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: