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: