Geraard of Cremona's translation of al-Khwarizmi's al-jabr: A critical edition

27 Pages·1986·2.009 MB·Latin,English

by  Barnabas B. Hughes

D, W, ROLLASON
soHum suum ac similis esse Altissimo grassatur, in execrabilissima reptilia,
bufonesetrubetas uel ranas deuolutusconuertitur; ettalisatellite,tali milite, tali
GERARD OF CREMONA'S TRANSLATION
exercitu pugnans uietus et confusus a mediocri certatrice conculcatur et
OF
exterminatur. ilia fortior in fide Dei effecta absterritum hostem tam in leone
AL-KHWARIZMI'S AL-JABR:
quam in uermiculo contemnit. Exercuit uitam uigiliis, orationibus aut
A CRITICAL EDITION
parsimonia. Tribus aut quattuor oblatis altaris aut exiguo legumine aut pomo
plerumque contenta erato In Quadragesima tribus diebus in ebdomada tantum
cibum capiebat. Cui cum quidam suggereret ut modestius abstineret ne
deficeret, hec tanquam de alio referebat: 'Noui, dornine mi, aliquem arnicum a Barnabas Hughes, O.F.M.
quarta f~ria usque in diem Resurrectionis Dominice nil gustasse et sine uHa
molestia leto et incolomi uigore perstitisse.' Hoc illam constat patrasse sub
alterius pretentione. lam ipsa in Domini misericordia propositum suum
compleuit, et in eodem loco sub beate Mildrethe patrocinio ubi certauit THE most significant mathematical innovations of the high Middle Ages
requieuit.
277 were the introduction of algebra into Western Europe through the
translations of al-Khwarizmi's al-Kitdb al-mukhata:;.ar ji /:zisdb al-jabr wa'l
University ofDurham. l11uqdbala (Liber algebre et almuchabala) and the foundation of abacist
arithmetic in the Liber abaci by Leonardo da Pisa.! The latter work has been
subjected to considerable study;2 more is certainly warranted. The translations
277 C now gives the chapter numbered 'xxxv' above.
ofijisdb al-jabr, however, have long beenin a process ofsorting andstudy. In
1838 Guillaume Libri published a faulty edition of Gerard's translation.3
TwelveyearslaterPrinceBaldassarreBoncompagnipresentedatranscriptionof
William ofLunis' translation which the Prince incorrectly accepted as that of
Gerard ofCremona.4 Louis Karpinski in 1915 offereda critical editionofa late
copyofRobertofChester's translation.SSeveralyearsago Ireported on sixteen
copies ofthe three translations;6 and in the near future my new critical edition
I B. Boncompagni,Scrilli di LeonardoPisano... I (Rome, 1857), pp. 1-459.
2 Apartfrom studiesinstandardhistoriesofmathematics,e.g., M. Cantor, Vorlesungen iiber
GeschichtederMathematik I(Leipzig, 1894),pp. 676-89,usefulinformationmaybefoundinK.
Vogel,'Fibonacci,Leonardo',Dictionary0/ScientificBiography4(New York, 197I),pp.604-13
(hereafter cited as DSB), particularly for the bibliography, and B. Boncompagni, 'Della vita e
delle opere di Leonardo Pisano matematico del secolo decimoterzo', Alii dell' Accademia
poll/ijicia de'nuovi lincei 5.I-3 (185I-52)(hereafter cited as A/Ii), which describes the codices
containing Liberabaci.
l G. Libri, Histoire des sciences mathemaliques en Italie depllis la renaissance des leI/res
jllsqll'dlajindll xVl/esiecle I(Paris,1838),pp. 253-97.
4 B. Boncompagni, 'Della vita e delle opere di Gherardo Cremonese, traduttore del secolo
duodecimo...',A/Ii 4(1850-5I) 4I2-35.
I L. Karpinski,Roberto/Chester'sLalin Translalion o/theAlgebra0/AI-Khuwarizmi(New
York, 19I5).Notehereasecondspellingfor 'al-Khwarizmi'andtherearemore;seefor instance
n. 8 below. My preference is based on the spelling used in DSB 7.358.
6 B. Hughes, The Medieval Latin Translations ofal-Khwarizmi's a/~iabr', Manuscripla 26
(1982) 31-37. This article corrects and adds to F. J. Carmody's Arabic Astronomical and
AstroloRicalSciences in Latin Translation. A CriticalBibliography (Berkeley, 1956), pp. 47-48.
212 B. HUGHES AL-KHWARIZMf'sAL-JABR 2lJ
of Robert of Chester's translation will be published.7 The core ofthe present deny him the creation of an entirely new approach to problem solving, the
article is a critical edition ofthe oldest extant copy of the translation made by standardization oftypes ofequations.
Gerard, together with variants found in the three older manuscripts which Combining variously three algebraic numbers, al-Khwarizmi constructs six
reproduce itmost faithfully. Remarks aboutthe other manuscriptcopies and an types ofequations, three which we will call simple, since he himselflabeledthe
analysis ofthe tract complete the article. last three composite. They are:
simple: ax2 = bx
ANALYSIS OF THE TEXT ax2 =c
=
bx c
According to the translation the treatise is divided into eightchapters and an composite: ax2 + bx =c
appendix. These discuss in turn decimal and algebraic numbers, six canonical ax2 + c =bx
firstandsecond degree equations, geometric demonstrationsfor three quadratic bx + C = ax2.
solutions, methods for multiplying with binomials, computing with roots,
The three simple equations are exemplified and solved with dispatch:
further examples for each type of equation, a variety of algebraic problems,
business problems involving proportion, and (as an appendix) additional x2 = 5x x=5
problems illustrating some ofthe standard equations. In the following analysis 5x2 =80 x2 = 16
of each section the discussion will employ modern terminology, such as 1/2X = 10 x = 20.
constant and coefficient, rather than the labored phraseology ofthe translator.
Apparentlythestudentwasexpectedtomemorizetheparadigms, for noexplicit
Thetextofthe lattercan always beconsulted to appreciate the efforts madeand
rules are offered for solving simple equations, save one: ifthe coefficientofthe
success realized by al-Khwarizmi as he sought to put new concepts and
unknown is greater or less than unity, divide or multiply all terms by the
techniques in old words.
inverse of the coefficient to reach unity.1O A geometric structure supports all
theseequations, both simple and composite, astheproofs ofthe methodsshow.
One may well wonder ifal-Khwarizmi had forgotten that he had written a
The strategy ofsetting one side ofan equation equal to zero did not occur until
tract on the decimal system entitled 0/1 Hindu Numerals.8 There he acknowl
the seventeenth century;lI the thinking of al-Khwarizmi and his successors
edgedthatthe decimal systemoriginated with the Hindus; but here in theLibel'
aligned number with geometric magnitude, a concept difficult to dispose of.12
algebre he credits himself with the discovery. As for algebraic numbers al The first example for composite equations is the oft-quoted x2 + lOx = 39
Khwarizmi setthe terminology: square(census), root(radix) and constant(two
and itissolvedbycompletingthesquare.Theroot 3isfound, ofcourse, butitis
names: numerus simplex and dragma). While he may have developed these
not the unknown; the unknown is the square, 9. In other problems the
ideas from astudyofDiophantos'Arilhmetic orEuclid'sElements,9 noonemay unknown is the root. AI-Khwarizmi seems to want his readers to be flexible in
7 Roberto{Chester's Latill Translatioll ofal-Khwarizml'sAL-JABR. A NewCriticalEdition
(forthcoming). Unlike Karpinski's edition, mine is based on the oldestLatin manuscripts. opinion is in line with that of Rodet who claims thatal-Khwiirizmi was 'purely and simply a
8 Edited by B. Boncompagni, Trallati d'arithmetica, vol. I: Algoritmi de /Illlnero Indorum discipleofthe Greekschool', SeeS, Gandz, 'TheSourcesofal-Khowarizmi's Algebra',Osiris I
(Rome, 1857),and by K. Vogel,Mohammed ibnMusa Alchwarizmi'sAlgorisl11us. Dasfriiheste (J936)263-77; W. Hartner, 'DJDAR', TheEncyclopaedia ofIslam, new edition, 5(Leiden, 1965),
Lehrbuch Z/l111 Recllllell mit illdischell ZifJern (AaJen, 1963). pp. 360-62; L. Rodet, 'L'Algebre d'AI-Kharizmi et les methodes indienne et grecque',Journal
9 T. L. Heath,DiophantosofAlexalldria.AStudyintheHistoryofGreekAlgebra, 2ndedition asiatique II ((878) 5-98.
(Cambridge, 1910)and(trans,)The ThirteellBooksofEuclid'sElemel1lS, 2ndrev. edition, 3vols. 10 See below, II.A.II-12: 'Similiter quoque quod fuerit maius censu aut minus, ad unum
(Cambridge, 1926: rpt. New York, 1956): and J. L. Heiberg and E. S. Stamatis, eds., Euclidis reduceturcensum:(Referencesaretochapterandlinenumbersofthetexteditedonpp. 233-6I)
Elemellta, 2nd rev. edition, 5 vols. (Leipzig, 1969-73). The question ofal-Khwiirizmi's sources below).
was addressed by Gandz who describes three schools of thought: Hindu influence, Greek (or II Credit for first setting an equation equal to zero probably belongs to Thomas Harriot
Greek-Hindu)resources, andSyriac-Persian fonts. Heexplicitly rulesouttheGreekbackground (J560-162I), author ofArtis allalyticae praxis... (London, 163I); see G. Loria, Storia delle
because he claims thatDiophantos'Arithmetica, the mostlikelysourceoftheory and problems matematichedall'alba della civilta altramolltodelsecolo XIX, 2ndedition(Milan, 1950), p. 445,
for al-Khwiirizmi, was translated only after the latter'sdeath. Gandz prefers theSyriac-Persian and J. Wallis,A Treatise oj'Algebra, Both Historical alldPractical... (London, 1685), p. 198.
fonts. Hartner, on the other hand, who presentsan informativedescription ofthe development 12 A. G. Molland, 'An ExaminationofBradwardine'sGeometry',ArchivelortheHistory of
ofalgebra in Islamic lands, opts for the Greek resource, particularly that of Diophantos. His theExact Sciellces 19(978) 113-75.
214 B. HUGHES AL-KHWARIZMI'SAL-JABR 215
what is to be sought. Following the example he reiterates the need to reduce, The meaning he intends for the word demonstrate, which appears as 'quod
where necessary, the coefficientofthe squared term to unity. Three additional demonstrare voluimus' attheend ofthe lasttwo proofs, is made explicitbythe
problems exemplify the first ofthe composite types, all solved by completing word which introduces the unit, causa. Rather than offer Euclidean proofs for
the square. It should be noted that al-Khwarizmi had no word for coefficient the methods, al-Khwarizmiconstructs aframework thatshowsvisibly why the
and that he expects his readers to understand that 'Media igitur radices' means methods produce the results. His approach is in fact pedagogical (to bring
'Halve the coefficient ofthe second degree term'. Moreover, he uses the word understanding) rather than logical (toorder understanding). All ofthis becomes
questio to signify our term equation. obvious in an analysis ofone demonstration.
Within the explanation accompanying the solution of the second type of Underlying the demonstration for the method of solving equations of the
composite equation, al-Khwarizmi discusses whether or notan equation in the second composite type is book 2, proposition 5 of Euclid's Elements: 'If a
form ax2 + c = bx can be solved. He says that if the square of half the straightline becutinto equal and unequal segments, the rectangle containedby
coefficient of the first degree term is less than the constant, the solution is the unequal segmentsofthe whole together with thesquare on the straightline
impossible. Furthermore, he remarks, if the same square equals the constant, between the parts ofthe section is equal to the square on the half.'14 Euclid of
then the root is immediately equal to half the coefficient.13 All of this, course proves the theorem synthetically; al-Khwarizmi on the contrary reaches
obviously, is a beginning of an analysis of the discriminant, ~/b2 - 4ac. itanalytically.The propositionisillustratedin thetextedited below(III, p. 239);
Additionally, and for thefirst time, heobservesthatthere may be asecondroot butit must beobserved that the diagram is acomposite picture showing all the
to an equation, which the student may find if he wishes: 'Quod si volueris....' stepstogether. The reader is expectedto draw the figure step bystepin order to
Mindful ofthe foregoing remarks, al-Khwarizrni shows that he is a careful appreciate the force ofthe demonstration. Here is how one should proceed to
teacher as he explains how to solve each ofthe three types ofequations. The solve x2 + 21 = lOx:
'0
rules are easily followed and well exemplified; in fact, a certain commonality IT_21_II
(1) Construct a square to represent Fig. 1
among thesteps becomesobvious. Regardlessofthetype, the first twostepsare
the areaofx2;(2) attach a rectangle to
thesame: halve the number ofroots andsquare the half. Then, for the first and ITj'
a side of the square to represent the
third types exemplified byx2 + lOx = 39 and x2 = 3x + 4 respectively, the
I"
constant term is added to the square; for the second type such as x2 + 21 = area 21; (3) thus added together by
juxtaposition, the two areas equal the ,s
lOx, the constantterm is subtracted from the square. Hence, as noted above, if Fig. 2
area of a rectangle of dimensions 10
the subtraction cannot be done, the equation cannot be solved. (Only much
by x, as shown in fig. 1. ag• ""
later, in the sixteenth century, would Cardano begin to tinker with what I
(4) Bisect the side oflength 10 at t
Descartes would call imaginary numbers whereby the second type can always
haveasolution.)Thefourth stepis thesamefor all types: take thesquarerootof and on the half construct square tklg h" d Fig. 3
(fig. 2) whose area is 25. (5) On hk '.
the sum or difference. Only the fifth and last step which directly produces the
(fig. 3) construct the square hkmn. •
value of x is unique for each type. For the first type, subtract the half of the m I
number of roots from the fourth step; for the second, subtract the square root With the constructions complete, al-Khwarizmi leads the reader through a
from the half; and for the third, add the half to the root. Clear, complete and chain ofreasoning which I will abbreviate.IS The area ofrectangle ahtp equals
concise: the rules need only be memorized. In the next section the student the area ofrectangle mldn, and therefore the area ofcomposite figure thnmlg
Comes to realize why the process always produces a solution. The didactic equals 21. Hence the area of square kmnh is 4, and segment hk =ah = 2.
= = = =
technique employed by al-Khwarizmi, therefore, is first to familiarize students But sinceeh ea + ah x + ah 5, then x 3. Ifthe lengthofsegment
with the canonical types ofequations and methods for solving them and then, x is known, the area ofthe previously unknown square x is 9. And that was
after some expertise had been realized, to demonstrate the reliability of the what was sought; the demonstration is complete. Through a series of visible
methods. constructionsand asequenceoflogicalsteps, therefore, thestudenthasbeen led
14 Heath, Elements 1.383.
13 S"P. h"lnw IIR.Sl-.'ili. IS See below, III.48-80.
to realize that the verbal technique, tantamount to completing the square, a~ _ r;]i2i _ ac
always produces a correct solution. (d) .j[j2 - V~ - b
The scope ofthe chapter on multiplication is limited to the multiplication of (e) (#)(Jb2) = ..j;;2i;i = ~= d
binomialsbymonomials andby binomials, thesecondterm ofabinomial being (f) (ajbi) (c..;d2) = (..j;;2i;i) (..;Aji) = ..;e=f
either positive or negative, the first term always positive. AI-Khwarizmi makes
it clear that the first term ofa binomial is in tens (articuli) and the second in Forseveral ofthese heexpectsthestudentto recall how to find the squareroots
units (unitates), and that if a binomial is multiplied by a binomial, four of numbers, whether they are perfect squares or not.
multiplications are required (each term of one by each term of the other) to' The last section on proofs (cause) offers intuitive explanations for the first
reach the final product. He notes that ifthe second terms are both positive or two problems whichintroducethe fifth chapter. Thisis donebycleveraddition
negative, theirproductisadded to thesumoftheotherpartialproducts; ifoneis and subtraction ofline segments set equal to the components ofthe leftsideof
positive and the other negative, their productis taken from the sum. He begins each problem:
with three specific examples- (l0 + I)(lO + 2), (lO - I)(lO - I), and(lO + (v'2QO- 10) + (20 -1200) = 10
2) (IO - 1)- which are worked out in detail. Since the student is presumed to J20ci)-
(20 - (y'2Oci - 10) = 30 - 2000.
know that each ofthese problems is only a reformation offamiliar factors (11
x 12, 9 x 9, and 12 x 9), heisforced to acceptthe reasonablenessoftherules But the solution ofthe third problem,
for multiplying and adding negative numbers. Then he gives examples of
[100 + (x2-20x)] + [50 + (lOx-2x2)] = 150-(x2 + lOx),
binomialsmultiplied by amonomial- (10- x) 10 and(10 + x) 10- and works
is offered verbally, much as it would be done today: similar terms on the left
these out in detail. Thereafter follow nine examples, eight of which have an
side ofthe equation are collected to yield the answer on the right. The verbal
unknown in the binomial:
explanation was required because al-Khwarizmi knew ofno way to combine
I. (J0 + x) (10 + x) 6. (10 + x)(x - 10) line segments and geometric squares to produce the answer. In view of the
2. (10 - x) (10 - x) 7. (10 + ~) (1/2- 5x) verbal explanation, however, one may wonder why he did not sum up by
remarking that the two previous problems solved by construction could be
4. (10 -x)(IO + x) 8. (10 + x) (x - 10)
resolved easily, in so many words, by collecting like terms.
5. (10 - x)x 9. (x + 10)(x - 10).
The discussion on radical numbers completes what may be called al
Repeatingthesixthexamplein whatIcall theeighth, hevariesthefirst factor to Khwarizml's elementary theory ofequations. The sixth chapter, on equations
maketheninthexamplewhich hesolves. Thethirdexampleisinterestingfor its (questiones), poses six problems each illustrating a different type of equation,
t) t)·
answer: (1 - (1 -~)=%+(~ . He closes this section by repeating the rule the techniques necessary to reduce each to its canonical form, and their
respective solutions. The equations are:
that, ifthe second terms ofthe binomials are opposite in sign, their product is (I) xl = x(lO - x)4
subtracted from the sum ofthe other partial products.
7
Three completed problems introduce the fifth chapter on computing with (2) I02 = 2Tx2
roots. Instead of explaining how these are solved as he did with solving (3) 10-x = 4
equation, al-Khwarizmi proceeds immediately to methods for multiplying and x
dividing radical numbers. While he explains the techniques by examples, the q q
(4) + I) + I) = 20
steps in performing various operations are perhaps best displayed in modern
generalizations: (5) (10 -x)2 + xl = 58
(a) a# = Ja2x2 = ax (6) (~)(~) = x + 24.
(b) aliJi = ..jaW = ok = d
Four technical words which describe operations necessary to putthe problems
..jdi _ f;2 _ a
(c) ..ji;i - v'~ - b into canonical forms appear in the solutions. They are:
218 B. HUGHES AL-KHWARlZMl'SAL-JABR 219
(I) reducere: to reduce the coefficient of the squared term to unity by (7) x2 =Y, y Y+ 2 "I2
multiplying all terms ofthe equation by the reciprocal ofthe coeffi
cient; (S) 0x0(-lOx-)x-)x -- 514"
(2) reintegrare: thesameasreducereexceptthatthecoefficientislessthan
unity; (9) (4x) (5x) = 2x2 + 36
(3) opponere: to subtract apositive term on one side ofan equation from (10) (32x - 3f = x
itselfand from its like term on the other side;
(4) restaurare: to add the absolutevalue ofanegative term from oneside (II) 3/2 =2x
ofan equation to itselfand to the other side. x + I
I
(12) x-To
< x +
Twelve additional problems VII. Questiones vade> reinforce much of
whathas Preceded. They areamixed bagcontainingasurprise. First, onlyfour The last chapter in Liber algebre is a short section on proportion applied to
ofthe model equations receive further exemplification: business problems, the well-known 'RuleofThree'. Following clear statements
ax2 = c: (example 9) aboutpossiblevariationsattendantuponthreegiven numbers with the fourthto
be found, i.e.,
bx = c: (examples 2and 7)
ax2 + bx = c: (example 12) ab xc and ab :c!'
ax2 + c = bx: (examples I, 2-6, S, 10-1I).
two examplesare worked through in detail. Interestingis the translator's use of
theexpressionnumerus ignotus for 'the number to be found', aphrasethatdoes
Secondly, thesurprise is new material: fractional equationsin examples(4), (S),
notappear in thetheory ofequations, as well as it might; there the unknown is
(7), (8), (1), and (2). Two methods for solving these fractional equations are
always referred to as res or census. A third example closes the chapter; it was
presented: first, the equivalent of cross-multiplication in (S), (7), (8), and (11);
probably included for its practical value since it focuses upon payment for six
second, the equivalent ofmultiplying each term ofthe equation by the lowest
days' work wherethesalaryissetfor onemonth's work. Withthis lastproblem
common denominator in (4) and (2). Furthermore, example (7) requires the
the older manuscript copies ofGerard's translation conclude Liber algebre.
reader to readjust his thinking; the object of the problem or unknown is a
Found only in Gerard's translation are the contents of the Appendix on
square. Since the initial equation will eventually become a quadratic, al
pp.2S7-61 below. Robert of Chester's version shows an appendix that
Khwiirizmi tells the student to treat it as res, the usual word for the first degree
summarizes the rules for solving the six types ofequations; William ofLunis
variable, X; otherwise, the problem produces a fourth degree equation which is
offers, as though hisown, nearly all ofthe algebraic section ofchapter IS, part
outside the scope ofthe text.16 Finally, the statements ofthe problems become
these equations: 3, of Fibonacci's Liber abaci; Rosen's Arabic source (MS. Oxford, Bodleian
Library Hunt 214, fols. 1-34) includes three additional chapters, on mensura
(I) x(lO - x) = 21 tion, legacies, and computation of returns.17 Gerard made it clear that he in
(2) (10 - x)2 - x2 = 40 corporated the material from another font, for he wrote: 'Liber hic finitur. In
(3) (10 - x)2 + x2 + (10 - x) - x 54 alio tamen Iibro repperi hec interpositasuprascriptis' (below, Appendix 2). His
(4) 10 - x + x = 21 statementcertainlysuggeststhat he recognizedthatthe material was not written
x 10 - x 6 byal-Khwarizmi, yethesawanothercopyofal-Jabr whichcontainedthesetof
(5) ~( 105:.x )=5(10 - x)
(6) (lO-x)2 = Six
17 TheAlgebra ofMohammed ben Musa (London, 1831; rpt. New York, 1969), a frequent
referenceforKarpinski;seeabove,n. 5. YetRosen'stranslation hasbeenseverelycriticizedbyJ.
Ruska, Zur iiltesten arabischen Algebra und Rechenkunst (Sitzungsberichte der Heidelberger
16 Thetechniqueofsubstitutingy forx2isusedextensivelyinLiberaugmelllietdiminutionis; Akademie der Wissenschaften, phil.-hist. Klasse 8; Heidelberg, 1917). Toomer gives the
see Libri, Histoire des sciences matlu!matiques en !talie 1.308 and passim. locations for three ofthe Arabic manuscripts; see his 'al-KhwarizmI',DSB 7.364.
220 B. HUGHES AL-KHWARIZMI'sAL-JABR 221
problems. Since one aspect of the value of the present critical edition is to (3) (x2-3x2-4x2- 4) =x2 + 12
appreciate a translation which from the sheer force of the number of extant
copies is assumed to have provided a major thrust toward the development of (4) x2(32) = 5
algebrain medieval Europe, Ijudged itimportantto includethe Appendix as a
cognate part ofal-Khwarizml's tract, although I have no evidence that he was (15) x2- y2 = 2 and-y2 =-I
x2 2
its author.
The Appendix is a selection oftwenty-one problems making a very uneven (16) x2 Ox) = 5xl
group. Abouthalfthesolutionsarestraightforward; the remainder do notcome (7) +(Xl - ~~ 3x =x2
so easily. Early on, the student is confronted with three quartic equations in a
row, (4}-(6), followed by a cubic. Although their solutions are shown to be (8) (xl - 4x) = 4x and Xl = 256
similar to si'mple types studied before, the student does have to refine his tools
(19) ';x2-x + x =2
for solving problems. Enough practice is offered, however, particularly for
=
thinking ofeel/sus in terms ofres or radix. (20) (xl - 3x)2 Xl
Problems (I5) and (19) are the most interesting, ifnotthe most difficult. The (2I) (x2) (~ Xl) = 5.
former begins with the squares of two unknowns, instead of the customary
'Divide ten into two parts'. Two relationships are established between them,
THE LATIN MANUSCRIPTS
which permit asubstitution from one equation into the other thereby reducing
the problem to one equation in one unknown. In problem (19), for the first
MarlUscriptcopies ofGerard's translation begin 'Hie(or Sic)post laudemdei
time, the student is confronted by a radical binomial in an equation. The
etipsius exaltationem inquit(or inquid)' and generally conclude'... cuius radix
wording ofproblemand solution, however, isobscure (thescribe's fault?); and
est quinque'. They are easily separated into two groups. Most of the seven
the medieval Latin reader may have ignored this partas unintelligible. This is a
manuscripts in the first set are from the thirteenth and early fourteenth
pity, since a new technique lies here: squaring both sides of an equation to
centuries and exhibit few significantvariations among themselves. The second
remove a radical term. Both problems are finally solved quite conventionally.
set ofeight manuscripts are later in composition, show many variations from
The problems in the Appendix may be expressed as follows:
the first set and among themselves, offer fewer or more problems, and suggest
(I) (I0 - x)2 == 81 that the terminology has been edited. The critical edition is based on the first
(2) lOx == (10 - x)2 group whose members are described below in detail; the various titles ofthe
2 I I tract are given immediately after identification ofthe codices. The members of
(3) 3(Sx2) == TX the second group are recognized as witnesses to the importance(or perhapsthe
(4) x2(4x2) == 20 availability)ofGerard's translationandare described inless detail. All and only
(~2) scientifil: works in the codices ofthe first group are itemized; some works are
(5) (x2) == 10
marked with an asterisk to signal atranslation to Gerard ofCremona. The diad
(6) (x2) (4x2) == 3x2 TK following a title refers to Thorndike and Kibre's Catalogue ofIncipits.18
(7) (x2) X == 3x2 Fonds ofthe Critical Edition
(8) Ox)(4x) == x2 + 44
C=Cambridge, Cambridge University Library Mm.2.18, fols. 65rb-69vb
(9) x (4x) == 3x2 + 50
('Uber maumeti filii moysi alchoarismi de algebraet almuchabala incipit').
(10) x2 + 20 == 12x France, c. 1360.
(1)(:~
(II) == x2
11 L. Thorndike and P. Kibre, A Catalogue ofIncipits ofMediaeval Scientific Writings in
II"'I) Latin, 2nd edition(Cambridge, Mass., 1963).
Contents: 42v-53v: Jordanus de Nemore, De numeris datis [TK 959]. (16) fols. 53v-70r: John of
(I) fols. 2r-49r: Jabir ibn Afla};1 al-Ishbili,Flores de almagesto* [TK 1403]. (2) fols. Seville, Algorismus [TK 1250]. (17) fol. 70r-v: Anon., Compotlls. (18) fols. 71r-72v:
49r-65r: Anon.,Liberdenumeriset/ineis rationalibus* [TK 33].(3) fols. 65rb-69vb: al RobertGrosseteste,Delineis angulisetfiguris [TK 1627].(19) fols. 72v-80r: Anon.,De
Khwiirizmi,Liberde algebra et almuchabala* [TK 624]. (4) fols. 69vb-76v: Abu Bakr numeris fractis [TK 1475]. (20) fols. 80r-86r: al-Khwiirizmi, Liber de algebra et
al-l;Iasan ibn al-Kha~ib, De mensuratione terrarum* [TK 281]. (5) fols. 76v-77r: Abu almuchabala* [TK 624]. (21) fols. 87r-91v: A};1mad ibn Yusuf al-Kammad, De
CUthman Sacid ibn YaCqub al-Dimashqi, De mensurationefigurarum superficialium et proportioneet proportionalitate [TK 1006].
corporearum* [TK 1390]. (6) fol. 77r: CAbd al-Ra};1man,De mensuratione* [TK 1387]. Thealgebrawasobviouslycopiedpiecemealbytwoscribes:the firstwas responsible
(7) fols. 77v-82r: Abraham ibn Ezra(?), Liberaugmenti et diminutionis [TK 238]. for fols. 80r-81vandthesecond for fols. 82r-86v.Themanuscriptissignificantforthree
ThecodexwascommissionedbyGeoffreydeWighton,O.F.M.,andpaidfor'byalms reasons. First,although itcontainsmorevariantsthantheotherthreemanuscripts used
given by his friends'.'9 Thomas Knyvett (d. 1622), Baron Escrick who discovered the for thecriticaledition,itsearlydatesuggestsastronginterestinal-Khwiirizmi'salgebra.
gunpowder plot, obtained the bookas his nameand motto within testify. Thereafter it Second, it is the only manuscript with the unusual spelling ofcensus, namely, sensus,
which occurs in the section copied by the first scribe. Third, a (near?) contemporary
passed into the library ofThomas Moore (1646-1714). Upon his death the collection
was purchased by King George 1 and presented to Cambridge University in 1715.20 gloss attributes the translation incorrectly to William of Lunis: 'Incipit liber gebre de
numerotranslatusa magistroGuillelmodelunisinquadrivialiscienciaperitissimo'(fol.
Items2through7mayhavebeencopieddirectlyfrom Paris,BibliothequeNationalelat.
80ra). Whilethe noteis excellenttestimony to the fact thatWilliam ofLunisdid trans
9335 orfrom itsexemplar,sincetheyareinexactlythesameorderastheyappearinthe
lateal-Khwiirizmi'sal-Jabr,itmiscreditsWilliamwiththistranslation. Heisresponsible
Pariscodex. The algebra, item 3, is the manuscript mentioned by Montfaucon. Byand
for an entirely different translation which spawned its own family ofcopies.21
large, itisaverygoodcopywithfew variationsfrom Parislat. 9335,notablykaficii for
cafficii (fol. 115rb) and only three omissions ofsignificant length (fols. 113ra, 113va, Bibliography: A. A. Bjornbo, Die mathematischen S. Marcohandschriften in Florenz,
116rb), the first and third due to homoeoteleuton. 2ndedition, ed. G. C. Garfagnini(Quaderni di storiae critica dellascienza, N.S.;
Pisa, 1976), pp. 88-92; B. B. Hughes, ed. and trans., Jordanus de Nemore. De
Bibliography:ACatalogueoftheManuscripts Preservedin theLibraryofthe University
numeris datis (Publications ofthe Center for Medieval and Renaissance Studies
ofCambridge 4 (Cambridge, 1861), pp. 132-38.
13; Berkeley, 1981), pp. 27-28; R. B. Thomson, 'Jordanus de Nemore: Opera',
= Mediaeval Studies 38 (1976) 97-144 paSSim.
F Florence, Biblioteca Nazionale Cony. soppr. J.Y.18 (Codex S. Marci
Florentini 216), fols. 80r-86v (no title). France/Italy, saec. Xlll ex. =
M Milan, Biblioteca Ambrosiana A 183 inf., fols. 115r-120r ('Incipit liber
Contellls: Mulumecti de algebra et almuchabila'). Northern Italy, saec. XIV in.
(I) fols. Ir-2r: Anon.,Liberde umbris.(3) fols. 4r-9v: Anon.,Liberysoperimetrorum
Contents:
[TK 1083 (3)]. (6) fols. Ilr-12v: Anon., (inc.) 'Perisimetra sunt quorum latera
coniunctim sunt...' [TK 1035]. (7) fols. 12v-16r: Anon., Practica geometrie [TK 870]. (I) fol. Ir: Anon., De compoto (fragmentum finis). (2) fols. Iv-7r: John of Sacro
(8) fols. 17r-29v:JordanusdeNemore,Detriangulis[TK760].(9) fols. 30r-32r:Anon., bosco, De spera [TK 1577]. (3) fols. 7v-13v: Jordanus de Nemore, De triangulis [TK
Liberdesinudemonstrato[TK477].(10) fols. 33ra: Anon.,Quadraturaperlunulas[TK 260].(4) fols. 14r-19v: Ptolemy,Planispheriwn[TK 1190].(5) fols. 20v-21r: al-Battani,
1058]. (II) fols. 33rb: Thabit ibn Qurra, De proportionibus [TK 1139]. (2) fols. 33v (excerptum inc.) 'integrorum multiplicantis'. (6) fols. 22r-23r: Campanus de Novara,
341': Campanus de Novara,Defigura sectoris [TK 280]. (13) fols. 37ra-39rb: Jordanus Almanach coniunctionum mediarum solis et lune. (7) fols. 24r-28v: al-Qabisi (trans.
deNemore,Demonstratio in algorisl1lum (inc.: 'Numerorumaliussimplex...')[TK958]. JohnofSeville),Adiudiciaastrorum(fragmentum).(8) fols. 29r-56r: Sahlibn Bishr,De
(J4) fols. 39rb-42va: Jordanus de Nemore, Tractatus minutiarum [TK 875]. (5) fols. significationetemporisadiudicia[TK1411].(9) fols. 56v-64v: Miishii"alliih(trans.John
ofSeville), De receptionibus [TK 774]. (10) fols. 64v-68v: Miisha"alliih (trans. John of
Seville),De revolutione annorum mundi[TK 362]. (II) fols. 68v-7Ir: Miishii"alliih, De
coniunctionibus planetarum [TK 729]. (12) fols. 71r-73r: Pseudo-Hippocrates, Liber
19 'Iste libel' est Fratris Galfridi de Wyghtonc quem fecit scribi de elemosinis amicorum astronomiae. (3) fols. 74r-76r: Thiibit ibn Qurra, Imagines [TK 285]. (14) fols. 76r
suorum' (fol. 11'). See A. B. Emden, A BiographicalRegisterofthe University ofOxfordtoA.D.
77v: Thiibit ibn Qurra, Super almagestum [TK 1570]. (5) fols. 77v-78v: Thiibit ibn
f500 3(Oxford, 1959), p. 2045.Thecodexis notmentioned byN. R. Ker,MedievalLibrariesof
Grew Britain, 2nd edition (London, 1964), even though Friar Geoffrey lived and died in Qurra,Demotu octavespere[TK 66il. (16) fol. 791': Anon.,Brevis tractatus desperico
England.
Many of the details here were graciously supplied by Jayne Cook, Assistant Under-
20
Librarian ofCambridge University Library, to whom my thanks. 2/ See n. 6above.
224 B. HUGHES AL-KHWARIZMI'SAL-JABR 225
corporeetsolido. (17) fols. 80-114 desunt. (18) fols. 115r-120r:al-Khwiirizmi,Liberde archum per cordam invenire'; (5) fols. 23v-25r: Thiibit ibn Qurra, Introductio in
algebra et almuchabala* [TK 624]. (19) fols. 120v-122v: Anon.,Algorismus [TK 990]. almagestum [TK 502]. (6) fols. 25r-28v: Theodosius ofBithynia,De locis habitabilibus
A noteworthy gathering oftreatises copied by several Italian (and French?) scribes [TK684].(7) fol. 28v:Anon.,Ordoquiestpostlibrumeuclidissecundumquodinvenitur
from the mid-thirteenthtothe mid-fourteenthcentury, the codexis witnessto astrong in scriptisIohanicii. (8) fols. 28v-30r: LiberArsamitisde mensura circuli. (9) fols. 30r
interestinscientifictopics. Thecopy ofLiberalgebredisplaysdifferencesin notationas 31v: AhmadibnYusuf,Dearcubussimilibus[TK624]. (10) fols. 31v-32v: ai-Kind!,De
well as improvements upon explanations. For instance, to represent 31/3 the scribe quinque essentiis* [TK 1376]. (II) fols. 32v-54v: Menelaos, De figuris spericis* [TK
wrote 1/ 3. For textual emendation, in place of 'Die: "Hie ... equales rei'" (below, 397]. (12) fols. 55v-63r: BanuMusa,Libertriumfratrum[TK 832].(13) fols.63v-64v:
3
VII.I02-104), there appears this clearer statement: 'Pro minori censu pone rem. Pro Anon., (inc.) 'Iste modus est sufficiens in arte eptagoni cadentis in circulo'. (14) fols.
maiori vero censu pone rem et duas dragmas. Quibus muitiplicatis per mediam 64v-75r: Ahmad ibn Yusuf, De proportione et proportionalitate [TK 1139]. (15) fols.
dragmamque provenitexdivisione minoris censi (sid per maiorem eteveniunt media 75r-82r: al-Kindi, De aspectibus [TK 10l3]. (16) fols. 82r-83v: Pseudo-Euclid, De
res et dragmam (Sid, id est, que equantur uni rei' (fol. 118va27-34). Seemingly, early speculis [TK 1084]. (17) fols. 84r-88v: ai-Kind!, De speculis [TK 1388]. (18) fols. 88v
fourteenth-century scholars were seeking clarifications and improvements upon texts 92r:Anon.,Deaspectibuseuclidis.(19) fols. 92v-IIOr:Mul,lammadibncAbdal-Biiqial
handedthem. Yetthecopy has numerous defects; a representativeselectionareshown Baghdadi,Commentaria in eUclidis elementis lib. X* [TK 333]. (20) fols. IIOvb-116va:
al-Khwiirizmi, Liberde algebra et almuchabala* [TK 624]. (2I) fols. 116v-125v: Abu
in the apparatus.
Baler al-I:Iasan ibn al-Khal?!b, De mensuratione terrarum* [TK 281]. (22) fols. 125v
Bibliography: P. Revelli, I codici ambrosiani di contenuto geografico (Milan, 1929),
126r: Abu CUthmiin Sacrd ibn YaCqub al-Dimashq!, De mensurationefigurarum [TK
pp. 24-25; A. L. Gabriel,A Summary Catalogue ofMicrofilms ofOne Thousand
1390]. (23) fol. 126r-v: Aderametus (cAbd al-Ral,lmiin?), De mensuratione [TK 1387].
Scientific Manuscripts in the Ambrosiana Library, Milan (Notre Dame, Ind.,
(24) fols. 126v-133v: Abraham ibn Ezra(?), Liber augmenti et diminutionis [TK 238].
1968), p. 44.
(25) fols. 135t-139v: al-Kind!, De gradibus medicine [TK 1228]. (26) fols. 140r-14Ir:
Anon.,Capitulumcognitionismansionislune. (27) fols. 14Ir-143r:ThiibitibnQurra,In
N =Paris, BibliotMque Nationale fr. 16965, fols. 2r-19v ('Liber mahumeti filii
motusaccessioniset recessionis. (28) fols. 143v-151v: al-Fariib!,Descienciis* [TK925].
moysi alchorismi de algebra et almuchabla incipit'). France, saec. XVI in.
(29) fols. 151v-160v: Harib ibn Zeid, De hortis et plantationibus.
Contents: Recognized as 'perhaps the most important manuscript of Gerard of Cremona's
(I) fols. 2r-19v: al-Khw~.rizm!, Liberde algebra et almuchabala* [TK 624]. (2) fols. works',22 thecodexisaveritablemineofmedievalresources,twenty-ninetractsinpure
20r-26v: Anon.,Excerptio uelExpositio compotiHerici. (10) fols. 379r-407r: Rudolph and applied mathematics. The 161 leaves are ofparchment, the two Columns oftext
of Spoleto, De proportione proportionum disputatio. (II) fols. 408r-448r: Anon., were written by a single hand, initials are red and blue, and an earlytable ofcontents
Arithmetica logarithmica. appearson fol. I. Thealgebra, item 20, isobviouslythe besttextofallthemanuscripts
Sometime in the Saint-Germain collection, the codex is an anthology of scientific reviewed: the wording is unambiguous, the diagrams are helpfully complete, and the
works copied in the sixteenth and seventeenth centuries, both in Latin and in French marginalia evince careful corrections by the same scribe who penned the text. (I
(twelve titles not identified above). The algebra was written in averyclear humanistic incorporated these corrections as well as others made by him, interlinear or over
hand, most probably copied from Paris, BibliotMque Nationale lat. 9335, and at one penned, into the text ofthe critical edition and noted them in italics). These features,
timethe manuscriptwaspartoftheLustierineLibrary.Therearenosignificantvariants reinforced by the manuscript's having been copied within perhaps fifty years of the
to recommend its use for the critical edition. translationand, conjecturally, from the final draft ofGerard, make Ptheexemplar for
all copies in its genre.
Bibliography: L. Delisle,Inventairegeneralet methodiquedes manuscritsfranraisde la
BibliothequeNationale 2(Paris, 1876; rpt. 1975), p. 235. Bibliography: L. Delisle,Inventairedes manuscritsconservesdlaBibliothequeImperiale
sous les n. 8823-lJ503 du fonds latin (Paris, 1863), no. 9335; A. A. Bj6rnbo,
P= Paris, BibliothequeNatlonale lat. 9335, fols. 11Ovb-116va('Liber maumeti 'lJber zwei mathematischen Handschriften aus dem vierzehnten Jahrhundert',
filii moysi alchoarismi de algebra' et almuchabala incipit'). Southern Bibliotheca mathematica, 3rd Ser., 3 (1902) 63-75 and corrections to this by
France/Italy, saec. Xlll in. Bj6rnbo in 'Handschriftenbeschreibung', Abhandlungen zur Geschichte der ma
thematischen Wissenschaften mitEinschluss ihrerAnwendungen 26.1 (I910) 138.
Contents:
(I) fols. Ir-19r: Thecdosius of Bithynia, De speris* [TK 1523]. (2) fols. 19r-2Iv:
Autolycus of Pitane, De motu spere* [TK 1151]. (3) fols. 22r-23r: Asculeus, De II R. H. Rouse, 'ManuscriptsBelongingto RicharddeFournival',ReViled'!listoiredes textes
ascensione signorum* [TK 1449]. (4) fol. 23v: Anon., (inc.) 'Cordam per archum et 3(J973) 256-57,
226 B. HUGHES AL-KHWARIZMI'sAL-JABR 227
Q=Paris, BibliothequeNationale lat. 7377A, fols. 34r-43v('Uber maumeti filii A comparison ofvariants between the textsofthe firstgroup ofmanuscripts
moysi alchoariximi de algebra et almuchabala incipif). France, saec. XIII. produces ten readings whereby the manuscripts can be separated into two
families, a and p, namely:
Contents:
(I) fols. Ir-33v: Anon.,Commentarius in decimum euclidislibrum. (2) fols.34r-43v: V-I: comprehendi potest de numeris ultime (1.13-14)
al-Khwarizmi,Liberde algebra etalmuchabala* [TK624].(3) fols. 43v-58v: Abu Bakr V-2: ad infinitam numerorum comprehensionem (1.13-14)
al-f;Iasan ibn al-Kha~ib, De mensuratione terrarum* [TK 281]. (4) fols. 58v-70v(?): V-3: questio est impossibilis m.B.55)
Abraham ibn Ezra(?), Liber augmenti et diminutionis [TK 238]. (5) fols. 7Iv-97v: V-4: questio est destructa(or destracta) m.B.55)
Anon., Scholiul/l de mensuratione pentagoni et decagoni. (6) fols. 99r-208r: Anon., V-5: dupla ergo radicem novem (Y.37-38)
Tractatus de aritlllnetica. V-6: multiplica ergo radicem novem (Y.37-38)
V-7: capitula numerationis et eorum modos(VI.2)
The Liber de algebra was copied directly from Paris lat. 9335 (P) during the third
V-8: capitula et eorum modos (VI.2)
quarterofthethirteenthcentury, possiblybyaParisianuniversityscribe,andthecodex
V-9: reintegres censum tuum (VII.l42-143)
was sometime in the Colbertine Library. Not only are there comparatively few
V-IO: reintegres novem radices (VII.142-143).
variations between the two manuscripts, but the corrections found in the margins of
Parislat. 9335 wereoftencopiedontothesamerelativeplacesinParislat. 7377A. Two
V-I and V-2 are different conclusions for the first paragraph ofthe tract; each
noteworthyvariationsinspellingoccur:kaficiiforcafficiiandcenteximeforcentessime.
This copy has nothing to offer the critical edition. ofthesevenmanuscriptsshowseitheroneortheotherexpression. V-3 and V-4
are different though meaningfully the same description of the possibility of
Bibliography: Catalogus codicul11 manuseriptorum Bibliothecae Regiae 4 (Paris, 1744),
solvinga particulartype ofan equation. Six ofthe manuscripts have oneorthe
p.349.
otherreading; the Vaticanmanuscriptlacks the clause in alongpassageomitted
v = possibly because ofhomoeoteleuton. Again, with respect to V-5 and V-6, all
VaticanCity, BibliotecaApostolica Vaticana Vat. lat. 5733, fols. 275r-287r
manuscripts haveeitheroneorthe othervariantreading. Thesame holds forV
('Incipit liber Mahumed filii Moysi Algorismi de algebra et almutabala
7 and V-8, and for V-9 and V-IO. The manuscripts with their respective
transcriptus a magistro Simone Cremonensi in Toleto de arabico in
variants Can be displayed in a matrix:
latinum'). Italy, saec. XVI in.
Manuscripts V-I V-2 V-3 V-4 V-5 V-6 V-7 V-8 V-9 V-IO
Contents:
Cambridge (C) x x x x x
(4) fols. 189r-195r: Hermes,Liberdequindecim stellis[TK768].(7) fols.21Ir-229v:
Paris fr. 16965 (N) x x x x x
Averroes (trans. Cal. Calonymus), Destructio destructionis. (12) fols. 275r-287r: al
Paris lat. 9335 (P) x x x x x
Khwarizrni, Liberde algebra et almuchabala* [TK 624].
Paris lat. 7377A (Q) x x x x x
This collection of scientific tracts centers on the work of Petrus Pomponatius of
Florence (F) x x x x x
Mantua (1462-1524) and was sometime part of the library of the gymnasium at
Milan (M) x x x x
Bologna. In general, the algebra text is reliableas far as itgoes; butitendswiththelast
Vatican (V) x x x x
ofthe Questiones varie, without the section 011 proportion nor with the set ofextra
problems. On a separate folio (274r) is a uniquetitle for the tract,Ars algebrae, which
The pattern recommends a clear division ofthe manuscripts into two families.
beginson fo!. 275ralong withthecompletelyerroneousascriptionofthe translationto
'magistroSimoneCremonensi'. The bodyofthe text is well-written inasimplecursive Hence, the members of the a family are C, N, P, and Q. Members of the fJ
handandisdividedintoclearlystatedsections,muchasIhavedone, withsubtitles.The family are F, M, and V.
only addition of any significance to the text is an insertion carefully placed within Regarding intrafamilial relationships: in the a family P is clearly the oldestin
parentheses(fo!. 282r) which I have included in the critical apparatus at VI.l8 below. thegroupandthe bestextantcopy. Qisa nearlyperfectreproduction ofP, with
While a good witness to the continuing interest in the work of al-Khwiirizmi, this only one omission ofany length (below, Appendix 115-116). N seems to be a
manuscript has nothing further to add to the critical edition. better copy of P, for it has the passage missing in Q. C has its own set of
Bibliography: R. Lemay, ed..PetriPomponatiiMantuaniLibriquinquedeJato, delibero omissions which are not found in N, P, or Q. and it does contain the sentence
arbitrio et de praedestinatione (Lucca, [1957]). pp. xxxi-xxxiii. missing in Q. These factors make a case for thegenealogy shown below for the
228 B. HUGHES AL-KHWARIZMI'sAL-JABR 229
members of the a family. The f3 family consists of three strikingly different Despite considerable water damage (humidity?) to the outer edges of the leaves,
manuscripts. They may be briefly described as exhibitors ofat leastone major much ofthe text can be read. It is clearly a member ofthe{J family, butseveral times
characteristic unique to each. F explains the word 'algebra' by oppositio and removed.Thereareanumberofeditorialchangesinthetextansproblems 1, 14,and 15
'almuchabala' by responsio. M has a unique substitution at VILI02-104. V are missing from the Appendix.
would have us believe from the title that the tract was translated by Simon of Bibliography: E. Narducci, Catalogo di manoscritti ora posseduti da D. Baldassarre
Cremona. Hence, none is a direct descendant of either of the other two Boncompagni, 2nd edition (Rome, 1892), p. 106 n. 179 (no. 265).23
members ofthe f3 family. A stemma sets forth the relationships:
(3) Madrid, Biblioteca Nacional 9119 (olim Aa 30), fols. 352v-360v ('Incipit liber
Mavmet filii Moysi Algorismi de Algebra et Almuchabala: Translatus a Magistro
Gerardo Cremonensi in Toleto: de arabico in latinum'). Italy, saec. xv ex.
{J inc.: Hic post laudem dei et ipsius exaltationem inquid...
a
P~ ~ expl.: _.. provenit .25. dragme cuius radix est .5.
F M V While the text has many characteristics typical of the {J family, it also evinces
~ numerouseditorialchanges,someofwhicharesimplyerroneous_ Furthermore,thetext
C N Q does not end with the twenty-first supplementary problem but continues on (fols.
360v12-363v) with sixteen additional algebraic problems, twenty-eight definitions for
Manuscript Witnesses arithmeticandgeometry,atractonextractionofroots,andeighteenproblemsandrules
for geometryand astronomy. Only alargeQ in themargin introducing the wordQuod
signals the beginning ofthis second appendix.
The eight manuscripts briefly described below witness to efforts ofscholars
to improve upon Gerard's translation. Each offers modifications in terminol Bibliography: J. L. Heiberg, 'Neue Studien zu Archimedes',Zeitschriftfiir Mathematik
undPhysik (Supplement 1890)5; M. Clagett,Archimedes in theMiddleAges,vo!'
ogy, addition or omission of problems, or additional textual material. Hence,
2.1-2: The Translations from the Greek by William ofMoerbeke (Philadelphia,
none ofthese manuscriptswas used to constructthe criticaledition. They merit
1976), pp. 69-71.
attention, however, as witnesses to the importance of Gerard's translation
which served as their foundation, not to overlook the burgeoning interest in (4) Milan, Biblioteca Ambrosiana P 81 sup. (oUm YS), fols. Ir-22r ('Machumeti de
algebra. AlgebraetAlmuchabala, idestrecuperationisetoppositionis. Liberincipit'). Italy,saec.
xv in.
(l) Berlin, Deutsche Staatsbibliothek Hamilton 692, fols. 279r-29lv ('In nomine dei <
inc.: H>ic post laudem dei et ipsius exaltationem inquid...
eterni. Incipitliber Mauchumeti in AlgebraetAlmuchabulaqui estorigoetfundamen
expl.: ... provenit radix de xxv et ilIa est v. Et cetera.
tum totius scientie arismetice'l. Italy, saec. XVI in.
Thiscopyisa memberofa family offour displaying thesame uniqueCharacteristics,
inc.: Hic post laudem dei et ipsius exaltationem inquit...
notably, the frequent use ofcosa for res and the addition oftwo lengthy paragraphs
expl.: ... et proueniunt 25, cuius radix est 5.
which begin 'Modus dividendi'. As a matter ofconvenience, I identify the set as 'the
Ingeneral this copy has the characteristicvariantsofthe{J family. Itisclassifiedas a Modus family'. Also characteristic ofthe family is the sentence 'Sed ut res gravis leuis
witnessbecauseofthenumeroussubtitleswhichwereaddedtothetext.Furthermoreit tibi fiat, sequaturidquod exquestionibus in textu propinquiscumilleeratperquorum
adds six problems to the Questiones varie and omits problems 1, 14, and 15 from the significationeminaliisconsimiliteroperaberis,sideusvoluerit'(fo!. 11d.Thecodexwas
Appendix. one time the property ofGian Vincenzo Pinelli (I535-160I).
Bibliography: H. Boese, Die lateinischen Handschriften del' Sammlung Hamilton zu
Ber/in (Wiesbaden, 1966), pp. 334-35.
II ThemanuscriptisnotrecordedbyV. RoseandF.SchiJlmann, Verzeichllisderlateinischen
Handschriften de,. Koniglichen Bibliolhek zu Berlin, 3vols. (Berlin, 1893-1919). However, the
(2) Berlin, Staatsbibliothek Preussischer Kulturbesitz Lat. quo 529, fols. 2r-16v ('...
microfilm shows two parts (whatappears to be the first and last third) ofa label:
Macumetii ... Algebra....). Italy, saec. xv med.
En~
< :gnil The obvious deduction led me to Narducci's catalogue whose description fits
inc.: H>ic post laudem dei et ipsius exaltationem inquit...
expl.: ... et proueniunt 25 cuius radix est quinque. exactly the contents ofthe manuscript shown on the microfilm.