in the
omyst ordre, so that the first be vnder the first, the second{e} vnder
the second{e}, And so of all{e} others. With{e}-draw therfor the first
of the lower{e} ordre fro the first of the ordre above his hede, and
that wolle be other more or lesse, oþ{er} egall{e}.
+---------------------------------+----+
| The remanent | 20 |
+---------------------------------+----+
| Wherof me shall{e} w{i}t{h}draw | 22 |
+---------------------------------+----+
| The nombre to be w{i}t{h}draw | 2 |
+---------------------------------+----+
yf it be egall{e} or even the figure sette beside, put in his place a
cifre. And yf it be more put away þ{er}fro als many of vnitees the lower
figure conteyneth{e}, and writ the residue as thus
+----------------------------------+---+---+
| The remanent | 2 | 2 |
+----------------------------------+---+---+
| Wherof me shall{e} w{i}t{h}-draw | 2 | 8 |
+----------------------------------+---+---+
| Þe nombre to be w{i}t{h}draw | | 6 |
+----------------------------------+---+---+
[*Fol. 50.]
+--------------------------+---+---+-----+---+---+---+---+---+---+
| Remane{n}s | 2 | 2 | 1 | 8 | 2 | 9 | 9 | 9 | 8 |
+--------------------------+---+---+-----+---+---+---+---+---+---+
| A quo sit subtraccio | 8 | 7 | 2 | 4 | 3 | 0 | 0 | 0 | 4 |
+--------------------------+---+---+-----+---+---+---+---+---+---+
| Numerus subt{ra}hend{us} | 6 | 5 |[{7}]|[6]| . | . | . | . | 6 |
+--------------------------+---+---+-----+---+---+---+---+---+---+
And yf it be lesse, by-cause the more may not be w{i}t{h}-draw ther-fro,
borow an vnyte of the next figure that is worth{e} 10. Of that .10. and
of the figure that ye wold{e} have w{i}t{h}-draw fro be-fore to-gedre
Ioyned{e}, w{i}t{h}-draw þe figure be-nethe, and put the residue in the
place of the figure put a-side as þ{us}:--
+----------------------------------+---+---+
| The remanent | 1 | 8 |
+----------------------------------+---+---+
| Wherof me shall{e} w{i}t{h}-draw | 2 | 4 |
+----------------------------------+---+---+
| The nombre to be w{i}t{h}-draw | 0 | 6 |
+----------------------------------+---+---+
[Sidenote: If the second figure is one.]
And yf the figure wherof me shal borow the vnyte be one, put it a-side,
and write a cifre in the place þ{er}of, lest the figures folowing faile
of thair{e} nombre, and þan worch{e} as it shew{i}t{h} in this figure
here:--
+--------------------------------+---+---+------+
| The remanent | 3 | 0 |9[{8}]|
+--------------------------------+---+---+------+
| Wherof me shal w{i}t{h}-draw | 3 | 1 | 2 |
+--------------------------------+---+---+------+
| The nombre to be w{i}t{h}-draw | . | . | 3 |
+--------------------------------+---+---+------+
[Sidenote: If the second figure is a cipher.]
And yf the vnyte wherof me shal borow be a cifre, go ferther to the
figure signyficatif{e}, and ther borow one, and reto{ur}nyng bak{e}, in
the place of eu{er}y cifre þ{a}t ye passid{e} ou{er}, sette figures of
.9. as here it is specified{e}:--
+----------------------------------+---+---+---+---+---+
| The remenaunt | 2 | 9 | 9 | 9 | 9 |
+----------------------------------+---+---+---+---+---+
| Wherof me shall{e} w{i}t{h}-draw | 3 | 0 | 0 | 0 | 3 |
+----------------------------------+---+---+---+---+---+
| The nombre to be w{i}t{h}-draw | | | | | 4 |
+----------------------------------+---+---+---+---+---+
[Sidenote: A justification of the rule given. Why it is better to
work from right to left. How to prove subtraction, and addition.]
And whan me cometh{e} to the nombre wherof me intendith{e}, there
remayneth{e} all{e}-wayes .10. ffor þe which{e} .10. &c. The reson why
þat for eu{er}y cifre left behynde me setteth figures ther of .9. this
it is:--If fro the .3. place me borowed{e} an vnyte, that vnyte by
respect of the figure that he came fro rep{re}sentith an .C., In the
place of that cifre [passed over] is left .9., [which is worth ninety],
and yit it remayneth{e} as .10., And the same reson{e} wold{e} be yf me
had{e} borowed{e} an vnyte fro the .4., .5., .6., place, or ony other so
vpward{e}. This done, withdraw the second{e} of the lower ordre fro the
figure above his hede of þe omyst ordre, and wirch{e} as before. And
note wele that in addicion or in subtracc{i}o{u}n me may wele fro the
lift side begynne and ryn to the right side, But it wol be more
p{ro}fitabler to be do, as it is taught. And yf thow wilt p{ro}ve yf
thow have do wele or no, The figures that thow hast withdraw, adde them
ayene to the omyst figures, and they wolle accorde w{i}t{h} the first
that thow haddest yf thow have labored wele; and in like wise in
addicio{u}n, whan thow hast added{e} all{e} thy figures, w{i}t{h}draw
them that thow first [*Fol. 50b] addest, and the same wolle reto{ur}ne.
The subtraccio{u}n is none other but a p{ro}uff{e} of the addicio{u}n,
and the contrarye in like wise.
[Headnote: Chapter IV. Mediation.]
[Sidenote: Definition of mediation. Where to begin. If the first
figure is unity. What to do if it is not unity.]
Mediacio{u}n is the fyndyng of the halfyng of eu{er}y nombre, that it
may be seyn{e} what and how moch{e} is eu{er}y half{e}. In halfyng ay oo
order of figures and oo nombre is necessary, that is to sey the nombre
to be halfed{e}. Therfor yf thow wilt half any nombre, write that nombre
by his differences, and begynne at the right, that is to sey, fro the
first figure to the right side, so that it be signyficatif{e} other
rep{re}sent vnyte or eny other digitall{e} nombre. If it be vnyte write
in his place a cifre for the figures folowyng, [lest they signify less],
and write that vnyte w{i}t{h}out in the table, other resolue it in .60.
mynvt{es} and sette a-side half of tho m{inutes} so, and reserve the
remen{au}nt w{i}t{h}out in the table, as thus .30.; other sette
w{i}t{h}out thus .{dī}: that kepeth{e} none ordre of place, Nathelesse
it hath{e} signyficacio{u}n. And yf the other figure signyfie any other
digital nombre fro vnyte forth{e}, oþ{er} the nombre is od{e} or
even{e}. If it be even, write this half in this wise:--
+-----------------+---+---+
| Halfed{e} | 2 | 2 |
+-----------------+----+--+
| to be halfed{e} | 4 | 4 |
+-----------------+---+---+
And if it be odde, Take the next even vndre hym conteyned{e}, and put
his half in the place of that odde, and of þe vnyte that remayneth{e} to
be halfed{e} do thus:--
+-----------------+---+---+
| halfed{e} | 2 | 3 | [di]
+-----------------+---+---+
| To be halfed{e} | 4 | 7 |
+-----------------+---+---+
[Sidenote: Then halve the second figure. If it is odd, add 5 to the
figure before.]
This done, the second{e} is to be halfed{e}, yf it be a cifre put it
be-side, and yf it be significatif{e}, other it is even or od{e}: If it
be even, write in the place of þe nombres wiped{e} out the half{e}; yf
it be od{e}, take the next even vnder it co{n}tenyth{e}, and in the
place of the Impar sette a-side put half of the even: The vnyte that
remayneth{e} to be halfed{e}, respect had{e} to them before, is worth{e}
.10. Dyvide that .10. in .2., 5. is, and sette a-side that one, and adde
that other to the next figure p{re}cedent as here:--
+-----------------+---+---+---+
| Halfed{e} | | | |
+-----------------+---+---+---+
| to be halfed{e} | | | |
+-----------------+---+---+---+
And yf þe addicio{u}n shold{e} be made to a cifre, sette it a-side, and
write in his place .5. And vnder this fo{ur}me me shall{e} write and
worch{e}, till{e} the totall{e} nombre be halfed{e}.
+------------------+---+---+---+---+---+----+----+---+
| doubled{e} | 2 | 6 | 8 | 9 | 0 | 10 | 17 | 4 |
+------------------+---+---+---+---+---+----+----+---+
| to be doubled{e} | 1 | 3 | 4 | 4 | 5 | 5 | 8 | 7 |
+------------------+---+---+---+---+---+----+----+---+
[Headnote: Chapter V. Duplation.]
[Sidenote: Definition of Duplation. Where to begin. Why. What to do
with the result.]
Duplicacio{u}n is ag{re}gacion of nombre [to itself] þat me may se the
nombre growen. In doublyng{e} ay is but one ordre of figures necessarie.
And me most be-gynne w{i}t{h} the lift side, other of the more figure,
And after the nombre of the more figure rep{re}sentith{e}. [*Fol. 51.]
In the other .3. before we begynne all{e} way fro the right side and fro
the lasse nombre, In this spice and in all{e} other folowyng we wolle
begynne fro the lift side, ffor and me bigon th{e} double fro the first,
omwhile me myght double oo thynge twyes. And how be it that me myght
double fro the right, that wold{e} be harder in techyng and in workyng.
Therfor yf thow wolt double any nombre, write that nombre by his
differences, and double the last. And of that doubly{n}g other
growith{e} a nombre digital, article, or componed{e}. [If it be a digit,
write it in the place of the first digit.] If it be article, write in
his place a cifre and transferre the article toward{e} the lift, as
thus:--
+------------------+----+
| double | 10 |
+------------------+----+
| to be doubled{e} | 5 |
+------------------+----+
And yf the nombre be componed{e}, write a digital that is part of his
composicio{u}n, and sette the article to the lift hand{e}, as thus:--
+------------------+----+
| doubled{e} | 16 |
+------------------+----+
| to be doubled{e} | 8 |
+------------------+----+
That done, me most double the last save one, and what groweth{e} þ{er}of
me most worche as before. And yf a cifre be, touch{e} it not. But yf any
nombre shall{e} be added{e} to the cifre, in þe place of þe figure
wiped{e} out me most write the nombre to be added{e}, as thus:--
+------------------+---+---+---+
| doubled{e} | 6 | 0 | 6 |
+------------------+---+---+---+
| to be doubled{e} | 3 | 0 | 3 |
+------------------+---+---+---+
[Sidenote: How to prove your answer.]
In the same wise me shall{e} wirch{e} of all{e} others. And this
p{ro}bacio{u}n: If thow truly double the halfis, and truly half the
doubles, the same nombre and figure shall{e} mete, such{e} as thow
labo{ur}ed{e} vpon{e} first, And of the contrarie.
+------------------+---+---+---+
| Doubled{e} | 6 | 1 | 8 |
+------------------+---+---+---+
| to be doubled{e} | 3 | 0 | 9 |
+------------------+---+---+---+
[Headnote: Chapter VI. Multiplication.]
[Sidenote: Definition of Multiplication. Multiplier. Multiplicand.
Product.]
Multiplicacio{u}n of nombre by hym-self other by a-nother, w{i}t{h}
p{ro}posid{e} .2. nombres, [is] the fyndyng of the third{e}, That so
oft conteyneth{e} that other, as ther ben vnytes in the oþ{er}. In
multiplicacio{u}n .2. nombres pryncipally ben necessary, that is to
sey, the nombre multiplying and the nombre to be multiplied{e},
as here;--twies fyve. [The number multiplying] is designed{e}
adu{er}bially. The nombre to be multiplied{e} resceyveth{e} a
no{m}i{n}all{e} appellacio{u}n, as twies .5. 5. is the nombre
multiplied{e}, and twies is the nombre to be multipliede.
+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
| Resultans |[{9}]| 1 | 0 || 1 | 3 | 2 | 6 | 6 | 8 | 0 | 0 | 8 |
+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
| Multiplicand{us}| . | . | 5 || . | . | 4 | . | 3 | 4 | 0 | 0 | 4 |
+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
| Multiplicans | . | 2 | 2 || . | 3 | 3 | 2 | 2 | 2 | . | . | . |
+-----------------+-----+---+---++---+---+---+---+---+---+---+---+---+
Also me may thervpon{e} to assigne the. 3. nombre, the which{e} is
[*Fol. 51b] cleped{e} p{ro}duct or p{ro}venient, of takyng out of one
fro another: as twyes .5 is .10., 5. the nombre to be multiplied{e},
and .2. the multipliant, and. 10. as before is come therof. And
vnderstonde wele, that of the multipliant may be made the nombre to
be multiplied{e}, and of the contrarie, remaynyng eu{er} the same some,
and herof{e} cometh{e} the comen speche, that seith{e} all nombre is
converted{e} by Multiplying in hym-self{e}.
+----+----+----+----+----+--------+----+----+----+-----+
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+----+----+----+----+----+--------+----+----+----+-----+
| 2 | 4 | 6 | 8 | 10 |10[{10}]| 14 | 16 | 18 | 20 |
+----+----+----+----+----+--------+----+----+----+-----+
| 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |
+----+----+----+----+----+--------+----+----+----+-----+
| 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 |
+----+----+----+----+----+--------+----+----+----+-----+
| 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
+----+----+----+----+----+--------+----+----+----+-----+
| 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 56 | 60 |
+----+----+----+----+----+--------+----+----+----+-----+
| 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 |
+----+----+----+----+----+--------+----+----+----+-----+
| 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 |
+----+----+----+----+----+--------+----+----+----+-----+
| 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 |
+----+----+----+----+----+--------+----+----+----+-----+
| 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 |
+----+----+----+----+----+--------+----+----+----+-----+
[Headnote: The Cases of Multiplication.]
[Sidenote: There are 6 rules of Multiplication. (1) Digit by digit.
See the table above. (2) Digit by article. (3) Composite by digit.]
And ther ben .6 rules of Multiplicacio{u}n; ffirst, yf a digit multiplie
a digit, considr{e} how many of vnytees ben betwix the digit by
multiplying and his .10. beth{e} to-gedre accompted{e}, and so oft
w{i}t{h}-draw the digit multiplying, vnder the article of his
deno{m}i{n}acio{u}n. Example of grace. If thow wolt wete how moch{e} is
.4. tymes .8., [{11}]se how many vnytees ben betwix .8.[{12}] and .10.
to-geder rekened{e}, and it shew{i}t{h} that .2.: withdraw ther-for the
quat{e}rnary, of the article of his deno{m}i{n}acion twies, of .40., And
ther remayneth{e} .32., that is, to some of all{e} the
multiplicacio{u}n. Wher-vpon for more evidence and declaracion the
seid{e} table is made. Whan a digit multiplieth{e} an article, thow most
bryng the digit into þe digit, of þe which{e} the article [has][{13}]
his name, and eu{er}y vnyte shall{e} stond{e} for .10., and eu{er}y
article an .100. Whan the digit multiplieth{e} a nombre componed{e},
þ{o}u most bryng the digit into aiþ{er} part of the nombre componed{e},
so þ{a}t digit be had into digit by the first rule, into an article by
þe second{e} rule; and aft{er}ward{e} Ioyne the p{ro}duccio{u}n, and
þ{er}e wol be the some totall{e}.
+----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
|Resultans | 1 | 2 | 6|| 7 | 3 | 6|| 1 | 2 | 0|| 1 | 2 | 0 | 8 |
+----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
|Multiplicand{us}| | | 2|| | 3 | 2|| | | 6|| | | | 4 |
+----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
|Multiplicans | | 6 | 3|| 2 | 3 | || | 2 | 0|| | 3 | 0 | 2 |
+----------------+---+---+--++---+---+--++---+---+--++---+---+---+---+
[Sidenote: (4) Article by article. (5) Composite by article.
(6) Composite by composite. How to set down your numbers. If the
result is a digit, an article, or a composite. Multiply next by
the last but one, and so on.]
Whan an article multiplieth{e} an article, the digit wherof he is
named{e} is to be brought Into the digit wherof the oþ{er} is named{e},
and eu{er}y vnyte wol be worth{e} [*Fol. 52.] an .100., and eu{er}y
article. a .1000. Whan an article multiplieth{e} a nombre componed{e},
thow most bryng the digit of the article into aither part of the nombre
componed{e}; and Ioyne the p{ro}duccio{u}n, and eu{er}y article wol be
worth{e} .100., and eu{er}y vnyte .10., and so woll{e} the some be
open{e}. Whan a nombre componed{e} multiplieth{e} a nombre componed{e},
eu{er}y p{ar}t of the nombre multiplying is to be had{e} into eu{er}y
p{ar}t of the nombre to be multiplied{e}, and so shall{e} the digit be
had{e} twies, onys in the digit, that other in the article. The article
also twies, ones in the digit, that other in the article. Therfor yf
thow wilt any nombre by hym-self other by any other multiplie, write the
nombre to be multiplied{e} in the ou{er} ordre by his differences, The
nombre multiplying in the lower ordre by his differences, so that the
first of the lower ordre be vnder the last of the ou{er} ordre. This
done, of the multiplying, the last is to be had{e} into the last of the
nombre to be multiplied{e}. Wherof than wolle grow a digit, an article,
other a nombre componed{e}. If it be a digit, even above the figure
multiplying is hede write his digit that come of, as it appereth{e}
here:--
+-----------------------+---+
| The resultant | 6 |
+-----------------------+---+
| To be multiplied{e} | 3 |
+-----------------------+---+
| Þe nombre multipliyng | 2 |
+-----------------------+---+
And yf an article had be writ ou{er} the fig{ur}e multiplying his hede,
put a cifre þ{er} and transferre the article toward{e} the lift hand{e},
as thus:--
+-------------------------+---+---+
| The resultant | 1 | 0 |
+-------------------------+---+---+
| to be multiplied{e} | | 5 |
+-------------------------+---+---+
| þe nombre m{u}ltipliyng | | 2 |
+-------------------------+---+---+
And yf a nombre componed{e} be writ ou{er} the figure multyplying is
hede, write the digit in the nombre componed{e} is place, and sette the
article to the lift hand{e}, as thus:--
+------------------------+---+---+
| Resultant | 1 | 2 |
+------------------------+---+---+
| to be multiplied{e} | | 4 |
+------------------------+---+---+
| the nombre multipliyng | | 3 |
+------------------------+---+---+
This done, me most bryng the last save one of the multipliyng into the
last of þe nombre to be multiplied{e}, and se what comyth{e} therof as
before, and so do w{i}t{h} all{e}, tille me come to the first of the
nombre multiplying, that must be brought into the last of the nombre to
be multiplied{e}, wherof growith{e} oþ{er} a digit, an article, [*Fol.
52b] other a nombre componed{e}. If it be a digit, In the place of the
ou{er}er, sette a-side, as here:
+--------------------------+---+---+
| Resultant | 6 | 6 |
+--------------------------+---+---+
| to be multiplied{e} | | 3 |
+--------------------------+---+---+
| the nombre m{u}ltipliyng | 2 | 2 |
+--------------------------+---+---+
If an article happe, there put a cifre in his place, and put hym to the
lift hand{e}, as here:
+-------------------------+---+---+---+
| The resultant | 1 | 1 | 0 |
+-------------------------+---+---+---+
| to be multiplied{e} | | | 5 |
+-------------------------+---+---+---+
| þe nombre m{u}ltiplying | | 2 | 2 |
+-------------------------+---+---+---+
If it be a nombre componed{e}, in the place of the ou{er}er sette
a-side, write a digit that[{14}] is a p{ar}t of the componed{e}, and
sette on the left hond{e} the article, as here:
+-----------------------------+---+-------+---+
| The resultant | 1 |3[{15}]| 2 |
+-----------------------------+---+-------+---+
| to be m{u}ltiplied{e} | | | 4 |
+-----------------------------+---+-------+---+
| þe nombr{e} m{u}ltiplia{n}t | | 3 | 3 |
+-----------------------------+---+-------+---+
[Sidenote: Then antery the multiplier one place. Work as before.
How to deal with ciphers.]
That done, sette forward{e} the figures of the nombre multiplying by oo
difference, so that the first of the multipliant be vnder the last save
one of the nombre to be multiplied{e}, the other by o place sette
forward{e}. Than me shall{e} bryng{e} the last of the m{u}ltipliant in
hym to be multiplied{e}, vnder the which{e} is the first multipliant.
And than wolle growe oþ{er} a digit, an article, or a componed{e}
nombre. If it be a digit, adde hym even above his hede; If it be an
article, transferre hym to the lift side; And if it be a nombre
componed{e}, adde a digit to the figure above his hede, and sette to the
lift hand{e} the article. And all{e}-wayes eu{er}y figure of the nombre
multipliant is to be brought to the last save one nombre to be
multiplied{e}, til me come to the first of the multipliant, where me
shall{e} wirche as it is seid{e} before of the first, and aft{er}ward{e}
to put forward{e} the figures by o difference and one till{e} they
all{e} be multiplied{e}. And yf it happe that the first figure of þe
multipliant be a cifre, and boue it is sette the figure signyficatif{e},
write a cifre in the place of the figur{e} sette a-side, as thus,
{et}c.:
+---------------------+---+---+---+
| The resultant | 1 | 2 | 0 |
+---------------------+---+---+---+
| to be multiplied{e} | | | 6 |
+---------------------+---+---+---+
| the multipliant | | 2 | 0 |
+---------------------+---+---+---+
[Sidenote: How to deal with ciphers.]
And yf a cifre happe in the lower order be-twix the first and the last,
and even above be sette the fig{ur}e signyficatif, leve it vntouched{e},
as here:--
+---------------------+---+---+---+---+---+
| The resultant | 2 | 2 | 6 | 4 | 4 |
+---------------------+---+---+---+---+---+
| To be multiplied{e} | | | 2 | 2 | 2 |
+---------------------+---+---+---+---+---+
| The multipliant | 1 | 0 | 2 | | |
+---------------------+---+---+---+---+---+
And yf the space above sette be void{e}, in that place write thow a
cifre. And yf the cifre happe betwix þe first and the last to be
m{u}ltiplied{e}, me most sette forward{e} the ordre of the figures by
thair{e} differences, for oft of duccio{u}n of figur{e}s in cifres
nought is the resultant, as here,
+-----------------------+---+---+---+---+---+
| Resultant | 8 | 0 | 0 | 8 | |
+-----------------------+---+---+---+---+---+
| to be m{u}ltiplied{e} | 4 | 0 | 0 | 4 | |
+-----------------------+---+---+---+---+---+
| the m{u}ltipliant | 2 | . | . | . | |
+-----------------------+---+---+---+---+---+
[*Fol. 53.] wherof it is evident and open, yf that the first figure of
the nombre be to be multiplied{e} be a cifre, vndir it shall{e} be none
sette as here:--
+-----------------------+---+---+--------+
| Resultant | 3 | 2 |0[{16}] |
+-----------------------+---+---+--------+
| To be m{u}ltiplied{e} | | 8 | 0 |
+-----------------------+---+---+--------+
| The m{u}ltipliant | | 4 | |
+-----------------------+---+---+--------+
[Sidenote: Leave room between the rows of figures.]
Vnder[stand] also that in multiplicacio{u}n, divisio{u}n, and of rootis
the extraccio{u}n, competently me may leve a mydel space betwix .2.
ordres of figures, that me may write there what is come of addyng other
with{e}-drawyng, lest any thynge shold{e} be ou{er}-hipped{e} and sette
out of mynde.
[Headnote: Chapter VII. Division.]
[Sidenote: Definition of division. Dividend, Divisor, Quotient.
How to set down your Sum. An example. Examples.]
For to dyvyde oo nombre by a-nother, it is of .2. nombres p{ro}posed{e},
It is forto depart the moder nombre into as many p{ar}tis as ben of
vnytees in the lasse nombre. And note wele that in makyng{e} of
dyvysio{u}n ther ben .3. nombres necessary: that is to sey, the nombre
to be dyvyded{e}; the nombre dyvydyng and the nombre exeant, other how
oft, or quocient. Ay shall{e} the nombre that is to be dyvyded{e} be
more, other at the lest even{e} w{i}t{h} the nombre the dyvysere, yf the
nombre shall{e} be mad{e} by hole nombres. Therfor yf thow wolt any
nombre dyvyde, write the nombre to be dyvyded{e} in þe ou{er}er
bordur{e} by his differences, the dyviser{e} in the lower ordur{e} by
his differences, so that the last of the dyviser be vnder the last of
the nombre to be dyvyde, the next last vnder the next last, and so of
the others, yf it may competently be done; as here:--
+------------------+---+---+---+
| The residue | | 2 | 7 |
+------------------+---+---+---+
| The quotient | | | 5 |
+------------------+---+---+---+
| To be dyvyded{e} | 3 | 4 | 2 |
+------------------+---+---+---+
| The dyvyser | | 6 | 3 |
+------------------+---+---+---+
+--------------+---+---+----+---+---++---+---+---++---+---+---+
| Residuu{m} | | | 8 || | || | 2 | 7 || | 2 | 6 |
+--------------+---+---+---++---+---++---+---+---++---+---+---+
| Quociens | | 2 | 1 || 2 | 2 || | | 5 || | | 9 |
+--------------+---+---+---++---+---++---+---+---++---+---+---+
| Diuidend{us} | 6 | 8 | 0 || 6 | 6 || 3 | 4 | 2 || 3 | 3 | 2 |
+--------------+---+---+---++---+---++---+---+---++---+---+---+
| Diuiser | 3 | 2 | || 3 | || | 6 | 3 || | 3 | 4 |
+--------------+---+---+---++---+---++---+---+---++---+---+---+
[Sidenote: When the last of the divisor must not be set below the
last of the dividend. How to begin.]
And ther ben .2. causes whan the last figure may not be sette vnder the
last, other that the last of the lower nombre may not be w{i}t{h}-draw
of the last of the ou{er}er nombre for it is lasse than the lower, other
how be it, that it myght be w{i}t{h}-draw as for hym-self fro the
ou{er}er the remenaunt may not so oft of them above, other yf þe last of
the lower be even to the figure above his hede, and þe next last oþ{er}
the figure be-fore þ{a}t be more þan the figure above sette. [*Fol.
53^2.] These so ordeyned{e}, me most wirch{e} from the last figure of þe
nombre of the dyvyser, and se how oft it may be w{i}t{h}-draw of and fro
the figure aboue his hede, namly so that the remen{au}nt may be take of
so oft, and to se the residue as here:--
[Sidenote: An example.]
+------------------+---+---+---+
| The residue | | 2 | 6 |
+------------------+---+---+---+
| The quocient | | | 9 |
+------------------+---+---+---+
| To be dyvyded{e} | 3 | 3 | 2 |
+------------------+---+---+---+
| The dyvyser | | 3 | 4 |
+------------------+---+---+---+
[Sidenote: Where to set the quotiente. Examples.]
And note wele that me may not with{e}-draw more than .9. tymes nether
lasse than ones. Therfor se how oft þe figures of the lower ordre may be
w{i}t{h}-draw fro the figures of the ou{er}er, and the nombre that
shew{i}t{h} þe q{u}ocient most be writ ou{er} the hede of þat figure,
vnder the which{e} the first figure is, of the dyviser; And by that
figure me most with{e}-draw all{e} oþ{er} figures of the lower ordir and
that of the figures aboue thair{e} hedis. This so don{e}, me most sette
forward{e} þe figures of the diuiser by o difference toward{es} the
right hond{e} and worch{e} as before; and thus:--
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
| Residuu{m} | | | | | | || | | | | . | 1 | 2 |
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
| quo{ciens} | | | | 6 | 5 | 4 || | | | 2 | 0 | 0 | 4 |
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
| Diuidend{us} | 3 | 5 | 5 | 1 | 2 | 2 || 8 | 8 | 6 | 3 | 7 | 0 | 4 |
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
| Diuisor | | 5 | 4 | 3 | | || 4 | 4 | 2 | 3 | | | |
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
+------------------+---+---+---+---+---+---+
| The quocient | | | | 6 | 5 | 4 |
+------------------+---+---+---+---+---+---+
| To be dyvyded{e} | 3 | 5 | 5 | 1 | 2 | 2 |
+------------------+---+---+---+---+---+---+
| The dyvyser | | 5 | 4 | 3 | | |
+------------------+---+---+---+---+---+---+
[Sidenote: A special case.]
And yf it happ{e} after þe settyng forward{e} of the fig{ur}es þ{a}t þe
last of the divisor may not so oft be w{i}t{h}draw of the fig{ur}e above
his hede, above þat fig{ur}e vnder the which{e} the first of the diuiser
is writ me most sette a cifre in ordre of the nombre quocient, and sette
the fig{ur}es forward{e} as be-fore be o difference alone, and so me
shall{e} do in all{e} nombres to be dyvided{e}, for where the dyviser
may not be w{i}t{h}-draw me most sette there a cifre, and sette
forward{e} the figures; as here:--
+------------------+---+---+---+---+---+---+---+
| The residue | | | | | | 1 | 2 |
|------------------+---+---+---+---+---+---+---+
| The quocient | | | | 2 | 0 | 0 | 4 |
|------------------+---+---+---+---+---+---+---+
| To be dyvyded{e} | 8 | 8 | 6 | 3 | 7 | 0 | 4 |
|------------------+---+---+---+---+---+---+---+
| The dyvyser | 4 | 4 | 2 | 3 | | | |
+------------------+---+---+---+---+---+---+---+
[Sidenote: Another example. What the quotient shows. How to prove
your division, or multiplication.]
And me shall{e} not cesse fro such{e} settyng of fig{ur}es forward{e},
nether of settyng{e} of þe quocient into the dyviser, neþ{er} of
subt{ra}ccio{u}n of the dyvyser, till{e} the first of the dyvyser be
w{i}t{h}-draw fro þe first to be divided{e}. The which{e} don{e}, or
ought,[{17}] oþ{er} nought shall{e} remayne: and yf it be ought,[{17}]
kepe it in the tables, And eu{er} vny it to þe diviser. And yf þ{o}u
wilt wete how many vnytees of þe divisio{u}n [*Fol. 53^3.] wol growe to
the nombre of the diviser{e}, the nombre quocient wol shewe it: and whan
such{e} divisio{u}n is made, and þ{o}u lust p{ro}ve yf thow have wele
done or no, Multiplie the quocient by the diviser, And the same
fig{ur}es wolle come ayene that thow haddest bifore and none other. And
yf ought be residue, than w{i}t{h} addicio{u}n therof shall{e} come the
same figures: And so multiplicacio{u}n p{ro}vith{e} divisio{u}n, and
dyvisio{u}n multiplicacio{u}n: as thus, yf multiplicacio{u}n be made,
divide it by the multipliant, and the nombre quocient wol shewe the
nombre that was to be multiplied{e}, {et}c.
[Headnote: Chapter VIII. Progression.]
[Sidenote: Definition of Progression. Natural Progression. Broken
Progression. The 1st rule for Natural Progression. The second rule.
The first rule of Broken Progression. The second rule.]
Progressio{u}n is of nombre after egall{e} excesse fro oone or tweyn{e}
take ag{r}egacio{u}n. of p{ro}gressio{u}n one is naturell{e} or
co{n}tynuell{e}, þ{a}t oþ{er} broken and discontynuell{e}. Naturell{e}
it is, whan me begynneth{e} w{i}t{h} one, and kepeth{e} ordure
ou{er}lepyng one; as .1. 2. 3. 4. 5. 6., {et}c., so þ{a}t the nombre
folowyng{e} passith{e} the other be-fore in one. Broken it is, whan me
lepith{e} fro o nombre till{e} another, and kepith{e} not the contynuel
ordir{e}; as 1. 3. 5. 7. 9, {et}c. Ay me may begynne w{i}t{h} .2., as
þus; .2. 4. 6. 8., {et}c., and the nombre folowyng passeth{e} the others
by-fore by .2. And note wele, that naturell{e} p{ro}gressio{u}n ay
begynneth{e} w{i}t{h} one, and Int{er}cise or broken p{ro}gressio{u}n,
omwhile begynnyth{e} w{i}th one, omwhile w{i}t{h} twayn{e}. Of
p{ro}gressio{u}n naturell .2. rules ther be yove, of the which{e} the
first is this; whan the p{ro}gressio{u}n naturell{e} endith{e} in even
nombre, by the half therof multiplie þe next totall{e} ou{er}er{e}
nombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10.
cometh{e} of, that is the totall{e} nombre þ{er}of. The second{e} rule
is such{e}, whan the p{ro}gressio{u}n naturell{e} endith{e} in nombre
od{e}. Take the more porcio{u}n of the oddes, and multiplie therby the
totall{e} nombre. Example of grace 1. 2. 3. 4. 5., multiplie .5. by .3,
and thryes .5. shall{e} be resultant. so the nombre totall{e} is .15. Of
p{ro}gresio{u}n int{er}cise, ther ben also .2.[{18}] rules; and þe first
is þis: Whan the Int{er}cise p{ro}gression endith{e} in even nombre by
half therof multiplie the next nombre to þat half{e} as .2.[{18}] 4. 6.
Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of all{e}
the p{ro}gressio{u}n, woll{e} folow. The second{e} rule is this: whan
the p{ro}gressio{u}n int{er}scise endith{e} in od{e}, take þe more
porcio{u}n of all{e} þe nombre, [*Fol. 53^4.] and multiplie by
hym-self{e}; as .1. 3. 5. Multiplie .3. by hym-self{e}, and þe some of
all{e} wolle be .9., {et}c.
[Headnote: Chapter IX. Extraction of Roots.]
[Sidenote: The preamble of the extraction of roots. Linear,
superficial, and solid numbers. Superficial numbers. Square numbers.
The root of a square number. Notes of some examples of square roots
here interpolated. Solid numbers. Three dimensions of solids. Cubic
numbers. All cubics are solid numbers. No number may be both linear
and solid. Unity is not a number.]
Here folowith{e} the extraccio{u}n of rotis, and first in nombre
q{ua}drat{es}. Wherfor me shall{e} se what is a nombre quadrat, and what
is the rote of a nombre quadrat, and what it is to draw out the rote of
a nombre. And before other note this divisio{u}n: Of nombres one is
lyneal, anoþ{er} sup{er}ficiall{e}, anoþ{er} quadrat, anoþ{er} cubik{e}
or hoole. lyneal is that þat is considred{e} after the p{ro}cesse,
havyng{e} no respect to the direccio{u}n of nombre in nombre, As a lyne
hath{e} but one dymensio{u}n that is to sey after the length{e}. Nombre
sup{er}ficial is þ{a}t cometh{e} of ledyng{e} of oo nombre into
a-nother, wherfor it is called{e} sup{er}ficial, for it hath{e} .2.
nombres notyng or mesuryng{e} hym, as a sup{er}ficiall{e} thyng{e}
hath{e} .2. dimensions, þ{a}t is to sey length{e} and brede. And for
bycause a nombre may be had{e} in a-nother by .2. man{er}s, þ{a}t is to
sey other in hym-self{e}, oþ{er} in anoþ{er}, Vnderstond{e} yf it be had
in hym-self, It is a quadrat. ffor dyvisio{u}n write by vnytes, hath{e}
.4. sides even as a quadrangill{e}. and yf the nombre be had{e} in
a-noþ{er}, the nombre is sup{er}ficiel and not quadrat, as .2. had{e} in
.3. maketh{e} .6. that is þe first nombre sup{er}ficiell{e}; wherfor it
is open þat all{e} nombre quadrat is sup{er}ficiel, and not
co{n}u{er}tid{e}. The rote of a nombre quadrat is þat nombre that is had
of hym-self, as twies .2. makith{e} 4. and .4. is the first nombre
quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of
the more quadrat .3. 1. 4. 2. 6. The most nombre quadrat 9. 8. 7. 5.
9. 3. 4. 7. 6. / the remenent ou{er} the quadrat .6. 0. 8. 4. 5. / The
first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The
second{e} caas .3. 8. 4. 5. The rote .6. 2. The third{e} caas .2. 8. 1.
9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas
.9. 1. 2. 0. 4. / The rote 3. 0. 2. The solid{e} nombre or cubik{e} is
þat þ{a}t comytħe of double ledyng of nombre in nombre; And it is
cleped{e} a solid{e} body that hath{e} þ{er}-in .3 [dimensions] þat is
to sey, length{e}, brede, and thiknesse. so þ{a}t nombre hath{e} .3.
nombres to be brought forth{e} in hym. But nombre may be had{e} twies in
nombre, for other it is had{e} in hym-self{e}, oþ{er} in a-noþ{er}. If a
nombre be had{e} twies in hym-self, oþ{er} ones in his quadrat, þ{a}t is
the same, þ{a}t a cubik{e} [*Fol. 54.] is, And is the same that is
solide. And yf a nombre twies be had{e} in a-noþ{er}, the nombre is
cleped{e} solide and not cubik{e}, as twies .3. and þ{a}t .2. makith{e}
.12. Wherfor it is opyn{e} that all{e} cubik{e} nombre is solid{e}, and
not {con}u{er}tid{e}. Cubik{e} is þ{a}t nombre þat comyth{e} of
ledyng{e} of hym-self{e} twyes, or ones in his quadrat. And here-by it
is open that o nombre is the roote of a quadrat and of a cubik{e}.
Natheles the same nombre is not q{ua}drat and cubik{e}. Opyn{e} it is
also that all{e} nombres may be a rote to a q{ua}drat and cubik{e}, but
not all{e} nombre quadrat or cubik{e}. Therfor sithen þe ledyng{e} of
vnyte in hym-self ones or twies nought cometh{e} but vnytes, Seith{e}
Boice in Arsemetrik{e}, that vnyte potencially is al nombre, and none in
act. And vndirstond{e} wele also that betwix euery .2. quadrat{es} ther
is a meene p{ro}porcionall{e}, That is opened{e} thus; lede the rote of
o quadrat into the rote of the oþ{er} quadrat, and þan wolle þe meene
shew.
[Sidenote: Examples of square roots.]
+-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
| Residuu{m} | | |0| || | | |4|| | |0| | || | | 0 | |
+-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
| Quadrand{e} |4|3|5|6||3|0|2|9||1|7|4|2|4||1| 9 | 3 |6|
+-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
| Duplum |1|2| | ||1|0| | ||2| |6| | || |[8]|[{19}]| |
+-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
| Subduplu{m} | |6| |6|| |5| |5||1| |3| |2|| | 4 | |4|
+-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+
[Sidenote: A note on mean proportionals.]
Also betwix the next .2. cubikis, me may fynde a double meene, that is
to sey a more meene and a lesse. The more meene thus, as to bryng{e} the
rote of the lesse into a quadrat of the more. The lesse thus, If the
rote of the more be brought Into the quadrat of the lesse.
[Headnote: Chapter X. Extraction of Square Root.]
[Sidenote: To find a square root. Begin with the last odd place.
Find the nearest square root of that number, subtract, double it,
and set the double one to the right. Find the second figure by
division. Multiply the double by the second figure, and add after
it the square of the second figure, and subtract.]
[{20}]To draw a rote of the nombre quadrat it is What-eu{er} nombre be
p{ro}posed{e} to fynde his rote and to se yf it be quadrat. And yf it be
not quadrat the rote of the most quadrat fynde out, vnder the nombre
p{ro}posed{e}. Therfor yf thow wilt the rote of any quadrat nombre draw
out, write the nombre by his differences, and compt the nombre of the
figures, and wete yf it be od{e} or even. And yf it be even, than most
thow begynne worche vnder the last save one. And yf it be od{e} w{i}t{h}
the last; and forto sey it shortly, al-weyes fro the last od{e} me
shall{e} begynne. Therfor vnder the last in an od place sette, me most
fynd{e} a digit, the which{e} lad{e} in hym-self{e} it puttith{e} away
that, þat is ou{er} his hede, oþ{er} as neigh{e} as me may: suche a
digit found{e} and w{i}t{h}draw fro his ou{er}er, me most double that
digit and sette the double vnder the next figure toward{e} the right
hond{e}, and his vnder double vnder hym. That done, than me most
fy{n}d{e} a-noþ{er} digit vnder the next figure bifore the doubled{e},
the which{e} [*Fol. 54b] brought in double setteth{e} a-way all{e} that
is ou{er} his hede as to reward{e} of the doubled{e}: Than brought into
hym-self settith{e} all away in respect of hym-self, Other do it as nye
as it may be do: other me may w{i}t{h}-draw the digit [{21}][last]
found{e}, and lede hym in double or double hym, and after in
hym-self{e}; Than Ioyne to-geder the p{ro}duccion{e} of them bothe, So
that the first figure of the last p{ro}duct be added{e} before the first
of the first p{ro}duct{es}, the second{e} of the first, {et}c. and so
forth{e}, subtrahe fro the totall{e} nombre in respect of þe digit.
[Sidenote: Examples.]
+------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
| The residue | | | | | || | | | | || | | |5| 4 |3|2|
+------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
| To be quadred{e} |4|1|2|0|9||1|5|1|3|9|| 9 |0| 0 |5| 4 |3|2|
+------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
| The double | |4|0| | || |2| |4| || |6| |0| | |0|
+------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
| The vnder double |2| |0| |3||1| |2| |3||[3]| |[0]| |[0]| |0|
+------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+
[Sidenote: Special cases. The residue.]
And if it hap þ{a}t no digit may be found{e}, Than sette a cifre vndre
a cifre, and cesse not till{e} thow fynde a digit; and whan thow hast
founde it to double it, neþ{er} to sette the doubled{e} forward{e}
nether the vnder doubled{e}, Till thow fynde vndre the first figure a
digit, the which{e} lad{e} in all{e} double, settyng away all{e} that is
ou{er} hym in respect of the doubled{e}: Than lede hym into hym-self{e},
and put a-way all{e} in regard{e} of hym, other as nygh{e} as thow
maist. That done, other ought or nought wolle be the residue. If nought,
than it shewith{e} that a nombre componed{e} was the quadrat, and his
rote a digit last found{e} w{i}t{h} vnder{e}-double other vndirdoubles,
so that it be sette be-fore: And yf ought[{22}] remayn{e}, that
shew{i}t{h} that the nombre p{ro}posed{e} was not quadrat,[{23}]
[[wher-vpon{e} se the table in the next side of the next leef{e}.]]
but a digit [last found with the subduple or subduples is]
[Sidenote: This table is constructed for use in cube root sums,
giving the value of ab.^2]
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 2 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 |
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 3 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 |
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 4 | 32 | 48 | 64 | 80 | 96 |112[{24}]| 128 | 144 |
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 5 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225 |
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 6 | 72 | 108 | 144 | 180 | 216 | 252 | 288 | 324 |
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 7 | 98 | 147 | 196 | 245 | 294 | 343 | 393 | 441 |
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 8 | 128 | 192 | 256 | 320 | 384 | 448 | 512 | 576 |
+---+-----+-----+-----+-----+-----+---------+-----+---------+
| 9 | 168 | 243 | 324 | 405 | 486 | 567 | 648 |729[{25}]|
+---+-----+-----+-----+-----+-----+---------+-----+---------+
[Sidenote: How to prove the square root without or with a remainder.]
The rote of the most quadrat conteyned{e} vndre the nombre
p{ro}posed{e}. Therfor yf thow wilt p{ro}ve yf thow have wele do or no,
Multiplie the digit last found{e} w{i}t{h} the vnder-double oþ{er}
vnder-doublis, and thow shalt fynde the same figures that thow haddest
before; And so that nought be the [*Fol. 55.] residue. And yf thow have
any residue, than w{i}t{h} the addicio{u}n þ{er}of that is res{er}ued{e}
w{i}t{h}-out in thy table, thow shalt fynd{e} thi first figures as thow
haddest them before, {et}c.
[Headnote: Chapter XI. Extraction of Cube Root.]
[Sidenote: Definition of a cubic number and a cube root. Mark off
the places in threes. Find the first digit; treble it and place it
under the next but one, and multiply by the digit. Then find the
second digit. Multiply the first triplate and the second digit, twice
by this digit. Subtract. Examples.]
Heere folowith{e} the extraccio{u}n of rotis in cubik{e} nombres;
wher-for me most se what is a nombre cubik{e}, and what is his roote,
And what is the extraccio{u}n of a rote. A nombre cubik{e} it is, as it
is before declared{e}, that cometh{e} of ledyng of any nombre twies in
hym-self{e}, other ones in his quadrat. The rote of a nombre cubik{e} is
the nombre that is twies had{e} in hy{m}-self{e}, or ones in his
quadrat. Wher-thurgh{e} it is open, that eu{er}y nombre quadrat or
cubik{e} have the same rote, as it is seid{e} before. And forto draw out
the rote of a cubik{e}, It is first to fynd{e} þe nombr{e} p{ro}posed{e}
yf it be a cubik{e}; And yf it be not, than thow most make extraccio{u}n
of his rote of the most cubik{e} vndre the nombre p{ro}posid{e} his rote
found{e}. Therfor p{ro}posed{e} some nombre, whos cubical rote þ{o}u
woldest draw out; First thow most compt the figures by fourthes, that is
to sey in the place of thousand{es}; And vnder the last thousand{e}
place, thow most fynde a digit, the which{e} lad{e} in hym-self cubikly
puttith{e} a-way that þat is ou{er} his hede as in respect of hym, other
as nygh{e} as thow maist. That done, thow most trebill{e} the digit, and
that triplat is to be put vnder the .3. next figure toward{e} the right
hond{e}, And the vnder-trebill{e} vnder the trebill{e}; Than me most
fynd{e} a digit vndre the next figure bifore the triplat, the which{e}
w{i}t{h} his vnder-trebill{e} had into a trebill{e}, aft{er}warde other
vnder[trebille][{26}] had in his p{ro}duccio{u}n, putteth{e} a-way
all{e} that is ou{er} it in regard{e} of[{27}] [the triplat. Then lade
in hymself puttithe away that þat is over his hede as in respect of hym,
other as nyghe as thou maist:] That done, thow most trebill{e} the digit
ayene, and the triplat is to be sette vnder the next .3. figure as
before, And the vnder-trebill{e} vnder the trebill{e}: and than most
thow sette forward{e} the first triplat w{i}t{h} his vndre-trebill{e} by
.2. differences. And than most thow fynde a digit vnder the next figure
before the triplat, the which{e} with{e} his vnder-t{r}iplat had in his
triplat afterward{e}, other vnder-treblis lad in p{ro}duct [*Fol. 55b]
It sitteth{e} a-way ałł that is ou{er} his hede in respect of the
triplat than had in hym-self cubikly,[{28}] [[it setteth{e} a-way all{e}
his respect]] or as nygh{e} as ye may.
+----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
| Residuu{m} | | | | | | | 5 || | | | | | 4|| 1|0|1 |9| |
+----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
| Cubicandu{s} | 8|3|6|5|4|3| 2 || 3|0|0|7|6| 7|| 1 1|6|6 |7| |
+----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
| Triplum | | |6|0| | | || | | |1|8| || | |4 | | |
+----------------+--+-+-+-+-+-+---++--+-+-+---+--++----+-+--+-+--+
| Subt{r}iplu{m} | 2| | |0| | |[3]|| | |6| | | 7|| 2| | |2| |
+----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+
[Sidenote: Continue this process till the first figure is reached.
Examples. The residue. Special cases. Special case.]
Nother me shall{e} not cesse of the fyndyng{e} of that digit, neither of
his triplacio{u}n, neþ{er} of the triplat-is [{29}]anteriorac{i}o{u}n,
that is to sey, settyng forward{e} by .2. differences, Ne therof the
vndre-triple to be put vndre the triple, Nether of the multiplicacio{u}n
þ{er}of, Neither of the subtraccio{u}n, till{e} it come to the first
figure, vnder the which{e} is a digitall{e} nombre to be found{e}, the
which{e} with{e} his vndre-treblis most be had{e} in tribles,
After-ward{e} w{i}t{h}out vnder-treblis to be had{e} into produccio{u}n,
settyng away all{e} that is ou{er} the hed{e} of the triplat nombre,
After had into hymself{e} cubikly, and sette all{e}-way that is ou{er}
hym.
+------------------+---+---+---+---++---+---+---+---+---+
| To be cubiced{e} | 1 | 7 | 2 | 8 || 3 | 2 | 7 | 6 | 8 |
+------------------+---+---+---+---++---+---+---+---+---+
| The triple | | | 3 | 2 || | | | 9 | |
+------------------+---+---+---+---++---+---+---+---+---+
| The vnder triple | | | 1 | 2 || |[3]| | 3 | 3 |
+------------------+---+---+---+---++---+---+---+---+---+
Also note wele that the p{ro}ducc{i}on comyng{e} of the ledyng of a
digite found{e}[{30}] [[w{i}t{h} an vndre-triple / other of an
vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h} out
vndre-triple other vndre-triplis in the p{ro}duct and ayene that
p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e} in
hym-self{e} cubicall{e}]] me may adde to, and also w{i}t{h}-draw fro of
the totall{e} nombre sette above that digit so found{e}.[{31}] [[as ther
had be a divisio{u}n made as it is opened{e} before]] That done ought or
nought most be the residue. If it be nought, It is open that the nombre
p{ro}posed{e} was a cubik{e} nombre, And his rote a digit founde last
w{i}t{h} the vnder-triples: If the rote therof wex bad{e} in
hym-self{e}, and afterward{e} p{ro}duct they shall{e} make the first
fig{ur}es. And yf ought be in residue, kepe that w{i}t{h}out in the
table; and it is open{e} that the nombre was not a cubik{e}. but a digit
last founde w{i}t{h} the vndirtriplis is rote of the most cubik{e} vndre
the nombre p{ro}posed{e} conteyned{e}, the which{e} rote yf it be had{e}
in hym-self{e}, And aft{er}ward{e} in a p{ro}duct of that shall{e} growe
the most cubik{e} vndre the nombre p{ro}posed{e} conteyned{e}, And yf
that be added{e} to a cubik{e} the residue res{er}ued{e} in the table,
woll{e} make the same figures that ye had{e} first. [*Fol. 56.] And yf
no digit after the anterioracio{u}n[{32}] may not be found{e}, than put
ther{e} a cifre vndre a cifre vndir the third{e} figure, And put
forward{e} þe fig{ur}es. Note also wele that yf in the nombre
p{ro}posed{e} ther ben no place of thowsand{es}, me most begynne vnder
the first figure in the extraccio{u}n of the rote. some vsen forto
distingue the nombre by threes, and ay begynne forto wirch{e} vndre the
first of the last t{er}nary other unco{m}plete nombre, the which{e}
maner of op{er}acio{u}n accordeth{e} w{i}t{h} that before. And this at
this tyme suffiseth{e} in extraccio{u}n of nombres quadrat or cubik{es}
{et}c.
[Sidenote: Examples.]
+-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
| The residue | | | | | | | 0|| | | | | | 1| 1|
+-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
| The cubicand{us} | 8 | 0| 0 | 0| 0| 0| 0|| 8| 2| 4| 2| 4| 1| 9|
+-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
| The triple | | |[{33}]| 0| 0| | || | | 6| | | | |
+-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
| The vndert{r}iple |[2]| | | 0| 0| | || 2| | | 6| 2| | |
+-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+
[Headnote: Table of Numbers, &c.]
[Sidenote: A table of numbers; probably from the Abacus.]
1 2 3 4 5 6
one. x. an. hundred{e}/ a thowsand{e}/ x. thowsand{e}/ An hundred{e}
7
thowsand{e}/ A thowsand{e} tymes a thowsand{e}/ x. thousand{e} tymes
a thousand{e}/ An hundred{e} thousand{e} tymes a thousand{e} A
thousand{e} thousand{e} tymes a thousand{e}/ this is the x place
{et}c.
[Ende.]
FOOTNOTES (The Art of Nombryng):
[1: MS. Materiall{e}.]
[2: MS. Formall{e}.]
[3: ‘the’ in MS.]
[4: ‘be’ in MS.]
[5: ‘and’ in MS.]
[6: ‘is’ in MS.]
[7: 6 in MS.]
[8: 0 in MS.]
[9: 2 in MS.]
[10: _sic._]
[11: ‘And’ inserted in MS.]
[12: ‘4 the’ inserted in MS.]
[13: ‘to’ in MS.]
[14: ‘that’ repeated in MS.]
[15: ‘1’ in MS.]
[16: Blank in MS.]
[17: ‘nought’ in MS.]
[18: 3 written for 2 in MS.]
[19: 7 in MS.]
[20: runs on in MS.]
[21: ‘so’ in MS.]
[22: ‘nought’ in MS.]
[23: MS. adds here: ‘wher-vpon{e} se the table in the next side of
the next leef{e}.’]
[24: 110 in MS.]
[25: 0 in MS.]
[26: double in MS.]
[27: ‘it hym-self{e}’ in MS.]
[28: MS. adds here: ‘it setteth{e} a-way all{e} his respect.’]
[29: ‘aucterioracio{u}n’ in MS.]
[30: MS. adds here: ’w{i}t{h} an vndre-triple / other of an
vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h}
out vndre-triple other vndre-triplis in the p{ro}duct and ayene
that p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e}
in hym-self{e} cubicall{e}’ /]
[31: MS. adds here: ‘as ther had be a divisio{u}n made as it is
opened{e} before.’]
[32: MS. anteriocacio{u}n.]
[33: 4 in MS.]
Accomptynge by counters.
[Transcriber’s Note:
The original text was printed as a single continuous paragraph, with
no break between speakers; all examples were shown inline. It has been
broken up for this e-text.]
[*116b]
¶ The seconde dialoge of accomptynge by counters.
_Mayster._
Nowe that you haue learned the commen kyndes of Arithmetyke with the
penne, you shall se the same art in cou{n}ters: whiche feate doth not
only serue for them that can not write and rede, but also for them that
can do bothe, but haue not at some tymes theyr penne or tables redye
with them. This sorte is in two fourmes co{m}menly. The one by lynes,
and the other without lynes: in that y^t hath lynes, the lynes do stande
for the order of places: and in y^t that hath no lynes, there must be
sette in theyr stede so many counters as shall nede, for eche lyne one,
and they shall supplye the stede of the lynes.
_S._ By examples I shuld better p{er}ceaue your meanynge.
_M._ For example of the [*117a.] ly[*]nes:
----1-0-0-0-0-0--
----1-0-0-0-0----
-X--1-0-0-0------
----1-0-0--------
----1-0----------
----1------------
[Sidenote: Numeration.]
Lo here you se .vi. lynes whiche stande for syxe places so that the
nethermost standeth for y^e fyrst place, and the next aboue it, for the
second: and so vpward tyll you come to the hyghest, which is the syxte
lyne, and standeth for the syxte place. Now what is the valewe of euery
place or lyne, you may perceaue by the figures whiche I haue set on
them, which is accordynge as you learned before in the Numeration of
figures by the penne: for the fyrste place is the place of vnities or
ones, and euery counter set in that lyne betokeneth but one: {and} the
seconde lyne is the place of 10, for euery counter there, standeth for
10. The thyrd lyne the place of hundredes: the fourth of thousandes:
{and} so forth.
_S._ Syr I do perceaue that the same order is here of lynes, as was in
the other figures [*117b] by places, so that you shall not nede longer
to stande about Numeration, excepte there be any other difference.
_M._ Yf you do vndersta{n}de it, then how wyll you set 1543?
_S._ Thus, as I suppose.
-------
-X--1--
----5--
----4--
----3--
_M._ You haue set y^e places truely, but your figures be not mete for
this vse: for the metest figure in this behalfe, is the figure of a
cou{n}ter round, as you se here, where I haue expressed that same summe.
-------------
-X--o--------
o
-------------
----o-o-o-o--
----o-o-o----
_S._ So that you haue not one figure for 2, nor 3, nor 4, and so forth,
but as many digettes as you haue, you set in the lowest lyne: and for
euery 10 you set one in the second line: and so of other. But I know not
by what reason you set that one counter for 500 betwene two lynes.
_M._ you shall remember this, that when so euer you nede to set downe 5,
50, or 500, or 5000, or so forth any other nomber, whose numerator
[*118a] is 5, you shall set one counter for it, in the next space aboue
the lyne that it hath his denomination of, as in this example of that
500, bycause the numerator is 5, it must be set in a voyd space: and
bycause the denominator is hundred, I knowe that his place is the voyde
space next aboue hundredes, that is to say, aboue the thyrd lyne. And
farther you shall marke, that in all workynge by this sorte, yf you
shall sette downe any summe betwene 4 and 10, for the fyrste parte of
that nomber you shall set downe 5, & then so many counters more, as
there reste no{m}bers aboue 5. And this is true bothe of digettes and
articles. And for example I wyll set downe this su{m}me 287965,
-X-----------
------o-o----
o
------o-o-o--
o
-X----o-o----
o
----o-o-o-o--
o
----o--------
o
-------------
which su{m}me yf you marke well, you nede none other exa{m}ples for to
lerne the numeration of [*118b] this forme. But this shal you marke,
that as you dyd in the other kynde of arithmetike, set a pricke in the
places of thousa{n}des, in this worke you shall sette a starre, as you
se here.
[Headnote: Addition on the Counting Board.]
[Sidenote: Addition.]
_S._ Then I perceave numeration, but I praye you, howe shall I do in
this arte to adde two summes or more together?
_M._ The easyest way in this arte is, to adde but 2 su{m}mes at ones
together: how be it you may adde more, as I wyll tell you anone.
Therfore when you wyll adde two su{m}mes, you shall fyrst set downe one
of them, it forseth not whiche, {and} then by it drawe a lyne crosse the
other lynes. And afterward set downe the other su{m}me, so that that
lyne may be betwene them, as yf you wolde adde 2659 to 8342, you must
set your su{m}mes as you se
-------------|-----------
o |
-X--o-o-o----|--o-o------
| o
----o-o-o----|--o--------
| o
----o-o-o-o--|-----------
| o
----o-o------|--o-o-o-o--
here. And then yf you lyst, you [*119a] may adde the one to the other in
the same place, or els you may adde them both together in a newe place:
which waye, bycause it is moste playnest, I wyll showe you fyrst.
Therfore wyl I begynne at the vnites, whiche in the fyrst su{m}me is but
2, {and} in y^e second su{m}me 9, that maketh 11, those do I take vp,
and for them I set 11 in the new roume, thus,
-------------|-------|-------
o | |
-X--o-o-o----|--o-o--|-------
| o |
----o-o-o----|--o----|-------
| o |
----o-o-o-o--|-------|-o-----
| |
-------------|-------|-o-----
Then do I take vp all y^e articles vnder a hundred, which in the fyrst
su{m}me are 40, and in the second summe 50, that maketh 90: or you may
saye better, that in the fyrste summe there are 4 articles of 10, and in
the seconde summe 5, which make 9, but then take hede that you sette
them in theyr [*119b] ryght lynes as you se here.
-----------|----------|-------------
o | |
-X--o-o-o--|--o-o-----|-------------
| o |
----o-o-o--|--o-------|-------------
| | o
-----------|----------|--o-o-o-o-o--
| |
-----------|----------|--o----------
Where I haue taken awaye 40 fro{m} the fyrste su{m}me, and 50 from y^e
second, and in theyr stede I haue set 90 in the thyrde, whiche I haue
set playnely y^t you myght well perceaue it: how be it seynge that 90
with the 10 that was in y^e thyrd roume all redy, doth make 100,
I myghte better for those 6 cou{n}ters set 1 in the thyrde lyne, thus:
----------
-X--------
----o-----
----------
----o-----
For it is all one summe as you may se, but it is beste, neuer to set 5
cou{n}ters in any line, for that may be done with 1 cou{n}ter in a
hygher place.
_S._ I iudge that good reaso{n}, for many are vnnedefull, where one wyll
serue.
_M._ Well, then [*120a] wyll I adde forth of hundredes: I fynde 3 in the
fyrste summe, and 6 in the seconde, whiche make 900, them do I take vp
{and} set in the thyrd roume where is one hundred all redy, to whiche I
put 900, and it wyll be 1000, therfore I set one cou{n}ter in the fourth
lyne for them all, as you se here.
-----------|-------|--------
o | |
-X--o-o-o--|--o-o--|--o-----
| |
-----------|-------|--------
| |
-----------|-------|--------
| |
-----------|-------|--o-----
Then adde I y^e thousandes together, whiche in the fyrst su{m}me are
8000, {and} in y^e second 2000, that maketh 10000: them do I take vp
fro{m} those two places, and for them I set one counter in the fyfte
lyne, and then appereth as you se, to be 11001, for so many doth amount
of the addition of 8342 to 2659.
----o-----
-X--o-----
----------
----------
----o-----
[*120b] _S._ Syr, this I do perceave: but how shall I set one su{m}me to
an other, not chaungynge them to a thyrde place?
_M._ Marke well how I do it: I wyll adde together 65436, and 3245,
whiche fyrste I set downe thus.
-------------|--------------
| o
-------------|--o-----------
| o
-X--o-o-o----|--------------
|
----o-o------|--o-o-o-o-----
|
----o-o-o-o--|--o-o-o-------
o | o
-------------|--o-----------
Then do I begynne with the smalest, which in the fyrst summe is 5, that
do I take vp, and wold put to the other 5 in the seconde summe, sauynge
that two counters can not be set in a voyd place of 5, but for them
bothe I must set 1 in the seconde lyne, which is the place of 10,
therfore I take vp the 5 of the fyrst su{m}me, {and} the 5 of the
seco{n}de, and for them I set 1 in the seco{n}d lyne, [*121a] as you se
here.
-------------|--------------
| o
-------------|--o-----------
| o
-X--o-o-o----|--------------
|
----o-o------|--o-o-o-o-----
|
----o-o-o-o--|--o-o-o-o-----
|
-------------|--o-----------
Then do I lyke wayes take vp the 4 counters of the fyrste su{m}me {and}
seconde lyne (which make 40) and adde them to the 4 counters of the same
lyne, in the second su{m}me, and it maketh 80, But as I sayde I maye not
conueniently set aboue 4 cou{n}ters in one lyne, therfore to those 4
that I toke vp in the fyrst su{m}me, I take one also of the seconde
su{m}me, and then haue I taken vp 50, for whiche 5 counters I sette
downe one in the space ouer y^e second lyne, as here doth appere.
-----------|--------------
| o
-----------|--o-----------
| o
-X--o-o-o--|--------------
|
----o-o----|--o-o-o-o-----
| o
-----------|--o-o-o-------
|
-----------|--o-----------
[*121b.] and then is there 80, as well w^t those 4 counters, as yf I had
set downe y^e other 4 also. Now do I take the 200 in the fyrste su{m}me,
and adde them to the 400 in the seconde summe, and it maketh 600,
therfore I take vp the 2 counters in the fyrste summe, and 3 of them in
the seconde summe, and for them 5 I set 1 in y^e space aboue, thus.
-----------|------------
| o
-----------|--o---------
| o
-X--o-o-o--|------------
| o
-----------|--o---------
| o
-----------|--o-o-o-----
|
-----------|--o---------
Then I take y^e 3000 in y^e fyrste su{m}me, vnto whiche there are none
in the second summe agreynge, therfore I do onely remoue those 3
counters from the fyrste summe into the seconde, as here doth appere.
----|-------------
| o
----|---o---------
| o
-X--|---o-o-o-----
| o
----|-o-----------
| o
----|---o-o-o-----
|
----|---o---------
[*122a.] And so you see the hole su{m}me, that amou{n}teth of the
addytio{n} of 65436 with 3245 to be 6868[1]. And yf you haue marked
these two exa{m}ples well, you nede no farther enstructio{n} in Addition
of 2 only summes: but yf you haue more then two summes to adde, you may
adde them thus. Fyrst adde two of them, and then adde the thyrde, and
y^e fourth, or more yf there be so many: as yf I wolde adde 2679 with
4286 and 1391. Fyrste I adde the two fyrste summes thus.
-------------|-----------|--------------
| | o
-X--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-o-o--|--o--------|--------------
[*122b.] And then I adde the thyrde thereto thus. And so of more yf you
haue them.
-------------|-----------|------------
| o | o
-X--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---------
[Headnote: Subtraction on the Counting Board.]
[Sidenote: Subtraction.]
_S._ Nowe I thynke beste that you passe forth to Subtraction, except
there be any wayes to examyn this maner of Addition, then I thynke that
were good to be knowen nexte.
_M._ There is the same profe here that is in the other Addition by the
penne, I meane Subtraction, for that onely is a sure waye: but
consyderynge that Subtraction must be fyrste knowen, I wyl fyrste teache
you the arte of Subtraction, and that by this example: I wolde subtracte
2892 out of 8746. These summes must I set downe as I dyd in Addition:
but here it is best [*116a (_sic_).] to set the lesser no{m}ber fyrste,
thus.
-------------|--------------
| o
-X--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-----------
Then shall I begynne to subtracte the greatest nombres fyrste (contrary
to the vse of the penne) y^t is the thousandes in this exa{m}ple:
therfore I fynd amongest the thousandes 2, for which I withdrawe so many
fro{m} the seconde summe (where are 8) and so remayneth there 6, as this
exa{m}ple showeth.
-------------+--------------
| o
-+-----------+--o-----------
o | o
----o-o-o----+--o-o---------
o |
----o-o-o-o--+--o-o-o-o-----
| o
----o-o------+--o-----------
Then do I lyke wayes with the hundredes, of whiche in the fyrste summe
[*116b] I fynde 8, and is the seconde summe but 7, out of whiche I can
not take 8, therfore thus muste I do: I muste loke how moche my summe
dyffereth from 10, whiche I fynde here to be 2, then must I bate for my
su{m}me of 800, one thousande, and set downe the excesse of hundredes,
that is to saye 2, for so moche 100[0] is more then I shuld take vp.
Therfore fro{m} the fyrste su{m}me I take that 800, and from the second
su{m}me where are 6000, I take vp one thousande, and leue 5000; but then
set I downe the 200 unto the 700 y^t are there all redye, and make them
900 thus.
-------------+--------------
| o
-+-----------+--------------
| o
-------------+--o-o-o-o-----
o |
----o-o-o-o--+--o-o-o-o-----
| o
----o-o------+--o-----------
Then come I to the articles of te{n}nes where in the fyrste su{m}me I
fynde 90, [*117a] and in the seconde su{m}me but only 40: Now
consyderyng that 90 can not be bated from 40, I loke how moche y^t 90
doth dyffer from the next summe aboue it, that is 100 (or elles whiche
is all to one effecte, I loke how moch 9 doth dyffer fro{m} 10) {and} I
fynd it to be 1, then in the stede of that 90, I do take from the second
summe 100: but consyderynge that it is 10 to moche, I set downe 1 in y^e
nexte lyne beneth for it, as you se here.
---------+------------
| o
-+-------+------------
| o
---------+--o-o-o-----
| o
---------+------------
| o
----o-o--+--o---------
Sauynge that here I haue set one counter in y^e space in stede of 5 in
y^e nexte lyne. And thus haue I subtracted all saue two, which I must
bate from the 6 in the second summe, and there wyll remayne 4, thus.
----+--------------
| o
-+--+--------------
| o
----+--o-o-o-------
| o
----+--------------
|
----+--o-o-o-o-----
So y^t yf I subtracte 2892 fro{m} 8746, the remayner wyll be 5854,
[*117b] And that this is truely wrought, you maye proue by Addition: for
yf you adde to this remayner the same su{m}me that you dyd subtracte,
then wyll the formar su{m}me 8746 amount agayne.
_S._ That wyll I proue: and fyrst I set the su{m}me that was subtracted,
which was 2892, {and} the{n} the remayner 5854, thus.
--------------+--------------
| o
-||--o-o------+--------------
o | o
-----o-o-o----+--o-o-o-------
o | o
-----o-o-o-o--+--------------
|
-----o-o------+--o-o-o-o-----
Then do I adde fyrst y^e 2 to 4, whiche maketh 6, so take I vp 5 of
those counters, and in theyr stede I sette 1 in the space, as here
appereth.
--------------+------------
| o
-||--o-o------+------------
o | o
-----o-o-o----+--o-o-o-----
o | o
-----o-o-o-o--+------------
| o
--------------+--o---------
[*118a] Then do I adde the 90 nexte aboue to the 50, and it maketh 140,
therfore I take vp those 6 counters, and for them I sette 1 to the
hundredes in y^e thyrde lyne, {and} 4 in y^e second lyne, thus.
------------+--------------
| o
-||--o-o----+--------------
o | o
-----o-o-o--+--o-o-o-o-----
|
------------+--o-o-o-o-----
| o
------------+----o---------
Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe,
and 9 in y^e second, that maketh 1700, therfore I take vp those 9
counters, and in theyr stede I sette 1 in the .iiii. lyne, and 1 in the
space nexte beneth, and 2 in the thyrde lyne, as you se here.
----------+--------------
| o
-||--o-o--+--o-----------
| o
----------+--o-o---------
|
----------+--o-o-o-o-----
| o
----------+--o-----------
Then is there lefte in the fyrste summe but only 2000, whiche I shall
take vp from thence, and set [*118b] in the same lyne in y^e second
su{m}me, to y^e one y^t is there all redy: {and} then wyll the hole
su{m}me appere (as you may wel se) to be 8746, which was y^e fyrst
grosse summe, {and} therfore I do perceaue, that I hadde well subtracted
before. And thus you may se how Subtraction maye be tryed by Addition.
----+--------------
| o
-X--+--o-o-o-------
| o
----+--o-o---------
|
----+--o-o-o-o-----
| o
----+----o---------
_S._ I perceaue the same order here w^t cou{n}ters, y^t I lerned before
in figures.
_M._ Then let me se howe can you trye Addition by Subtraction.
_S._ Fyrste I wyl set forth this exa{m}ple of Additio{n} where I haue
added 2189 to 4988, and the hole su{m}me appereth to be 7177,
--------------+-----------+----------
| | 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
-----o-o-o-o--+--o-o-o----+--o-o-----
[*119a] Nowe to trye whether that su{m}me be well added or no, I wyll
subtract one of the fyrst two su{m}mes from the thyrd, and yf I haue
well done y^e remayner wyll be lyke that other su{m}me. As for example:
I wyll subtracte the fyrste summe from the thyrde, whiche I set thus in
theyr order.
--------------+----------
| o
-||--o-o------+--o-o-----
|
-----o--------+--o-------
o | o
-----o-o-o----+--o-o-----
o | o
-----o-o-o-o--+--o-o-----
Then do I subtract 2000 of the fyrste summe fro{m} y^e second su{m}me,
and then remayneth there 5000 thus.
-------------+----------
| o
-X-----------+-----------
|
----o--------+--o-------
o | o
----o-o-o----+--o-o-----
o | o
----o-o-o-o--+--o-o-----
Then in the thyrd lyne, I subtract y^e 100 of the fyrste summe, fro{m}
the second su{m}me, where is onely 100 also, and then in y^e thyrde lyne
resteth nothyng. Then in the second lyne with his space ouer hym,
I fynde 80, which I shuld subtract [*119b] from the other su{m}me, then
seyng there are but only 70 I must take it out of some hygher summe,
which is here only 5000, therfore I take vp 5000, and seyng that it is
to moch by 4920, I sette downe so many in the seconde roume, whiche with
the 70 beynge there all redy do make 4990, & then the summes doth stande
thus.
--------------+--------------
|
-||-----------+--o-o-o-o-----
| o
--------------+--o-o-o-o-----
| o
--------------+--o-o-o-o-----
o | o
-----o-o-o-o--+--o-o---------
Yet remayneth there in the fyrst su{m}me 9, to be bated from the second
summe, where in that place of vnities dothe appere only 7, then I muste
bate a hygher su{m}me, that is to saye 10, but seynge that 10 is more
then 9 (which I shulde abate) by 1, therfore shall I take vp one counter
from the seconde lyne, {and} set downe the same in the fyrst [*120a] or
lowest lyne, as you se here.
-----+--------------
|
-||--+--o-o-o-o-----
| o
-----+--o-o-o-o-----
| o
-----+--o-o-o-------
| o
-----+--o-o-o-------
And so haue I ended this worke, {and} the su{m}me appereth to be y^e
same, whiche was y^e seconde summe of my addition, and therfore I
perceaue, I haue wel done.
_M._ To stande longer about this, it is but folye: excepte that this you
maye also vnderstande, that many do begynne to subtracte with counters,
not at the hyghest su{m}me, as I haue taught you, but at the
nethermoste, as they do vse to adde: and when the summe to be abatyd,
in any lyne appeareth greater then the other, then do they borowe one of
the next hygher roume, as for example: yf they shuld abate 1846 from
2378, they set y^e summes thus.
--------------+------------
|
-||--o--------+--o-o-------
o |
-----o-o-o----+--o-o-o-----
| o
-----o-o-o-o--+--o-o-------
o | o
-----o--------+--o-o-o-----
[*120b] And fyrste they take 6 whiche is in the lower lyne, and his
space from 8 in the same roumes, in y^e second su{m}me, and yet there
remayneth 2 counters in the lowest lyne. Then in the second lyne must 4
be subtracte from 7, and so remayneth there 3. Then 8 in the thyrde lyne
and his space, from 3 of the second summe can not be, therfore do they
bate it from a hygher roume, that is, from 1000, and bycause that 1000
is to moch by 200, therfore must I sette downe 200 in the thyrde lyne,
after I haue taken vp 1000 from the fourth lyne: then is there yet 1000
in the fourth lyne of the fyrst summe, whiche yf I withdrawe from the
seconde summe, then doth all y^e figures stande in this order.
-----+------------
|
-||--+------------
| o
-----+------------
|
-----+--o-o-o-----
|
-----+--o-o-------
So that (as you se) it differeth not greatly whether you begynne
subtractio{n} at the hygher lynes, or at [*121a] the lower. How be it,
as some menne lyke the one waye beste, so some lyke the other: therfore
you now knowyng bothe, may vse whiche you lyst.
[Headnote: Multiplication by Counters.]
[Sidenote: Multiplication.]
But nowe touchynge Multiplicatio{n}: you shall set your no{m}bers in two
roumes, as you dyd in those two other kyndes, but so that the multiplier
be set in the fyrste roume. Then shall you begyn with the hyghest
no{m}bers of y^e seconde roume, and multiply them fyrst after this sort.
Take that ouermost lyne in your fyrst workynge, as yf it were the lowest
lyne, setting on it some mouable marke, as you lyste, and loke how many
counters be in hym, take them vp, and for them set downe the hole
multyplyer, so many tymes as you toke vp counters, reckenyng, I saye
that lyne for the vnites: {and} when you haue so done with the hygheest
no{m}ber then come to the nexte lyne beneth, {and} do euen so with it,
and so with y^e next, tyll you haue done all. And yf there be any nomber
in a space, then for it [*121b] shall you take y^e multiplyer 5 tymes,
and then must you recken that lyne for the vnites whiche is nexte beneth
that space: or els after a shorter way, you shall take only halfe the
multyplyer, but then shall you take the lyne nexte aboue that space, for
the lyne of vnites: but in suche workynge, yf chau{n}ce your multyplyer
be an odde nomber, so that you can not take the halfe of it iustly, then
muste you take the greater halfe, and set downe that, as if that it were
the iuste halfe, and farther you shall set one cou{n}ter in the space
beneth that line, which you recken for the lyne of vnities, or els only
remoue forward the same that is to be multyplyed.
_S._ Yf you set forth an example hereto I thynke I shal perceaue you.
_M._ Take this exa{m}ple: I wold multiply 1542 by 365, therfore I set
y^e nombers thus.
------------+--------------
|
-||---------+--o-----------
| o
-----o-o-o--+--------------
o |
-----o------+--o-o-o-o-----
o |
------------+--o-o---------
[*122a] Then fyrste I begynne at the 1000 in y^e hyghest roume, as yf it
were y^e fyrst place, & I take it vp, settynge downe for it so often
(that is ones) the multyplyer, which is 365, thus, as you se here:
-----------+-----------+------------
| |
-----------+-----------+--o-o-o-----
| | o
-----------+-----------+--o---------
| | o
-X---------+-----------+------------ [<-]
| o |
----o-o-o--+-----------+------------
o | |
----o------+--o-o-o-o--+------------
o | |
-----------+--o-o------+------------
where for the one counter taken vp from the fourth lyne, I haue sette
downe other 6, whiche make y^e su{m}me of the multyplyer, reckenynge
that fourth lyne, as yf it were the fyrste: whiche thyng I haue marked
by the hand set at the begynnyng of y^e same,
_S._ I perceaue this well: for in dede, this summe that you haue set
downe is 365000, for so moche doth amount [*122b] of 1000, multiplyed by
365.
_M._ Well the{n} to go forth, in the nexte space I fynde one counter
which I remoue forward but take not vp, but do (as in such case I must)
set downe the greater halfe of my multiplier (seyng it is an odde
no{m}ber) which is 182, {and} here I do styll let that fourth place
stand, as yf it were y^e fyrst:
------------+-----------+--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------+---------+------------
as in this fourme you se, where I haue set this multiplycatio{n} with
y^e other: but for the ease of your vndersta{n}dynge, I haue set a
lytell lyne betwene them: now shulde they both in one su{m}me stand
thus.
------------+-----------+--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------+----------------
[*123a] Howe be it an other fourme to multyplye suche cou{n}ters i{n}
space is this: Fyrst to remoue the fynger to the lyne nexte benethe y^e
space, {and} then to take vp y^e cou{n}ter, {and} to set downe y^e
multiplyer .v. tymes, as here you se.
---------+---------+-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-----+-
| | | o | o | o | o | o |
[->]-X-o-o-o-+---------+-------+------+------+------+------+------+-
o | | | | | | | |
---o-----+-o-o-o-o-+-------+------+------+------+------+------+-
o | | | | | | | |
---------+-o-o-----+-------+------+------+------+------+------+-
Which su{m}mes yf you do adde together into one su{m}me, you shal
p{er}ceaue that it wyll be y^e same y^t appeareth of y^e other worki{n}g
before, so that [*123b] bothe sortes are to one entent, but as the other
is much shorter, so this is playner to reason, for suche as haue had
small exercyse in this arte. Not withstandynge you maye adde them in
your mynde before you sette them downe, as in this exa{m}ple, you myghte
haue sayde 5 tymes 300 is 1500, {and} 5 tymes 60 is 300, also 5 tymes 5
is 25, whiche all put together do make 1825, which you maye at one tyme
set downe yf you lyste. But nowe to go forth, I must remoue the hand to
the nexte counters, whiche are in the second lyne, and there must I take
vp those 4 counters, settynge downe for them my multiplyer 4 tymes,
whiche thynge other I maye do at 4 tymes seuerally, or elles I may
gather that hole summe in my mynde fyrste, and then set it downe: as to
saye 4 tymes 300 is 1200: 4 tymes 60 are 240: and 4 tymes 5 make 20: y^t
is in all 1460, y^t shall I set downe also: as here you se.
o
-----------+-------+-----------+--------------
| | |
-----------+-------+--o-o-o-o--+--o-----------
| | o |
-X---------+-------+--o-o------+--o-o-o-o-----
| | o | o
----o-o-o--+-------+-----------+--o-----------
o | | |
[->] ----o------+-------+-----------+--------------
o | | |
-----------+--o-o--+-----------+--------------
[*124a] whiche yf I ioyne in one summe with the formar nombers, it wyll
appeare thus.
o
---------+-------+----------
| | o
---------+-------+--o-------
| |
---------+-------+--o-o-----
| |
--o-o-o--+-------+-o--------
o | |
[->] --o------+-------+----------
o | |
---------+--o-o--+----------
Then to ende this multiplycation, I remoue the fynger to the lowest
lyne, where are onely 2, them do I take vp, and in theyr stede do I set
downe twyse 365, that is 730, for which I set [*124b] one in the space
aboue the thyrd lyne for 500, and 2 more in the thyrd lyne with that one
that is there all redye, and the reste in theyr order, {and} so haue I
ended the hole summe thus.
o
---------+-----+------------
| | o
---------+-----+--o---------
| |
---------+-----+--o-o-------
| | o
--o-o-o--+-----+--o-o-o-----
o | |
--o------+-----+--o-o-o-----
o | |
---------+-----+------------
Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes
incarnation) beyng multyplyed by 365 (which is the nomber of dayes in
one yeare) dothe amounte vnto 562830, which declareth y^e no{m}ber of
daies sith Chrystes incarnatio{n} vnto the ende of 1542[{1}] yeares.
(besyde 385 dayes and 12 houres for lepe yeares).
_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:
----------+-----------+---------
| |
-||--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.]
[Illustration: (Numbers as described in text)]
[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]