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: