The Earliest Arithmetics in English

Appendix I.

+A Treatise on the Numeration of Algorism.+

To alle suche even nombrys the most have cifrys as to ten. twenty. thirtty. an hundred. an thousand and suche other. but ye schal vnderstonde that a cifre tokeneth nothinge but he maketh other the more significatyf that comith after hym. Also ye schal vnderstonde that in nombrys composyt and in alle other nombrys that ben of diverse figurys ye schal begynne in the ritht syde and to rekene backwarde and so he schal be wryte as thus--1000. the sifre in the ritht side was first wryte and yit he tokeneth nothinge to the secunde no the thridde but thei maken that figure of 1 the more signyficatyf that comith after hem by as moche as he born oute of his first place where he schuld yf he stode ther tokene but one. And there he stondith nowe in the ferye place he tokeneth a thousand as by this rewle. In the first place he tokeneth but hymself. In the secunde place he tokeneth ten times hymself. In the thridde place he tokeneth an hundred tymes himself. In the ferye he tokeneth a thousand tymes himself. In the fyftye place he tokeneth ten thousand tymes himself. In the sexte place he tokeneth an hundred thousand tymes hymself. In the seveth place he tokeneth ten hundred thousand tymes hymself, &c. And ye schal vnderstond that this worde nombre is partyd into thre partyes. Somme is callyd nombre of digitys for alle ben digitys that ben withine ten as ix, viii, vii, vi, v, iv, iii, ii, i. Articules ben alle thei that mow be devyded into nombrys of ten as xx, xxx, xl, and suche other. Composittys be alle nombrys that ben componyd of a digyt and of an articule as fourtene fyftene thrittene and suche other. Fourtene is componyd of four that is a digyt and of ten that is an articule. Fyftene is componyd of fyve that is a digyt and of ten that is an articule and so of others . . . . . . But as to this rewle. In the firste place he tokeneth but himself that is to say he tokeneth but that and no more. If that he stonde in the secunde place he tokeneth ten tymes himself as this figure 2 here 21. this is oon and twenty. This figure 2 stondith in the secunde place and therfor he tokeneth ten tymes himself and ten tymes 2 is twenty and so forye of every figure and he stonde after another toward the lest syde he schal tokene ten tymes as moche more as he schuld token and he stode in that place ther that the figure afore him stondeth: lo an example as thus 9634. This figure of foure that hath this schape 4 tokeneth but himself for he stondeth in the first place. The figure of thre that hath this schape 3 tokeneth ten tyme himself for he stondeth in the secunde place and that is thritti. The figure of sexe that hath this schape 6 tokeneth ten tyme more than he schuld and he stode in the place yer the figure of thre stondeth for ther he schuld tokene but sexty. And now he tokeneth ten tymes that is sexe hundrid. The figure of nyne that hath this schape 9 tokeneth ten tymes more than he schulde and he stode in the place ther the figure of 6 stondeth inne for thanne he schuld tokene but nyne hundryd. And in the place that he stondeth inne nowe he tokeneth nine thousand. Alle the hole nombre of these foure figurys. Nine thousand sexe hundrid and foure and thritti.

Appendix Ii.

Carmen de Algorismo.

Hec algorismus ars presens dicitur[{1}]; in qua Talibus Indorum[{2}] fruimur his quinque figuris. 0. 9. 8. 7. 6. 5. 4. 3. 2. 1. Prima significat unum: duo vero secunda: Tercia significat tria: sic procede sinistre 4 Donec ad extremam venies, qua cifra vocatur; [{3}][Que nil significat; dat significare sequenti.] Quelibet illarum si primo limite ponas, Simpliciter se significat: si vero secundo, 8 Se decies: sursum procedas multiplicando.[{4}] [Namque figura sequens quevis signat decies plus, Ipsa locata loco quam significet pereunte: 12 Nam precedentes plus ultima significabit.] [{5}]Post predicta scias quod tres breuiter numerorum Distincte species sunt; nam quidam digiti sunt; Articuli quidam; quidam quoque compositi sunt. 16 [Sunt digiti numeri qui citra denarium sunt; Articuli decupli degitorum; compositi sunt Illi qui constant ex articulis digitisque.] Ergo, proposito numero tibi scribere, primo 20 Respicias quis sit numerus; quia si digitus sit, [{5}][Una figura satis sibi; sed si compositus sit,] Primo scribe loco digitum post articulum fac Articulus si sit, cifram post articulum sit, 24 [Articulum vero reliquenti in scribe figure.] Quolibet in numero, si par sit prima figura, Par erit et totum, quicquid sibi continetur; Impar si fuerit, totum sibi fiet et impar. 28 Septem[{6}] sunt partes, non plures, istius artis; Addere, subtrahere, duplare, dimidiare; Sexta est diuidere, set quinta est multiplicare; Radicem extrahere pars septima dicitur esse. 32 Subtrahis aut addis a dextris vel mediabis; A leua dupla, diuide, multiplicaque; Extrahe radicem semper sub parte sinistra.

Addere si numero numerum vis, ordine tali 36 Incipe; scribe duas primo series numerorum Prima sub prima recte ponendo figuram, Et sic de reliquis facias, si sint tibi plures. Inde duas adde primas hac condicione; 40 Si digitus crescat ex addicione priorum, Primo scribe loco digitum, quicunque sit ille; Si sit compositus, in limite scribe sequenti Articulum, primo digitum; quia sic iubet ordo. 44 Articulus si sit, in primo limite cifram, Articulum vero reliquis inscribe figuris; Vel per se scribas si nulla figura sequatur. Si tibi cifra superueniens occurrerit, illam 48 Deme suppositam; post illic scribe figuram: Postea procedas reliquas addendo figuras.

A numero numerum si sit tibi demere cura, Scribe figurarum series, vt in addicione; 52 Maiori numero numerum suppone minorem, Siue pari numero supponatur numerus par. Postea si possis a prima subtrahe primam, Scribens quod remanet, cifram si nil remanebit. 56 Set si non possis a prima demere primam; Procedens, vnum de limite deme sequenti; Et demptum pro denario reputabis ab illo, Subtrahe totaliter numerum quem proposuisti. 60 Quo facto, scribe supra quicquit remanebit, Facque novenarios de cifris, cum remanebis, Occurrant si forte cifre, dum demseris vnum; Postea procedas reliquas demendo figuras. 64

Si vis duplare numerum, sic incipe; solam Scribe figurarum seriem, quamcumque voles que Postea procedas primam duplando figuram; Inde quod excrescet, scribens, vbi iusserit ordo, 72 Juxta precepta que dantur in addicione. Nam si sit digitus, in primo limite scribe; Articulus si sit, in primo limite cifram, Articulum vero reliquis inscribe figuris; 76 Vel per se scribas, si nulla figura sequatur: Compositus si sit, in limite scribe sequenti Articulum primo, digitum; quia sic jubet ordo: Et sic de reliquis facias, si sint tibi plures. 80 [{8}][Si super extremam nota sit, monadem dat eidem, Quod tibi contingit, si primo dimidiabis.]

Incipe sic, si vis aliquem numerum mediare: Scribe figurarum seriem solam, velud ante; 84 Postea procedens medias, et prima figura Si par aut impar videas; quia si fuerit par, Dimidiabis eam, scribens quicquit remanebit; Impar si fuerit, vnum demas, mediare, 88 Nonne presumas, sed quod superest mediabis; Inde super tractum, fac demptum quod notat unum; Si monos, dele; sit ibi cifra post nota supra. Postea procedas hac condicione secunda:[{9}] 92 Impar[{10}] si fuerit hic vnum deme priori, Inscribens quinque, nam denos significabit Monos prædictam: si vero secunda dat vnam, Illa deleta, scribatur cifra; priori 96 Tradendo quinque pro denario mediato; Nec cifra scribatur, nisi inde figura sequatur: Postea procedas reliquas mediando figuras, Quin supra docui, si sint tibi mille figure. 100 [{11}][Si mediatio sit bene facta probare valebis, Duplando numerum quem primo dimidiasti.] Si super extremam nota sit monades dat eidem Quod contingat cum primo dimiabis Atque figura prior nuper fuerit mediando.]

Si tu per numerum numerum vis multiplicare, Scribe duas, quascunque volis, series numerorum; 104 Ordo tamen seruetur vt vltima multiplicandi Ponatur super anteriorem multiplicantis; [{12}][A leua relique sint scripte multiplicantes.] In digitum cures digitum si ducere, major 108 Per quantes distat a denis respice, debes Namque suo decuplo tociens delere minorem; Sicque tibi numerus veniens exinde patebit. Postea procedas postremam multiplicando, 112 Juste multiplicans per cunctas inferiores, Condicione tamen tali; quod multiplicantis Scribas in capite, quicquid processerit inde; Set postquam fuerit hec multiplicata, figure 116 Anteriorentur seriei multiplicantis; Et sic multiplica, velut istam multiplicasti, Qui sequitur numerum scriptum quicunque figuris. Set cum multiplicas, primo sic est operandum, 120 Si dabit articulum tibi multiplicacio solum; Proposita cifra, summam transferre memento. Sin autem digitus excrescerit articulusque, Articulus supraposito digito salit ultra; 124 Si digitus tamen, ponas illum super ipsam, Subdita multiplicans hanc que super incidit illi Delet eam penitus, scribens quod provenit inde; Sed si multiplices illam posite super ipsam, 128 Adiungens numerum quem prebet ductus earum; Si supraimpositam cifra debet multiplicare, Prorsus eam delet, scribi que loco cifra debet, [{12}][Si cifra multiplicat aliam positam super ipsam, 132 Sitque locus supra vacuus super hanc cifra fiet;] Si supra fuerit cifra semper pretereunda est; Si dubites, an sit bene multiplicando secunda, Diuide totalem numerum per multiplicantem, 136 Et reddet numerus emergens inde priorem.

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