The Algebra of Abu Kamil
Author(s): L. C. Karpinski
Source: The American Mathematical Monthly, Vol. 21, No. 2 (Feb., 1914), pp. 37-48
Published by: Mathematical Association of America
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THE
AMERICAN
MATHEMATICAL MONTHLY
VOLUMEXXI
FEBRUARY,1914
NUMBER2
THE ALGEBRA OF ABU KAMIL.
By L. C. KARPINSKI, University
ofMichigan.
Popularity
of AlgebraAmongthe Arabs. The greatinterestin the study
ofalgebraamongtheArabsoftheMiddleAgesis attestedbythenumerous
works
writtenupon this subject by Mohammedan mathematicians. The period of
this popularityextends fromthe ninthto the fifteenthcentury. The students
ofalgebraincludedpoets,philosophers,
and kings. The popularOmarKhayyam
was too excellent
a mathematician
and toofinea poetto woothemuseofalgebra
in verse. But histreatisein proseon algebra,editedwithFrenchtranslation
by
Franz Woepcke,is as good a titleto fameas his verse. In the libraryof the
Escurialis preserved
a poemwritten
sixhundred
yearsagobya nativeofGranada,
Mohammedal-Qasim,and treating
ofalgebra. Needlessto saythecontentdoes
not comparewiththe proseof OmarKhayyam. The King of SaragossaJusuf
al-Mutamin(reigned1081-1085)was a devotedstudentof the mathematical
sciences. The titleof one of his workswouldseemto indicateits algebraical
nature. Even in the studyof law a knowledge
of algebraseemsto have been
forvariousquestionsofinheritance
necessary,
weretreatedby thisscience. On
the whole,however,
in thathappyday the questionofpracticalapplicationsof
mathematics
did not loom so largeupon the horizonof the Arabicscientist.
That fortunate
individualfeltentirely
freeto pursuethissubjectas one of the
naturaland inevitableactivitiesof the humanintellect.
The FirstGreatArabicWriteron Algebra. The firstsystematic
treatiseon
MohammedibnMusa, al-Khowarizalgebrawas theproductofan Arabicwriter,
mi (c. 825 A. D.). Commentaries
werewritten
uponthisworkbyArabicauthors
and twodifferent
Latin translations
weremadeofit. One ofthesetranslations
was published by Libri in his Histoiredes sciencesmathematiques
en Italie, Vol. I,
ofthis
253-297,Paris,1838. This workis out ofprint. Whilethe authorship
versionis uncertain
thecomposition
maysafelybe placedin thetwelfth
century,
as severalwritersofthatcenturyappearto have studiedthisworkand use the
of this translation.Manuscriptcopiesare somewhatcommonin
terminology
37
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38
THE ALGEBRA OF ABU KAMIL
European libraries. French, German, and Italian writersof the fifteenthand
sixteenthcenturiesdrew inspiration,and copy, fromthese pages. The other
Latin translationwas effectedin 1144 A. D. (1183 of the Spanish era) by an
Englishman,Robert of Chester,the firsttranslatorof the Koran. This version
is found,so far as known,only in three manuscripts.1 The text with English
translation and notes is being published in Vol. XI, Universityof Michigan
Studies,HumanisticSeries. The Arabic text with English translationwas published by F. Rosen, London, 1831, but unfortunatelythis last is out of print.
Adequate treatmentof this initial work in the science of algebra is found in
Cantor's VorlesungenuiberGeschichte
der Mathematik,volume I.
The Second Great Arabic Writeron Algebra. The name and fame of AlKhowarizmiis knownto most studentsof mathematics. But fate has not dealt
so kindlywith the second, in point of time, of the great Arabic writersin this
field. Abu Kamil Shoja' ben Aslam2wrotein the late ninthor earlytenthcentury
a much more extensivetreatise on algebra than that of Al-Khowarizmi. Two
commentariesof the tenth centuryhave been noted by Arabic historians. In
addition to these a commentaryin Arabic was prepared by a Spanish Arab,
Al-Khoreshi,and a Spanish translationby an unknown Christian Spaniard.3
This was followedby a Hebrew translation,which is preservedin manuscript,
by Mordechai Finzi (c. 1473) of Mantua. Not alone by these translationsand
commentarieswas the influenceof Abu Kamil exertedupon later writersin this
science; for two prominentalgebraists made large use of his treatise without,
however,mentioninghis name. These two men are Al-Karkhi (died c. 1029),
whose treatiseentitledAl-fakhriwas analyzed by Woepeke (Extraitdu Faklhri,
Paris, 1853), and Leonard of Pisa (1202). Nor mustit be understoodthat there
is any suggestionhereof plagiarism,forundoubtedlyat the timethat Al--Karkhi
wrote,Abu I?amil's methodswere so well knownthat any writercould employ
his resultsas commonproperty,while the phraseologyof Leonard of Pisa, in the
statementof manyproblemsdrawnfromAbu Kamil, is plainlyintendedto carry
the idea of a citation. Al-Karkhireproducednot only many problemsgiven by
Abu Kamil but also the geometricalsolutions devised by this writer,while
Leonard of Pisa drewcopiouslyfromAbu Kamil forthe problemsin the sectionof
his Liber Abbaci entitled "Expliciunt introductionesalgebre et almuchabale;
Incipiunt questioneseiusdem."4
Abu Kamil's Works. Abu Kamil Shoja'ben Aslam ben Mohammed ben
Shoja, the reckonerfrom Egypt, was an excellent and learned arithmetician.
He wrote: The book of fortune,The book of the key to fortune,The book on
algebra, The book of extracts,The book of omens (by the flightof birds), On
1 Karpinski," RobertofChester'sTranslation
oftheAlgebraofAl-Khowarizmi,"
Bibliotheca
mathematica,
thirdseries,XI, 125-131.
2 Karpinski," The Algebraof Abu Kamil Shoja'benAslam,"Bibliotheca
mathematica,
third
series,XII (1912),40-55. To thisarticlethereaderis referred
formorecompletebibliographical
references
and forcitationsfromthe Latin text.
3 H. Suter,in a personalcommunication
to the author.
4Scritti di LeonardoPisano, publishedby PrinceBoncompagni,
Rome,1857,I, 410-459.
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THE
ALGEBRA
OF ABU
KAMIL
39
completionand diminution,The book on the rule of double false position,The
book of surveyingand geometry,The book of the adequate (possiblyin arithmetic).' Such is the account of the lifeand worksof Abu Kamil. as given in the
Kitab al-Fihrist (987 A. D.) of An-Nadim, who includes this writerin his list
of the later (i. e., nearly contemporarywith An-Nadim) reckonersand arithmeticians. Aside fromthe worksincluded in this list a treatiseon the mensuration of the pentagon and the decagon exists in Latin and Hebrew translations,
and a furtherarithmeticalworkis preservedin the originalArabic as well as in
Latin and Hebrew. The workon the pentagonand decagon has been published
in Italian by Sacerdote,2and in German by Suter.' The arithmeticalwork,in
whichhe deals with firstdegreeindeterminateequations, has been translatedby
Suter.4 Otherinformationabout the lifeand activityof this great Arab we do
not have. The titlesmentionedindicatethat,like so many of the later scientists
in Europe, Abu Kamil was interestedin magic,doubtlessincludingastrology,and
in omens.
The Sources of Information. The manuscriptupon which I have based
my study of Abu Kamil's algebra is the Paris manuscript7377A of the Bibliotheque nationale.5 The treatisecoversfolios71v_93v,and containsbetween35,000
and 50,000 words.6
In the opening sentence the author refersto his famous predecessorin this
field,Mohammed ibn Musa al-Khowarizmi,and a littlelater again refersto the
algebra ofthe same writer. Several referencesare made to Euclid but aside from
these two names no othershave been noted in this algebra.
Quotations from the Manuscript. The followingtranslation of selected
passages fromthe algebra is a very freeone, as an attemptis made to preserve
the mathematical significance. Additions to clarifythe meaning,and modern
notations,are put in parentheses.
The firstthingwhichis necessary
forstudentsofthisscienceis to understand
thethreespecies
whichare notedby Mohammedibn Musa al-Khowarizmi
in his book. These are roots,squares
and number. A rootis anything
whichcan be multiplied
by itself,composedofone,and numbers
above one,and fractions. A squareis thatwhichresultsfromthemultiplication
byitselfofroot,
composedofunits,and unitsand fractions. A numberis a quantityby itselfwhichdoesnothave
thenameofrootor square,but is proportional
to thenumberofunitswhichit contains. These
threespeciesareproportional
to each otherby turnin twos. Thus,squaresequal to roots,squares
equal to number,and rootsequal to number(ax' = bx,ax2 = n, bx = n).
An illustrationof squares equal to roots is this:
A squareis equal to fiveof its roots(X2 = 5x). The explanationof thisis that a square is
fivetimesthe rootof it. The rootof the square is alwaysthe same as the numberof the roots
to whichthe square is equal. In this questionthis root is fiveand the square is twenty-five
1 I translate
fromtheGermantranslation
of the passagesin the Fihristdealingwithmathematicalscientists,
by Suter,in Abhandl.z. Geschichte
d. math.Wissen.,VI, 1-87.
2 Festschrift
Steinschneiders,
Leipzig,1896,169-194.
3 Bibliotheca
X, thirdseries,15-42.
mathematica,
4 Das BuchderSeltenheiten
derRechenkunst,
in Bibl. math.,3d ser.,XI, 100-120.
5 I am indebtedto thelibrarian
forpermission
to have photographic
reproductions
made.
6To givean idea of the extentof this work,if the Latin textwereprintedin the MONTHLY
it wouldtake between75 and 100 pages.
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40
THE
ALGEBRA
OF ABU KAMIL
whichis the same as fiveof its roots. That the rootof the square is the same as the numberof
the root this I will explain.
We place forthe square a square surfaceabgdwhose sides are ab, bg,gd,and da (see Fig. 1).
Of this square any side multipliedby one in numberis the same as the area of a surfacewhose
lengthis theside or rootofthesauare. The side ab multipliedbv one whichis begivesthesurface
ae whichis therootof thesquare ag. The surfaceag is the same as fiveroots
or fivetimes the root of itself. Therefore,it is five times the surfaceae.
Divide then,the surfaceag by equidistantlines into fiveequal parts,that is
m the surfacesae, ck,sr,nh,and mg. The linesbe,ek,kr,rh and hgare equal.
el
Moreover the line be is one. Thereforethe line bgwill be fivewhichis the
root of the square and the square itselfis twenty-five
which is the surfaceag.
d This is what we wishedto explain.
.L
.z
FIG. 1.
If it is proposedthat one half a sauare shouldequal 10 roots (<x2-lOx)
then the wholesquareis 20 roots. The rootof the squareis 20 and the square
400. Similarlyif it is proposedthat 5 squares equal 20 roots (5X2= 20x) then the rootof the
square is 4 and the square is 16. And whethermore than a square or less than a square (is
proposed) you reduce to 1 square. You operate in the same mannerwith that which is connectedwiththe square, the roots.
A squarewhichis equal to a numberis illustrated:
A square equals 16 (x2 = 16). Therefore,it itselfis 16 and the root of it is 4. Similarlyif
given5 squaresequal 45 drachmas(5x2 = 45), then1 square is a fifthof45 whichis 9 and theroot
So whethermoreor less than a square (is given) it is reduced to 1 square and
of it is 3. . .
similarlyis treatedthat whichis joined or equal to themof numbers.
Roots whichare equal to numbersare illustrated:
the rootitselfis 4 and the square is 16. Similarlyagain
A rootequals 4 (x = 4). Therefore,
if it is proposedthat 5 rootsshouldequal 30 (5x = 30) thenthe root is equal to 6 and the square
36. . ..
We constructnow thesethreespecies,that is roots,squares and numbers,joinedby twosand
proportionalto the third. This gives,a square and rootsequal to number,a square and number
equal to roots,and rootsand numberequal to squares.
(In modernnotation
ax2 + bx = n,
ax2 + n
=
bx,
bx + n
=
ax2.)
First Typeof QuadraticEquation. An illustrationof squares and roots
equal to numberis the following:
A square and 10 roots equal to 39 drachmas (X2+ lOx = 39). The explanationof this is
that thereis some square to whichif you add the same as 10 of its rootsthe numberof it results
in 39 drachmas. In this questionthereare two ways (methods)of whichone leads you to know
therootofthe square and the otherto knowthe square oftheroot. We willset forthand explain
each of theseby a geometricalfigurewhichwillbe understoodby thosewho understandthe book
of Euclid, that is the Elements. The methodwhichleads us to know the rootof the square has
been narratedby Mohammedibn Musa al-Khowarizmiin his book. This is that you divide the
rootsin half,thatis in twoequal parts. In thisquestionthisgives5. You multiplythisby itself,
giving25, whichyou add to 39. This gives 64 of whichnumberyou take the root whichis 8;
fromthis you take the half of the roots,that is 5, and thereremains3, whichis the root of the
square. The square is 9.
The methodindeedwhichleads you to knowthe square is thatyou multiply10 rootsby itself,
giving100. This you multiplyby 39, whichis equal to the square and roots,giving3,900. Now
divide 100 in halves, giving50, whichyou multiplyby itself,giving2,500. This add to 3,900,
giving6,400,and of thisnumbertake the root,whichis 80. This you subtractfrom50, whichis
the halfof 100, and from39, whichis equal to the square and the roots,togethergiving89, and 9
remains,whichis the square.'
I The textto thispointis foundin the Paris MS. on folio71v-72n;givenin my articlein the
loc. cit.,p. 42-44.
BibliothecaMathematica,
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ALGEBRA
THE
41
KAMIL
OF ABU
In modernnotationthe steps ofthis solutionare indicatedas follows,in which
both the generalcase and this special problemare given in parallel columns.
bx = n,
(1)
x2+
(2)
b2x2
(3)
b2x2 ? b3x?
(4)
bx + b- = N?
+
x2 +?Ox=39,
)24
(b2
+ b3x = nb2,
b2
=
/
nb2
100X2 +
nb2?
+ -,
lOOx2
4
Jooox
= 3,900,
+ 2500= 6,400,
lOOOx
+
lox+ 50 = 80.
Onlythepositiverootis taken,as thenegativerootleadsto thenegativesolutionoftheequation.
Subtracting(4) from(1), memberformember,
x2 -
X2-50=39-80,
2= n-i/nb'+2
nb'?
X2
=n
b2 _ nlb2+
+ b-
50?=39 -80,
b4
x2
=
50 + 39 -80,
X2
=
9.
Followingthis the problemis proposed:
2X2+ lOx= 48.
This is a problemwhichis also given by Al-Khowarizmiand the same is true of
manyfurtherproblemsgivenby Abu Kamil. The latterauthorgivesthe method
of solutionpresentedby Al-IKhowarizmi
and also the method,above illustrated,
leadingto the value of X2 directly. Furtherthan this he presentsthe geometrical
solution for the firstmethod as given by his more famous predecessor. Abu
Kamil adds the geometricalsolutioncorresponding
to his own method,the second
of those given above. This seems to be worthpresentinghere.
ofthesquare
In orderto showyouthemethod(thedoor,literally)
whichleads to a knowledge
we place forthesquarethelineab (see Fig. 2). To thiswe add 10 rootsofitselfwhichare representedby thelinebg,givingthelineag, whichis 39. We wishto knowthenthevalue oftheline
ab. We construct
thenupon the line bga square surface,whichis the surfacedegb. This then
thenthe
willbe lOOx2 (in modernnotation),that is, 100 timesthe lineab. . . . We construct
surfaceah thesame as thesquaresurfacebe,thatis equal to the
of the line ab in e
square,whichis thesameas the multiplication
one of the unitswhichit containstaken 100 times. Then we
completethe surfacean whichis 3,900,sincethelineag is 39 and
am is 100,whichlinescontainthissurface. Moreoverthesurface db
b
ah is the same as the surfacebe. Therefore,
thesurfacedn is
of the linene by
3,900,whichis producedby the multiplication
VA
a
eg,as thelineegis thesameas theline ed and the linegnis 100
FIG. 2.
wedividetheline
sinceit is the same as the lineam. Therefore
gn in twoequal partsat thepoint1,to whichline so dividedthe
of thewholelinene by egtogether
with
line ge is added in length. Therefore,
themultiplication
the productof the line gl by itselfis the same as the productof the line le by itself,as Euclid
says in the secondbook' of hisElements. The productofne by egis 3,900and theproductofgl
1 Euclid'sElements,
II, 6, " If a straightlineis bisectedand a straightlinebe addedto it in a
straightline,the rectanglecontainedby thewholewiththe added straightline and the added
withthe square on the half,is equal to the square on the straightline
straightline,together
line."
madeup ofthehalfand theadded straight
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42
THE ALGEBRA OF ABU KAMIL
byitselfis 2,500,whichbeingcombinedmake 6,400,whichis thesame as theproductof the line
theproductofthelinele byitselfis 6,400,ofwhichtherootis 80, which
le by itself. Therefore,
is thelinele. But thelinegeis thesameas thelinegb,and so thelineIg and thelinegbmake80.
When we take away the line Ig and the linedg,whichare 80, fromthelinesag, 39, and gl,50,
whichis 9. This is thesquareand thatis whatwe desiredto
whichmake89,thelineabremains,
explain.'
Second Type of Quadratic Equations. The firstproblem which Abu Kamil
presentsof the second tyTpeof compositequadratic equations is again the same
as that given by Al-Khowarizmi,
x2+ 21 = lOx.
For this type of equation also Abu Kamil presentstwo solutions,the one leading
directlyto the value of the root of the equation and the otherto the value of the
square of the root. Furtherthe author notes that there are two solutionsfor
both the root and the square in thistype of quadratic equation. The reasonthat
this was noted by both of these Arabic writersis that in this type of equation the
two rootsare positive,whereasin the othertwo typesthe one root is positiveand
the other root is negative. Negative quantities as such are not accepted by
these Arabic mathematiciansand this, of course,accounts forthe threetypes of
quadratic equations. Express mentionis also made of the factthat if the square
of halfthe coefficient
of the rootsis less in magnitudethan the constantthen the
problem is impossible,fromthe Arabic standpoint,or contradictory. Further
also, the author mentionsthat when this square is equal to the given constant
the root of the equation is the same as one half the given coefficient.
In the above problem,
X2+ 21 = lOx,
the analyticalsolutionleadingto the value ofthe square ofthe unknownquantity
proceedsin substance as follows:
Multiply10 by 10, giving100. Multiplythisby 21, giving2,100. Then take halfof 100,
giving50, whichyou multiplyby itself,giving2,500. Fromthissubtract2,100,leaving400 of
whichtherootis 20,whichyousubtractfrom50, thehalfof100,leaving30. Fromthisyousubtract21,leaving9, whichis thesquare. Andifyouwish,add 20 to 50,giving70,fromwhichyou
subtract21, leaving49, whichis the square and the rootof it is 7.2
Al-Khowarizmigave the geometricalfigurecorrespondingonly to the smaller
root of the equation. Thus in the equation, x2+ n = bx, in which
b
x2
lb2
14-n
one of these roots,when both are real, is necessarilyless than b/2and the other
is greater. Abu Kamil givesthe geometricalexplanationcorrespondingto both,
thus completingthe work of Al-Khowarizmi. Thus he says that when you
assume that the square is less (in numericalvalue) than the number which is
given,then the solutionappears by subtraction,and when you assume the same
IIn the manuscriptfolio 72v73r ; in Bib. Math., loc. cit., p. 46.
2 Manuscript folio,73r Bib. Math.,
i loc. cit.,p. 47.
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ALGEBRA
THE
OF ABU
43
KAMIL
greaterthan the constantthe solutionappears by addition. Furtherhe gives a
geometricalsolution for the type in which the square of the coefficientof x is
equal to the given constant. The figurein this case consistssimplyof two equal
squares with a side in common. We followwith a demonstration,originalwith
Abu IKamil,leading to value of x2 in the equation, x2 + 21 = lOx.
x2 (Fig. 3), and we add to it the drachmaswhichaccom. . .Place thelineab to represent
thelineag is 10 rootsofthelineab. We construct
panyit,21,andletit be thelinebg. Therefore,
thesquareagedand thisis 100timesthelineab . . . since
thenuponthelineag,a squaresurface,
the lineag is 10 rootsofthelineab, and 10 rootsmultiplied
by itselfgives100 squares(x2). We
oonstruct
thenthesurfaceah equal to thesquareagedand let onesideofit be thelineab. Therethe surfacebn is
fore,the otherside bhis 100. Then we completethe surfacean and therefore
thenthe surfacemyequal to the
2,100sincethelinebgis 21 and the linegn 100. We construct
surfaceae, but the surfaceae is equal to the surfaceah. Take away,then,the surfacethand
thereremainsthe area cn equal to thearea ca. Addingthenthesurfaceby,thewholesurfaceay
the area ay is 2,100whichis the productof
is equal to thesurfacebn,whichis 2,100. Therefore
gyby yn,sinceynis equal to yt,sincethearea tnis a square. We dividethenthelinegnin two
the line ng is dividedintotwo equal partsby the point1
equal partsat the point1. Therefore
theproductofgyby yntogether
and intotwounequalpartsby thepointy. Therefore
withthe
square upon ly is the same as the productof In by itself.' But In by itselfmakes2,500,sinceit
is 50, and theproductofgyby ynis 2,100. The squareon thelineIyis thenleftas 400 and the
lineIy 20. But In is 50 and as it (iy) is to be subtractedthelineynis leftas 30. Sincetheline
bg is_21the remaining
lineab is 9 whichis the square. This is whatwe wishedto explain.2
e.
g
g
FIG. 3.
I
d[
FIG. 4.
Having given the explanation "by subtraction," that is to say, when the
square rootof 2,500 - 2,100 is subtractedfrom50, Abu IKamilproceedsto explain
"by addition." gn is again dividedinto two equal partsbut the point of division
falls between y and n (Fig. 4). This correspondsto the assumptionthat the
square is greaterthan the numberwhich accompanies it. This has also been
noted above.
The productof gyby yn togetherwiththe square of yl equals the productof Ig by itself.
the square of Iy
Ig multipliedby itselfgives2,500and the line ynby yggives2,100. Therefore
is 400 and lyis 20. SinceIn is 50,thewholelineynis 70. But ynis thesameas ytand ytequals
ag. Therefore
ag is 70. But bgis 21 whenceab is 49 whichis thesquare. Thisis whatwe desired
to explain.3
In the third type of quadratic equations Abu Kamil again follows Al-Khowarizmiin takingthe equation 3x + 4 = x2. Three explanationsare presented
forfindingthe root of the square. The firstis equivalent to the demonstration
presentedby Al-Khowarizmibut the figureis not completed. Instead of this
the proofis made to depend upon the second book of Euclid. The second gives
the
'-Thisis by Euclid VI, 5, " If a straightline be cut into equal and unequal segments,
withsquareon the straight
rectanglecontainedby the unequalsegmentsof thewhole,together
linebetweenthepointsofsection,is equal to thesquareon thehalf."
2Manuscript folio,74r; Bib. Math., loc. cit., pp. 48-49.
3 Manuscript
folio,74r;Bib. Math.,loc.cit.,p. 49.
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44
THE
ALGEBRA
OF ABU
KAMIL
the completedfigureand explainsas in the algebraof Al-Khowarizmi.In his
x2whichby theconthirdfigure(Fig. 5) thesquareabdgis placed to represent
ah
and yd are taken1'
the
lines
4.
Then
3x
+
of
equals
the
problem
ditions
completedcuttingofffromtheoriginalsquare
unitsin lengthand therectangles
havein common.
3x lessthelittlesquare,12 on a side,whichthesetworectangles
thatghis
It
follows
or
6-.
is
4
(11)2
+
which
remains
hnyz
the
square
Hence
4.
is
x)
(or
and
ay
2'
Xt
a
FIG. 5.
FIG. 6.
Abu Kamil places (Fig. 6) ab for
To arriveat the value Ofx2 geometrically
ag
equal to thethreeroots. Then
leaving
off
4,
and
cuts
to
gb
equal
thesquare
he continues:
Uponag a squareis constructed
agedwhoseareais 9X2or9 timesthelineab. ah is constructed
equal to thesquareae. Fromthisit followsthatan is 9 and thatgcwhichequals an is 9. Since
bgis 4 thearea ofghis 36. Take ay equal to an and drawylparallelto de. The area ygequals ac
and the area ae equals ah. Hence ye equals cb whichis 36. But ya is 9 sincean is 9. Divide
ay intotwoequal partsby thepointm. Now ydis added in lengthto ay. Hencead by dyplus
by itself,as EUCLID says in his secondbook. But
ymmultipliedby itselfequals mdmultiplied
ad by dy is ye or 36. ymmultipliedby itselfis 201. Addingyou obtain56{. Thereforemd
by itselfgives561 and mdis 72. But am is 41 and gbis 4. Henceab is 16 and thisis
multiplied
whatwe desiredto demonstrate.'
theresultsofthealgebraup to thispoint,statingthat
AbuKamilsummarizes
that is, restoration
all questionsthatcan be solvedby algebraand almuchabala,
mustreduceto one or the otherof the six typesof equations
and opposition,
mentioned.
and
and Division. In the next sectionthe multiplication
Multiplication
addiboth
involving
Binomials,
is
considered.
divisionof algebraicquantities
are givenparticularconsideration.The writeruses res
tion and subtraction,
forthe firstpowerof the unknown. This is combined
and radix,indifferently,
by additionor subtractionwith a pure number. The secondpowerof the
unknownis designatedby census,but thiswordis sometimesused simplyfor
any unknownquantityas in problemsbelow. Similarusage obtains in the
et diminutionisby one
work of Leonard of Pisa and in the Liber augnmenti
I, 304-371.
Abraham,publishedby Libri,Histoire,
whichsummarizesthe work
about multiplication
Aftera generalstatement
explanationoftheproblems.
witha geometrical
Abu Kamilcontinues
presented,
the factthat2x
foursmallsquaresis used to illustrate
Thus a figure
containing
3x
multipliedby 6 drachmas.
by 2x gives4x2. The nextproblemis
multiplied
' Manuscriptfolio,75r; Bib. Math.,loc.cit.,pp. 49-50.
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45
THE ALGEBRA OF ABU KAMIL
In this sectionthe constantis referredto drachmas althoughin otherplaces the
word numerusis also used. Figures are shown for 10 + x by x, 10 - x by x,
10 + x bv 10 - x and othersimilarexamples.
Radical Expressions. The addition and subtractionof radicals, involving
quadratic irrationalitiesonly,is effectedby means ofthe relationsgivenin modern
symbols by the equalities Va - i/b = Va + b - 2 Vab. Thus, to subtract
the square root of 8 fromthe square root of 18 the rule is simply: "subtract
24 from26, as you know, and 2 remains. The root of this is the root of 8 subtracted fromthe root of 18." This problem,by the way, is given by Al-Karkhi.
Leonard of Pisa1 employsthe same methodbut uses 18 and 32, instead of 8 and
18, combiningthem by addition and subtractionin the same way as Abu Kamil.
Al-Karkhi2proceeds to the considerationof cube roots in a similarway but
our author takes up fora finalproblemin roots the addition of 1/10 to the I/2
which,he states, does not lead to a simpleresult since 10 2 = 5 which is not a
square anld aso 2 :10 = 15 which is not a square. This sectionconcludeswith
the statementsthat i1O+iV2= J12+2 /20 and 1/10- /2=J12-21V/20.
Problems. The section dealing with problems is introducedin much the
same wav that Al-Khowarizmidoes the correspondingsection,and many of the
problemsare taken fromthe older work. The firstproblemis to divide ten into
two partssuch that the square ofthe largerpart shouldequal 12 timesthe product
of the two parts. This leads to the equation
__
or
2 =
_
_
_
_
12X(10
-
_
_
_
)
22x2 = 15x.
Thus the problemleads to the firstof the six types of quadratic equations.
The above problemis the second presentedby Leonard of Pisa3 in the section
with the heading: Expliciunt introductiones
algebreet almuchabale. Incipiunt
questioneseiusdem. Furthermorethis problem does not appear in Al-Karkhi4
in the list given by Woepeke. The conclusionis reasonable that Leonard had
some source of informationconcerningthe algebra of Abu Kamil other than
Al-Karkhi'swork,and even more particularlyin view of the long series of problems foundin Leonard's workwhichcorrespondto those given by Abu Kamil.
The second and thirdquestionshere are found to be the same as the following
two questionsin Leonard of Pisa,5but with sligbtlydifferent
numbers. Thus instead of 6{x2 = 100, Leonard and Al-Khowarizmigive 2 x2 = 100 and in place of
x/(10 - x) = 4 as given by Abu Kamil and Al-Khowarizmi,Leonard sets this
same fractionequal to 2X. Our space does not permit an examinationof all
similaritiesof this kind.
1 Scrittidi LeonardoPisano, publishedby Boncompagni,
Vol. I: II liberabbaci,Roma,1857,
p. 363-365.
2 F. Woepcke,
Extraitdu Fakhri,Paris,1853,p. 57-59.
3 Liberabbaci,p. 410.
4Woepcke, loc. cit.,p. 75-137.
5 Liberabbaci,p. 410.
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46
THE ALGEBRA OF ABU KAMIL
becauseof its similarity
interest
A problemgivenon folio81Vis ofparticular
whichhas given rise to discussion. The
to a problemof Al-Khowarizmi's'
problemis to divide50 amongsomenumberof menand thento add threemen
and again distribute50 (drachmas)amongthem (equally). In the second
instanceeach manreceives32 - (meaning3 1/2+ -) drachmaslessthanin the
problemas follows:
first, Leonardstatesthecorresponding
Distribute60 among(a certainnumberof) menand each willreceivesomething.Add two
60 and theneach man willreceive2I denarsless than
menand amongthemall again distribute
at first.
The statement
ofAbu Kamil is:
Andifwe tellyou,distribute
50 among(a certainnumberof)menand eachreceivesa certain
amount (res). Add threemen and distributeagain 50 drachmasamong them. Each then
receives3' 4 drachmasless thanbefore.
explanation.
Bothfollowwitha geometrical
employedby this
Fractions. The notationof fractions
ProblemsInvolving
employed
laterbyLeonard.
the
system
peculiar
translator
Kamil
resembles
ofAbu
as in the
Thus on fol.83"the squareof 2 and of 1 are givenin combinations
following
translation:
1
11
1
1
1
2 and - whichmultipliedby itself,gives and -9p. To thiswe add 11 drachmasand W01
1
1
1
ei
ei
-
giving 11 drachmas and 4 and 6 and 9-2 and 9
Of this we take the root or 3 and 3 and
2
1
g g standsfor plus81 whereas9 2 as wellas 2 represents
The fraction
simply
1
Se~99
5192
18 So also Leonard uses 8 8 for64 This systemis explainedby Leonard3
1
1
10
5
1 57
7
is
whowritesfor14 theform2 7 and 2 6 10 for10+ 60 + 120 - The difference
onlyin theorderofreading.
Al-Khowarizmi
givesseveralproblemsofthisnature:4
and thesumbe subtracted
be added to threedirhems,
To finda squareof whichif one-third
fromthesquare,theremainder,
by itselfrestoresthesquare.
multiplied
thesquare. So alsoLeonard5
The solutioninvolvestheuseofx (root)to represent
in a similarproblemstates:". . . placeforthatsquarex (res)." Ourversionof
ofthesamekind:
Abu Kamilhas severalproblems
Andifwe tellyou thereis a quantity(census) fromwhichwhenyou subtract3 ofitselfand
2 drachmas,theproductoftheremainder
by itselfgivesthe quantityand 24 drachmas.
The solutionis obtainedbyplacingx (res)forthequantity(census).
1 Libri,Histoiredessciences
en Italie,Vol. 1, Paris,1838: " LiberMaumeti. . .
mathematiques
benMusa, London,1831,
de algebraet almuchabala,"p. 286. Rosen,TheAlgebraofMohammed
explanation.
does not presentany zeometrical
D. 63-64. Al-Khowarizmi
2 Liberabbaci,p. 447.
4Rosen,loc.cit.,p. 56, 57.
5 Liberabbaci,p. 422.
3 Liberabbaci,p. 24, 25.
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THE ALGEBRA OF ABU KAMIL
47
The longest discussion of any algebra problem in Leonard is that of the
:
following
Divide 10 in twoparts,and dividethe largerby thesmaller,and thesmallerby thelarger.
Sum theresultsofthedivisionand thisequals thesquarerootof5.
Leonard presents several solutions. In the first he arrives at the equation
V5x4 + 2x2+ 100 = 20x + l/500x2. The coefficientof x2 is made equal to
unity by multiplyingthrough by 1/5- 2. The same equation is found in
Abu Kamil and the treatmentis the same. The value of x is 5- -225- V50000
which is found by both writers. Leonard goes on to find the value of
as 1/125 - 10. Al-Khowarizmiand Al-Karkhi do not give
4225-V50000
this problem although they do give one upon which this is based, namely: to
divide 10 into two parts such that the sum of each divided by the otheris 2*6.
Al-Karkhi2discusses four solutions.
ifthisexpressionbe permissible.
Leonard ofPisa occasionallysolvesincorrectly,
Thus the problem:' To finda numbersuch that, if the square root of 3 be added
to it and then the square root of 2 be added, the product of the two sums will
be 20. Algebraically (x + 1/3)(x + v'2) = 20. The product of the two
+ V/8x2. This leads to the incorrect
binomials is given as x2+ 6 + V/12x2z
value \19 + '/24 - /'3- 1/2 for x. Our manuscript of Abu Kamil gives
-4-,
for x. The problems4
the correct value, -\211+ V1/ - V6 followingthis in the algebra of Leonard, withthe exceptionof the two finalones,
are all taken fromAbu Kamil and in the same orderin whichthey occur in the
workof the Arabic writer.
The Italian writerfrequentlyuses an expressionwhichmightbe supposed to
in
referto Abu Kamil: " Operate accordingto algebra,"5 . . . " Operate therefore
exa
similar
this according to algebra, etc." However our manuscriptemploys
pression. Thus in a problemarrivingat 16x4= 256x2,this Latin versionof Abu
Kamil reads: Fac secundumalgebra in eis et erunt16 census census equales 256
censibus. census census (xl) ergo equatur. 16 censibus et census equatur 16
dragmis,ergo res equatur 4 dragmis . . ." (fol. 91"). In the discussionof the
same problemLeonard concludesone formof solutionwiththe words: "Age ergo
in eis secundum algebra, et inuenies, census census equari 16 censibus: quare
census est 16, et radix eius est 4, ut dictum est." Both writerspresentseveral
solutions,includinga geometricalone, of the problem in question which is to
divide 10 into two parts such that if fromthe largeryou subtracttwo of its roots
and to the smalleradd two of its roots the quantities are then equal. The expression secundumalgebrarefersto the Book of Algebra and Almucabala by
Mohammed ben Musa.
I
2
Liberabbaci,p. 434-438.
Woepcke, loc. cit., p. 91, 92.
3 Liberabbaci,p.
445.
4 Liberabbaci,445-459.
5 Liberabbaci,438.
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48
A CURIOUS CONVERGENT SERIES
Treatise on the Pentagon and Decagon. The algebra terminateswith the
firsteight lines on fol. 93V,and is followed by the work on the pentagon and
decagonby the same author. Althoughthis treatiseon the pentagonand decagon
by Abu Kamil is geometricalin its nature yet the treatmentand the solutions
are algebraical,includinga fourthdegree equation (X4 = 8000x2- i'51 200 000)
as well as mixed quadratics with irrationalcoefficients.The twelfthproblemis
in the Latin: "Et si dicemus tibi trianguliequilateriet equianguli mensuraest
cum perpendiculariipsius est 10 ex numero,quanta sit perpendicularis?"' This
suggeststhe similarproblemsin Greek of unknowndate and authorpresentedby
Heiberg and Zeuthen,2as well as the similar problems given by Diophantos.
In theseGreekproblemsalso linesand areas are summedquite contraryto ancient
Greek usage. A furtherpoint of interestis that in the equation x2+ 75 = 75x,
to whichthe solutionof the thirteenthproblemleads, Abu Kamil gives only one
solutionwhereasin the algebra he recognizesthat this equation has two positive
roots. The openingsentenceof thistreatiseon the pentagonand decagon makes
referenceto the algebra as immediatelyprecedingit, whichindeed is the fact in
the Hebrew and Latin manuscriptsthat are preserved.
Conclusion. The algebra terminateswith a general statementto the effect
that by the methods taught in this book many more problems can be easily
solved. In trueArabicfashion,the closingwordsare: " Whencepraise and glory
be to the only Creator."
Let us summarizethe resultsof our study. The most importantconclusion
of this investigationof Abu Kamil's algebra is that Al-Karkhi and Leonard
of Pisa drewextensivelyfromthisArabicwriter. Throughthemthisman,though
himselfcomparativelyunknownto modernwriters,exerteda powerfulinfluence
on the early developmentof algebra. Abu Kamil deserves somewhatthe same
recognitionfrommodernmathematiciansand historiansof science as that which
Leonard of Pisa and Al-Khowarizmihave received. We may hope that the
futurewill be more just than the past in accordingto Abu Kamil a prominent
place among the mathematiciansof the middle ages.
A CURIOUS CONVERGENT SERIES.
ofIllinois.
By A. J. KEMPNER, University
It is well knownthat the series
1
001
1
1
1
2
3
n=i
n
diverges. The object of this Note is to prove that if the denominatorsdo not
1 "If we say to you that an equilateraland equiangulartriangle,
withits altitude,is
together
measuredby 10,whatis the altitude?"
VIII, third
Analytik,in Biblioth.mathem.,
Aufgabenderunbestimmten
2Einige griechische
series,118-134.
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