Chapter 7

+----+----+----+----+----+--------+----+----+----+-----+ | 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.]