And so haue I ended this worke, and the summe appereth to be ye same, whiche was ye 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 summe, 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 ye summes thus.
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120 b. *And fyrste they take 6 whiche is in the lower lyne, and his space from 8 in the same roumes, in ye second summe, 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 ye figures stande in this order.
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So that (as you se) it differeth not greatly whether you begynne subtraction at the hygher lynes, or at 121 a. *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.
Multiplication by Counters.
Multiplication. But nowe touchynge Multiplication: you shall set your nombers 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 nombers of ye 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 nomber then come to the nexte lyne beneth, and do euen so with it, and so with ye next, tyll you haue done all. And yf there be any nomber in a space, then for it 121 b. *shall you take ye 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 chaunce 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 counter 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 example: I wold multiply 1542 by 365, therfore I set ye nombers thus.