Chapter 12

[2: “Hæ necessariæ figuræ sunt Indorum characteros.” _MS. de numeratione._ Bib. Sloan. Mus. Brit. 513, fol. 58. “Cum vidissem Yndos constituisse IX literas in universo numero suo propter dispositionem suam quam posuerunt, volui patefacere de opere quod sit per eas aliquidque esset levius discentibus, si Deus voluerit. Si autem Indi hoc voluerunt et intentio illorum nihil novem literis fuit, causa que mihi potuit. Deus direxit me ad hoc. Si vero alia dicam preter eam quam ego exposui, hoc fecerunt per hoc quod ego exposui, eadem tam certissime et absque ulla dubitatione poterit inveniri. Levitasque patebit aspicientibus et discentibus.” MS. U.L.C., Ii. vi. 5, f. 102.]

[3: From Eg. 2622.]

[4: 8 C. iv. inserts Nullum cipa significat: dat significare sequenti.]

[5: From 12 E. 1.]

[6: En argorisme devon prendre Vii especes . . . . Adision subtracion Doubloison mediacion Monteploie et division Et de radix eustracion A chez vii especes savoir Doit chascun en memoire avoir Letres qui figures sont dites Et qui excellens sont ecrites. --MS. _Seld. Arch._ B. 26.]

[7: From 12 E. 1.]

[8: From 12 E. 1.]

[9: 8 C. iv. inserts Atque figura prior nuper fuerit mediando.]

[10: _I.e._ figura secundo loco posita.]

[11: So 12 E. 1; 8 C. iv. inserts--

[12: 12 E. 1 inserts.]

[13: 12 E. 1 inserts to l. 174.]

[14: 12 E. 1 omits, Eg. 2622 inserts.]

[15: 12 E. 1 inserts.]

[16: 8 C. iv. inserts-- Hinc illam dele duplans sub ei psalliendo Que sequitur retrahens quicquid fuerit duplicatum.]

INDEX OF TECHNICAL TERMS[1*]

[Footnote 1*: This Index has been kindly prepared by Professor J. B. Dale, of King’s College, University of London, and the best thanks of the Society are due to him for his valuable contribution.]

[Transcriber’s Note: The Technical Terms and Glossary (following) refer to page and line numbers in the printed book. Information in [[double brackets]] has been added by the transcriber to aid in text searching.]

+algorisme+, 33/12; +algorym+, +augrym+, 3/3; the art of computing, using the so-called Arabic numerals. The word in its various forms is derived from the Arabic _al-Khowarazmi_ (i.e. the native of Khwarazm (Khiva)). This was the surname of Ja’far Mohammad ben Musa, who wrote a treatise early in the 9th century (see p. xiv). The form _algorithm_ is also found, being suggested by a supposed derivation from the Greek ἀριθμός (number).

+antery+, 24/11; to move figures to the right of the position in which they are first written. This operation is performed repeatedly upon the multiplier in multiplication, and upon certain figures which arise in the process of root extraction.

+anterioracioun+, 50/5; the operation of moving figures to the right. [[written anteriorac{i}o{u}n or anterioracio{u}n]]

+article+, 34/23; +articul+, 5/31; +articuls+, 9/36, 29/7,8; a number divisible by ten without remainder. [[also articull{e}]]

+cast+, 8/12; to add one number to another. ‘Addition is a _casting_ together of two numbers into one number,’ 8/10.

+cifre+, 4/1; the name of the figure 0. The word is derived from the Arabic _sifr_ = empty, nothing. Hence _zero_. A cipher is the symbol of the absence of number or of zero quantity. It may be used alone or in conjunction with digits or other ciphers, and in the latter case, according to the position which it occupies relative to the other figures, indicates the absence of units, or tens, or hundreds, etc. The great superiority of the Arabic to all other systems of notation resides in the employment of this symbol. When the cipher is not used, the place value of digits has to be indicated by writing them in assigned rows or columns. Ciphers, however, may be interpolated amongst the significant figures used, and as they sufficiently indicate the positions of the empty rows or columns, the latter need not be indicated in any other way. The practical performance of calculations is thus enormously facilitated (see p. xvi).

+componede+, 33/24; +composyt+, 5/35; with reference to numbers, one compounded of a multiple of ten and a digit. [[written componed{e}]]

+conuertide+ = conversely, 46/29, 47/9. [[written co{n}u{er}tid{e} or {con}u{er}tid{e}]]

+cubicede+, 50/13; +to be c.+, to have its cube root found. [[written cubiced{e}]]

+cubike nombre+, 47/8; a number formed by multiplying a given number twice by itself, _e.g._ 27 = 3 × 3 × 3. Now called simply a cube. [[written cubik{e} ...]]

+decuple+, 22/12; the product of a number by ten. Tenfold.

+departys+ = divides, 5/29. [[written dep{ar}tys]]

+digit+, 5/30; +digitalle+, 33/24; a number less than ten, represented by one of the nine Arabic numerals. [[written digitall{e}]]

+dimydicion+, 7/23; the operation of dividing a number by two. Halving. [[written dimydicioñ]]

+duccioun+, multiplication, 43/9. [[written duccio{u}n]]

+duplacion+, 7/23, 14/15; the operation of multiplying a number by two. Doubling. [[written duplacioñ or duplacioɳ with fancy “n”]]

+i-mediet+ = halved, 19/23.

+intercise+ = broken, 46/2; intercise Progression is the name given to either of the Progressions 1, 3, 5, 7, etc.; 2, 4, 6, 8, etc., in which the common difference is 2. [[written int{er}cise]]

+lede into+, multiply by, 47/18. [[words always separated, as “lede ... into”]]

+lyneal nombre+, 46/14; a number such as that which expresses the measure of the length of a line, and therefore is not _necessarily_ the product of two or more numbers (_vide_ Superficial, Solid). This appears to be the meaning of the phrase as used in _The Art of Nombryng_. It is possible that the numbers so designated are the prime numbers, that is, numbers not divisible by any other number except themselves and unity, but it is not clear that this limitation is intended.

+mediacioun+, 16/36, 38/16; dividing by two (see also +dimydicion+). [[written mediacioɳ with fancy “n”, generally without “u”]]

+medlede nombre+, 34/1; a number formed of a multiple of ten and a digit (_vide_ componede, composyt). [[written medled{e} ...]]

+medye+, 17/8, to halve; +mediete+, halved, 17/30; +ymedit+, 20/9.

+naturelle progressioun+, 45/22; the series of numbers 1, 2, 3, etc. [[written naturell{e} p{ro}gressio{u}n]]

+produccioun+, multiplication, 50/11. [[written produccio{u}n]]

+quadrat nombre+, 46/12; a number formed by multiplying a given number by itself, _e.g._ 9 = 3 × 3, a square.

+rote+, 7/25; +roote+, 47/11; root. The roots of squares and cubes are the numbers from which the squares and cubes are derived by multiplication into themselves.

+significatyf+, significant, 5/14; The significant figures of a number are, strictly speaking, those other than zero, _e.g._ in 3 6 5 0 4 0 0, the significant figures are 3, 6, 5, 4. Modern usage, however, regards all figures between the two extreme significant figures as significant, even when some are zero. Thus, in the above example, 3 6 5 0 4 are considered significant.

+solide nombre+, 46/37; a number which is the product of three other numbers, _e.g._ 66 = 11 × 2 × 3. [[usually written solid{e}]]

+superficial nombre+, 46/18; a number which is the product of two other numbers, _e.g._ 6 = 2 × 3. [[written sup{er}ficial or sup{er}ficiall{e}]]

+ternary+, consisting of three digits, 51/7. [[written t{er}nary]]

+vnder double+, a digit which has been doubled, 48/3.

+vnder-trebille+, a digit which has been trebled, 49/28; +vnder-triplat+, 49/39. [[written vnder-trebill{e}, vnder-t{r}iplat]]

+w+, a symbol used to denote half a unit, 17/33 [[shown in e-text as superscript ʷ]]

GLOSSARY

[Transcriber’s Note:

Words whose first appearance is earlier than the page cited in the Glossary are identified in double-bracketed notes. To aid in text searching, words written with internal {italics} are also noted, and context is given for common words.]

+ablacioun+, taking away, 36/21 [[written ablacio{u}n]] +addyst+, haddest, 10/37 +agregacioun+, addition, 45/22. (First example in N.E.D., 1547.) [[written ag{r}egacio{u}n]] +a-ȝenenes+, against, 23/10 +allgate+, always, 8/39 +als+, as, 22/24 +and+, if, 29/8; +&+, 4/27; +& yf+, 20/7 +a-nendes+, towards, 23/15 +aproprede+, appropriated, 34/27 [[written ap{ro}pred{e}]] +apwereth+, appears, 61/8 +a-risyȝt+, arises, 14/24 +a-rowe+, in a row, 29/10 +arsemetrike+, arithmetic, 33/1 [[written arsemetrik{e}]] +ayene+, again, 45/15

+bagle+, crozier, 67/12 +bordure+ = ordure, row, 43/30 [[written bordur{e}]] +borro+, _inf._ borrow, 11/38; _imp. s._ +borowe+, 12/20; _pp._ +borwed+, 12/15; +borred+, 12/19 +boue+, above, 42/34

+caputule+, chapter, 7/26 [[written caputul{e}]] +certayn+, assuredly, 18/34 [[written c{er}tayɳ]] +clepede+, called, 47/7 [[written cleped{e}]] +competently+, conveniently, 35/8 +compt+, count, 47/29 +contynes+, contains, 21/12; [[written {con}tynes]] _pp._ +contenythe+, 38/39 [[written co{n}tenyth{e}]] +craft+, art, 3/4

+distingue+, divide, 51/5

+egalle+, equal, 45/21 [[written egall{e}]] +excep+, except, 5/16] +exclusede+, excluded, 34/37 [[written exclused{e}]] +excressent+, resulting, 35/16 [[written exc{re}ssent]] +exeant+, resulting, 43/26 +expone+, expound, 3/23

+ferye+ = ferþe, fourth, 70/12 +figure+ = figures, 5/1 [[written fig{ure}]] +for-by+, past, 12/11 +fors; no f.+, no matter, 22/24 +forseth+, matters, 53/30 +forye+ = forþe, forth, 71/8] +fyftye+ = fyftþe, fifth, 70/16

+grewe+, Greek, 33/13

+haluendel+, half, 16/16; +haldel+, 19/4; _pl._ +haluedels+, 16/16 +hayst+, hast, 17/3, 32 +hast+, haste, 22/25 [[in “haue hast to”]] +heer+, higher, 9/35 +here+, their, 7/26 [[in “in her{e} caputul{e}”]] +here-a-fore+, heretofore, 13/7 [[written her{e}-a-for{e}]] +heyth+, was called, 3/5 +hole+, whole, 4/39; +holle+, 17/1; +hoole+, of three dimensions, 46/15 +holdyþe+, holds good, 30/5 +how be it that+, although, 44/4

+lede+ = lete, let, 8/37 +lene+, lend, 12/39 +lest+, least, 43/27 [[in “at the lest”]] +lest+ = left, 71/9 [[in “the lest syde”]] +leue+, leave, 6/5; _pr. 3 s._ +leues+, remains, 11/19; [[first in 10/40]] +leus+, 11/28; _pp._ +laft+, left, 19/24 +lewder+, more ignorant, 3/3 [[written lewd{er}]] +lust+, desirest to, 45/13 +lyȝt+, easy, 15/31 +lymytes+, limits, 34/18; +lynes+, 34/12; +lynees+, 34/17; Lat. limes, _pl._ limites.

+maystery+, achievement; [[written mayst{er}y]] +no m.+, no achievement, i.e. easy, 19/10 +me+, _indef. pron._ one, 42/1 [[first in 34/16]] +mo+, more, 9/16 +moder+ = more (Lat. majorem), 43/22 +most+, must, 30/3 [[first in 3/12 and many more]] +multipliede+, +to be m.+ = multiplying, 40/9 +mynvtes+, the sixty parts into which a unit is divided, 38/25 [[written mynvt{es}]] +myse-wroȝt+, mis-wrought, 14/11

+nether+, nor, 34/25 [[in “It was, nether is”]] +nex+, next, 19/9 +noȝt+, nought, 5/7 [[first in 4/8]] +note+, not, 30/5

+oo+, one, 42/20; +o+, 42/21 [[first in 34/27; 33/22]] +omest+, uppermost, higher, 35/26; +omyst+, 35/28 +omwhile+, sometimes, 45/31 [[first in 39/17]] +on+, one, 8/29 [[in “on vnder an-oþ{er}”]] +opyne+, plain, 47/8 [[written opyn{e}]] +or+, before, 13/25 [[in “or þou be-gan”]] +or+ = þe oþ{er}, the other, 28/34 [[in “or by-twene”]] +ordure+, order, 34/9; row, 43/1 [[word form is “order”]] +other+, or, 33/13, 43/26; [[in “art other craft” on 33/13, “other how oft” on 43/26; note also “one other other” on 35/24]] +other . . . or+, either . . . or, 38/37 [[in “other it is even or od{e}” on 38/37; there are earlier occurrences]] +ouerer+, upper, 42/15 [[written ou{er}er]] +ouer-hippede+, passed over, 43/19 [[written ou{er}-hipped{e}]]

+recte+, directly, 27/20 [[in “stondes not recte”; also on 26/31 in “recte ou{er} his hede”]] +remayner+, remainder, 56/28 +representithe+, represented, 39/14 [[written rep{re}sentith{e}]] +resteth+, remains, 63/29 [[first in 57/29 and others]] +rewarde+, regard, 48/6 [[written reward{e}]] +rew+, row, 4/8 +rewle+, row, 4/20, 7/12; [[in “place of þe rewle”, “þe rewle of fig{ure}s”]] +rewele+, 4/18; +rewles+, rules, 5/33

+s.+ = scilicet, 3/8 [[in “s. Algorism{us}”]] +sentens+, meaning, 14/29 +signifye(tyf)+, 5/13. The last three letters are added above the line, evidently because of the word ‘significatyf’ in l. 14. But the ‘Solucio,’ which contained the word, has been omitted. +sithen+, since, 33/8 +some+, sum, result, 40/17, 32 [[first in 36/21 in “me may see a some”, then in “the same some” and “to some of”]] +sowne+, pronounce, 6/29 +singillatim+, singly, 7/25 +spices+, species, kinds, 34/4 [[first in 5/34 and others]] +spyl+, waste, 14/26 +styde+, stead, 18/20 +subtrahe+, subtract, 48/12; _pp._ +subtrayd+, 13/21 +sythes+, times, 21/16

+taȝt+, taught, 16/36 +take+, _pp._ taken; +t. fro+, starting from, 45/22 [[in “fro oone or tweyn{e} take”]] +taward+, toward, 23/34 +thouȝt+, though, 5/20 +trebille+, multiply by three, 49/26 [[written trebill{e}]] +twene+, two, 8/11 [[first in 4/23]] +þow+, though, 25/15 [[in “þow þ{o}u take”]] +þowȝt+, thought; +be þ.+, mentally, 28/4 +þus+ = þis, this, 20/33 [[in “þus nombur 214”]]

+vny+, unite, 45/10

+wel+, wilt, 14/31 [[in “If þ{o}u wel”]] +wete+, wit, 15/16; +wyte+, know, 8/38; _pr. 2 s._ +wost+, 12/38 +wex+, become, 50/18 +where+, whether, 29/12 [[written wher{e} in “wher{e} in þe secunde, or”]] +wher-thurghe+, whence, 49/15 [[written Wher-thurgh{e}]] +worch+, work, 8/19; [[first in 7/35]] +wrich+, 8/35; +wyrch+, 6/19; _imp. s._ +worch+, 15/9; [[first in 9/6]] _pp._ +y-wroth+, 13/24 +write+, written, 29/19; [[first in 6/37 in “hast write”, “be write”]] +y-write+, 16/1 +wryrchynge+ = wyrchynge, working, 30/4 [[written wryrchyng{e}]] +w^t+, with, 55/8

+y-broth+, brought, 21/18 +ychon+, each one, 29/10 [[written ychoɳ]] +ydo+, done, added, 9/6 [[first in 8/37 in “haue ydo”; 9/6 in “ydo all to-ged{er}”]] +ylke+, same, 5/12 +y-lyech+, alike, 22/23 +y-myȝt+, been able, 12/2 +y-nowȝt+, enough, 15/31; +ynovȝt+, 18/34 +yove+, given, 45/33 +y^t+, that, 52/8 +y-write+, _v._ +write.+ +y-wroth+, _v._ +worch.+

* * * * * * * * * * * * * *

MARGINAL NOTES:

+Headnotes+ have been moved to the beginning of the appropriate paragraph. Headnotes were omitted from the two Appendixes, as sidenotes give the same information.

+Line Numbers+ are cited in the Index and Glossary. They have been omitted from the e-text except in the one verse selection (App. II, _Carmen de Algorismo_). Instead, the Index and Glossary include supplemental information to help locate each word.

+Numbered Notes+:

Numbered sidenotes show page or leaf numbers from the original MSS. In the e-text, the page number is shown as [*123b] inline; mid-word page breaks are marked with a supplemental asterisk [*]. Numbers are not used.

Footnotes give textual information such as variant readings. They have been numbered sequentially within each title, with numbers shown as [{1}] to avoid confusion with bracked text--including single numerals--in the original. Editorial notes are shown as [1*]. When a footnote calls for added text, the addition is shown in the body text with [[double brackets]].

+Sidenotes+ giving a running synopsis of the text have been moved to the beginning of each paragraph, where they are shown as a single note.

ERRORS AND ANOMALIES (Noted by Transcriber):

Introduction:

dated Mij^c [_In this and the remainder of the paragraph, the letter shown as ^c is printed directly above the preceding j._]

The Crafte of Nombrynge:

sursu{m} {pr}ocedas m{u}ltiplicando [_Italicized as shown: error for “p{ro}cedas”?_] Sidenote: Our author makes a slip here [_Elsewhere in the book, numerical errors are corrected in the body text, with a footnote giving the original form._] ten tymes so mych is þe nounb{re} [_text unchanged: error for “as”?_] 6 tymes 24, [{19}]þen take [_misplaced footnote anchor in original: belongs with “6 times 24”_] Fn. 7: ‘Subt{ra}has a{u}t addis a dext{ri}s [_open quote missing_]

The Art of Nombryng:

oone of the digitis as .10. of 1.. 20. of. 2. [_text unchanged: error for “as .10. of .1. 20. of .2.”?_] sette a-side half of tho m{inutes} [_text unchanged: error for “the”?_] and. 10. as before is come therof [_text unchanged: error for “and .10.”?_] Sidenote: Where to set the quotiente [_spelling (1922) unchanged_] Sidenote: Definition of Progression. [_f in “of” illegible_] Sidenote: ... giving the value of ab.^2 [_That is, “a(b^2).”_]

Accomptynge by counters:

For example of the [*117a.] ly[*]nes [_final . in sidenote missing or invisible_] [_also in 121b, 122a] which in the fyrst summe is 5 [_invisible “5” supplied by transcriber_] [*116a (_sic_).] [_Editor’s “sic”: page numbering jumps back to 116 instead of the expected 123, and continues from 116._] [*123a] ... set downe y^e multiplyer .v. tymes, as here you se [_Diagram shown as printed, with 35500 for 36500 in one column, and apparent misplaced “thousands” marker_] 365 (which is the nomber of dayes ... [_open ( missing_]

The arte of nombrynge by the hande:

for 1 the fynger is clasped in [_In at least one printing of the text, “clasped” is misprinted as “elasped”_] but this teacyed me not [_text unchanged_]

Appendix I: A Treatise on the Numeration of Algorism:

_See Introduction and Glossary for ſ:f and þ:y errors_

Appendix II: Carmen de Algorismo:

_In this selection, errors that are not explained in footnotes were assumed to be typographic._

l. 99 Postea procedas [procdeas] l. 163 Articuli digitum post in digitum mixti duc [post iu]

End of Project Gutenberg's The Earliest Arithmetics in English, by Anonymous