º
º
|
º
º
T
in
- º
º º
º
º
º
º
T
---
º
ºº::
º
º
º
º
º
º º
º
-*.*.*
º º
º
IRBY COGHILL NI
CHOIS

- ºtſ * -
ºf º





A comparative STUDY OF FRACTIONS
A the
EARLY TREATISES
- on the
HINDU ART OF RECKONING.
by
IRBY COGHILL NICHOLS.
A dissertation submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in the
University of Michigan.





º
-
-
CONTENTS.
Page
PREFACE. . . . . . . . . . . . . . . . . e -e-----------e-º-º-º-º-e e -º-º-º-º-e-e l
BTBLTOGRAPHY. . . . . . . . . . . . . . . . • * * * * * * * * * * * * * * * * * * *-e - 3
CHAPTER T. Development of fractions in ancient,
and medieval times . . . . . . . . . . . . . . . . . . . 16
CHAPTER TT. Addition of fractions . . . . . . . . . . . . . . . . 65
CHAPTER III. Subtraction of fractions . . . . . . . . . . . . . 86
CHAPTER TV. Duplation and Mediation ºf trººp. 94.
CHAPTER V. Multiplication of fractions. . . . . . . . . . 98
CHARTER WT. Division of fractions . . . . . . . . . . . . . . . . 133
CHAPTER VTT - Coric lusion . . . . . . . . . . . . . . . * * * * * * * * * * * * 153



PREFACE. .
The development of the mathematical science during the
period from the introduction of the Hindu-Arabic numerals to
the time of printing has not received the systematic study
which the more ancient and more modern mathematics have re-
ceived. Recently, however, many medieval documents on astron-
omy, astrology, arithmetic and algebra have been published so
that it is now possible to undertake somewhat systematic studies
- &
in the development of the medieval mathematical eeurees.
Jºe ºv º tº -
This research is a study of the fundamental operations
With fractions as found in the medieval treatises on arithme-
tic employing the numerals now in use. It is a continuation
of a similar study on integers recently published by Dr. Suzan
R. Benedict. Accordingly the sources of material for the pre-
sent work are found to be largely the same as those for Dr.
Benedict's work on integers; that is, the files of the
BTELtopheca MATHEMATICA, ABHANDLUNGEN ZUR GESCHICHTE DER MATH-
EMATISCHEN WISSENSCHAFTEN, ZEITSCHRIFT FUR MATHEMATTK UND
PHYSIK, and the treatises of Colebrooke, silverijºs, weepeke,
Poncompagni, Steinschneider, and other scholars known through
*-
the translation and publication of mathematical manuscripts.
Bease I have not thought it necessary to give very full
bibliographical roºks, except in a few instances where
additional sources have been described.
The operations with fractions have been studied,
first, With reference to the individual Works, and , then,
in a comparative way, with reference to all of the works.
2.
Finally an attempt has been made to trace the origin, develop-
ment and spread of these operations. In chapter I, at the
conclusion of the remarks on each of the Latin treatises, a
list of technical terms, alphabetically arranged, has been
added. In other places, the Latin equivalent of terms is fre-
quently given. The infinitive form is given for verbs; the
nominative form for nouns .
The Works examined have been found for the most part
in the Library of the University of Michigan. I am indebted
to the Library of Smith College for the loan of the Arithmetic
of Lewis ben Gerson; to Mr. G. A. Plimpton of New York for the
loan of the arithmetics of Johannes de Liverius and of Joannes
de Gmunden; to Professor A. Ziwet for Eisenlohr's translation
of the Ahmes Papyrus (1891); and, particularly, to Dr. L. C.
Karpinski for unrestricted access to his private library, from
Which I have obtained the ALGORITMI DE NUMERO INDORUM, the
LIBER ALGORISMI DE PRATICA ARISMETRICE, Peurbach's arithmetic
(1534), photographic copies of the algorisms of John Killing-
Worth and of Joannes de Muris , and other shorter Works, I
Wish also to acknowledge a feeling of very deep appreciation
of Professor Karpinski's scholarly direction and ready assist-
ance in carrying on the research itself a
- 1
-------



BIBLIOGRAPHY.
T. EIN MATHEMATISCHES HANDBUCH DER **. by A.
Eisenlohr, 2nd edition, 1891. *
| From the private library of Professor A. Ziwet,
Ann Arbor.
This work is a translation and commentary by Dr.
August Eisenlohe of the Ahmes papyrus of the British museum.
ºlºr, this famous work is divided into an intro-
duction and five parts, treating respectively arithmetic,
calculation of volumes, geometry, calculation of the pyra-
mids, and a calculation of practical examples. The first
part, arithmetic, is further subdivided into five chapters,
treating in order division of the number 2, distribution of
broad (among 10), the completion (SEQUEM) caledation, solution
of first degree equations (AHAU RECKONING), and distribution
into unequal parts (TUNNU).
rt. GATTáD'Hºva of Brahmagupta.
Brahmagupta was an Astronomer of western India, who
flourished about 628 A.D. The 12th chapter of his Astron-
omical work, BRAHMA-SIDD'HANTA is on (Arithmetic. It is sub-
divided into ten sections, fractions being treated in sections
I and X. (ALGEBRA WITH ART THMETIC AND MENSURATION FROM THE -
ÅANSCRIT OF BRAHMAGUPTA AND BHKSKARA, by Henry Thomas Cole-
brook. London, 1817, pp. 277-324)
Trr. Algoºrmit be numero intoRun.
This 12th Century Algorism is probably the earliest
Latin translation of the Arithmetic of the Arabic mathematician
º

(4)
Mohammed ibn Musa Al-Khowarizmi, who lived at Bagdad in the
first half of the 9th century. As published it treats of
integers and, briefly, of fractions. It is about 5000 words
in length. (TRATTATI d'ARITIMETICA I," Prince Balaeºlºrſ,
|| From the private iterary of Dr. L. C. Karpinski. ||
Boſsompagni. Rome, 1857. )
~
TV. THE GANITA-SARA-SANGRAHA OF MAHAWIRACARYA.
This is an English translation * * * arithmetic
written about 850 A.D. and used widely in southern India. It
is divided into nine chapters, treating in orderi terminology,
arithmetical operations, fractions, miscellaneous problems on
fractions, rule of three, problems, measurement of areas, ex-
cavations, and shadows. (THE GANITA-SARA-SANGRAHA OF MAHAVIR-
ACARYA, with English translations and notes by M. Rangacarya,
Madras, 1912. )
vſ. THE TRISATTKA OF SRTDHARACARYA.
This Hindu arithmetic was written about 1020, A. D. ,
translated into English by N. Ramanujacharya of Madras , and
published with introduction and notes by G. R. Kaye. It treats
of integers, extraction or roots, fractions and several other
subjects. (THE TRISATIKA OF SRIDHARACARYA, BIBLIOTHECA MATH-
EMATICA, XIII.3, pp. 203-217. )
VT. THE ARITHMETIC OF AL-NASAWI.
This Arabic arithmetic is entitled THE SATTSFACTORY
ONE. Its introduction has been translated into French and


(5)
º ſº
its contents, outlined by M. F. Woepcke (JOURNAL ASIATIQUE I6,
/* - º º -
pp. 492-500). But H. Suter has given a better description of
it (BIBLIOTHECA MATHEMATICA, VII.3, pp. 113-119). It treats of
integers, common fractions and séxagesimals, including square
and cube roots. -
The author Ali ibn Ahmed, Abu 'l-Hassan, Al-Nasawi
died about 1030,
WTT. ATI-KAFT FII, HISAP OF AL-KARKHI.
Abī Bekr Muhammed Ben Alnusun Alkarkhi, an Arabic /
scholar of Bagdad, probably flourished between 1010 and 1016
A. D. , the supposed date of this work and of his work on Gulge-
bra, AL-FAKHRI. In the German translation, this arithmetic
is about 30000 words in length and treats of integers and
fractions, applications to merchantile life, proportion, ex-
traction of roots, mensuration and algebra. (ºr rººtsº
DEs ABU BEKR ALHUSEIN ALKARKHI, by Adolf Hochheim. Halle
1878-1879. )
VIII • THE ARTI HMETIC OF AL-HAşşAR.
This work is a descript ion in German of an Arabic * * *
arithmetic of the lzth Century. It is divided into seven
chapters treating integers, fractions, and extraction of
roots. (DAS RECHENBUCH DES ABU zakarºº-Haggas, ºn Hein-
rich Suter tri-zurieh, Bibliotheca Mathematica, 113, pp. 12-40)
Al-Hassar is the surname of the Arabic Writer Abu
Bekrº Mohammed ibn Abdallahs
TX. THE LTTAVATI OF BHASKARA.
This Hindu & Pithmetic was written before 1150 A.D.
(6)
by Bhāskara, an astronomer and Mathematician of Ujjain, Western
India, and was translated into English and published in isiº by
Henry Thomas colebrooke. It is divided into thirteen chapters
and treats of integers, extraction of roots, fractions, and a
variety of other subjects. (ALGEBRA WITH ARITHMETIC AND
MENSURATION FROM THE SANSCRIT OF BRAHMEGUPTA AND BHASCARA,
Henry Thomas Colebrooke, London, 1817, pp. 1-127. )
X, ANONYMOUS - JOANNIS HIS PALENSIS LIBER ALGORISMT DE PRATTCA
ARISMETRICE.
Johannes Hispalensis was a prominent Spanish Jew who
died probably in 1157. He studied at Toledo and translated
Arabic works, among which is this one, LIBER ALGORISMI DE
PRATICA ARISMETRICE. The work itself claims to be Al-
Khowarizmi's. It is a treatise on the algorism with an addi-
tion of many excerpts, unrelated to each other or to the
algorism. It contains about 20000 Words and treats of inte-
gers and fractions, both common and sexagesimal, and of
square root. (TRATTATI Bºrºrica II, * Bºonvagni.
| | From the private libary of Dr. L. c. Karpinski.I.
Rome, 1857. )
An abstract of a shorter edition of this work en-
titled INTRODUCTORIUS LIBER QUT ET Hiſvºrs DIC ITUR IN
MATHEMATICAM DISCIPLINAM was published in 1883 by Enrico
%rauce. (BULLETIN DES SCIENCES MATHEMATIQUES ET
ASTRONOMIQUES, Tome 18, pp. 247-256. Referred to in
Bibliotheca Mathematica, V, p. 410-411; VII, pp. 86-87; IX,
pp. 323-325. )


(7)
XI. ANONYMOUS - A TWELFTH CENTURY ALGORISM.
This is an anonymous algorism of unknown authorship
put ascribed to Adelard of Bath who flourished in the early
part of the twelfth century. It contains about 3000 words,
and treats of integers, fractions and square root. (ABHAND-
LUNGEN ZUR GESCHICHTE DER MATHEMATISCHEN WISSENSCHAFTEN, M.
Curtze, Vol. VIII, pp. 1-27.)
XII. THE ARITHMETIC OF RAUUL DE LAON.
This work is primarily an explanation of the abacus
reckoning in which Hindu-Arabic numerals are used e It is
included here because it deals with Roman fractions, also a
(DER ARITHMETISCHE TRACTAT DES RUDOLPH VON LAON von Dr. A.
Nagl, ABH. ZUR GESCH. DER MATH. , V, pp. 85-133)
Raoul de Laon (d. 1131) lived and taught in Paris.
XITT - SEFER HA-MISPAR OF RABBT BEN ESRA.
Rabbi Abraham ben Ezra (1092 |93-1167 was a Very
learned scholar of Spain. He died in Rome. This Hebrew
arithmetic of his has been translated into German, with notes,
by Mortiz Silberberg. It treats of integers, fractions, pro-
portion and extraction of square root. (DAS BUCH DER ZAHL
DES R. ABRAHAM IBN ESRA, M. Silberberg, Frankfurt A.M., 1895. )
XIV. LIBER ABAC T OF TEONARD OF PISA.
-
- t
Leonard of Pisa, also known as Fåbonacci, "the most
noteworthy mathematical genius of the Middle Ages, was born
W
at Pisa about 1175. * This great work of his was written in
THE HINDU-ARABIC NUMERALS, Smith-Karpinski, p. 128. ||
(8)
1202, after he had traveled in Egypt, Syria, and Greece. It
is supposed to contain all that he had learned of arithmetic.
It was revised in 1228. This revised text was the basis of a
publication by Prince Boncompagni in 1857. It contains 459
pages, divided into fifteen chapters, as follows:
1-V, Numeration and operations upon integers.
VI-VII, Common fractions.
WITI-XT Applications.
XII Series and proportion.
XIII Rule of raise position.
XTV Square and cube roots
XV Geometry and Algebrae
(IL LIBER ABACT DI LEONARDO PISANO da Baldas sare Boncompagni,
Roma, 1857. )
To wº º º 2. º dºwn Mºs -
xv. DAS-BRUCHRECHNEN DES*jöRDANUs RARIUS.
This work is a description and analysis by G.
Eneström of a portion of an Algorism” ascribed to Jordanus
|| The portion dealing with integers has already been
described and analyzed by Eneström. See #BER bre
DEMONSTRATIO JORDANI DE ALGORISMO, Bibliotheca Math-
----------
ematica VII.3, pp. 24-37; and UBER EINE DEM JORDANUS
NEMORARTUS Zugsscherseene Kurze ALGORISMESSCHRIFT,
ibid, VIII.3, pp. 147-153.
Nemorarius. As its name indicates, it treats of fractions,
and Was probably written early in the 13th Century, since
Jordanus is said to have died in 1237. (DAS BRUCHRECHNEN
DES JORDANUS NEMORARIUS, von G. Eneström, Bibliotheca Math-
ematica, XTV3, QQ - 41-54. )


9 - .
- ( ) - º - º º -
xvi. DER-ALGORISMUS DE MINUTIIs Dºs Mºfºrºs GERNARDUs,
This is the second part of an algorism attributed to
an unknown Magister Gernardus of the 13th Century - The first
part treats of integers. From a manuscript in the Vatican,
Gustav Eneström has published and described the whole algor-
1am. (Part 1, Bibliotheca Mathematica, XIIIs, pp. 269-332,
part ll, Ibid., XIV3, pp. 99-149). Each part is subdivided
into many short propositions.
Eneström has also made a comparison between it and
the two works, already treated; of Jordanus Nemorarius:
---- tº -º-º:
fººtº sº.
|This research, 9 • 48.
DEMONSTRATIO DE MINUTIIS and TRACTATUS MINUTIARTM, which
probably are the basis of this Work.
This second part begins:
Deinceps ad minucias procedat negorium et eia ( ; )
and ends : – -
Hec sunt gue de minuciis scienda et ideo collingenda.
---
putavi et eia ( ; ) finit. . "
XVII. TALKHIS OF TBN AL-BANNA.
This 13th century Arabic arithmetic is in two parts:
part 1 treating integers, common fractions andboots; part 2,
treating Algebra. Al-Banna (c. 1251- c. 1340), the author,
Was a very learned man. He taught mathematics in Morocco in
the first half of the 13th century. His real. name Was Aboul.
Abbas Ahmed ibn Mohammed ibn Uthman Alazádi. (LE TALKHYS
D' IBN ALBANNA, tºadsit, #. ARISTIDE MARRE. Atti dell'
. - - - . . º
Accademia Pontificia de' Nuovi Lincer, 1864, Vol. XVII, pp.
289-319. )

(10)
XVIII. SEFER MAASSET CHOSCHEB of Lewi Ben Gerson. *
|| From the library of Smith college. || -
This Hebrew arithmetic was written in 1321 by Lewi
ben Gerschon (known also as Leo Hebraeus), A Jewish Rabbi,
celebrated as an astronomer and mathematician. His arithme-
tic is divided into two parts: part 1 treats of algebra;
part 2, of integers, or common and sexagesimal fractions and
of extraction of square and cube roots. (SEFER MAASSET
CHOSCHEB, DTE PRAXIS DES RECHNERS. EIN HEBRATSCH ARITHMETISCHES
iſ ſº ºf ſy
WERK DES LEVT BEN GERSCHON AUS DEM JAHRE 1321, UBERTRAGEN, VON
- M.
Dr. Gerson Lange. Frankfºrt A.M. 1909.)
Dr. Joseph Carlebach has published a description of
this same work (Lewi ben Gerson als. Mathematiker, Inaugural
Dissertation Heidelberg, 1910. )
XIx. THE ARITHMETIC OF PLANUDEs.
Maximus Planudes, the author of this arithmetic, was
a Greek monk of the 14th century. He was ambassador to Venice
in 1337, but lived much of his life in constantinople. His
arithmetic was published in Greek in 1865 by C. I. Gerhardt,
and in German, in 1878, by Herman wasons. It treats of
integers, sexagesimal fractions, and extraction of square root
with the aid of sexagesimal fractions. (DAS RECHENBUCH DES
Žº -
MAXIMUS PLANUDES, * Dr. Hermann Wäschke. Halle, 1878. )
XX. LES DEUX LETTRES ARITHMETIQUES DE NICOLAS RHABDAS.
These two letters of Nicholas Artavasde de Smyrne,
commonly called Nicolas Rhabdas of Smyrna (c. 1340) were

(11)
º
ſº
Written from constantinople ºiative to the operations of
arithmetic, the second dealing particularly with fractions.
They are now in the National Library of Paris, M. Paul -
Tannery has published a description of them together with
the original Greek text and a French translation of the
same. (NATIGES ET EXTRATTS DES MANUSCRIIS DE LA BIBLIO-
TÉquE NATIONALE, Paris, 1886, pp. 121-253.)
XXI. PROSDOCIMO DE BELDAMANDI, AND LIVERIUS. *
| From the library of Geo. A. Plimpton, New York. ||
"This rare work was written for the Latin schools,
and is a good example, the first to appear in print, of the
non-commercial algorisms of the fifteenth century." It
treats of the fundamentals operations with integers includ-
ing square root, and with fractions, both common and sexage-
simal. Prosdocimo treats the former, Liverius the latter.
The treatment of fractions begin{s: 'Incipit Algor-
ismus de mill nuti is tam vulgaribus quam physicis magill stri
---
Ioannes de T.ir if s Giaiji i , , / As a
- -- Tºi rºle rij. s º i. C uli o ºf - - º, º º - ºf 4 st* * Jºaº - 3. * º
* ... º. -
- - * . - º, ºr ******
Johannes de Liverius (Liveriis, Lineriis) was a º ****,
Zs. * * **** -
* º,
º
C.
Sicilian Writer on astronomy. He flourished, 1300-1350.
(PROSDOMCIMI DE BELDAMANDIS ALGo ||RISMI TRACTATUS** 1483
edition. Smith, RARA MATHEMATICA p. 13.)
XXII. QUADRIPARTITUM NUMERORUM OF JEAN DE MEURS.
This work, written in verse with a prose commentary,
is an algorism of the 14th century. It is divided into four
books, treating arithmetic and algebras Fractions are treat-
ed in the second book.

(12)
The treatment of the fractions of de Muris given in
the present work is based on notes loaned by Dr. L. C. Kar-
pinski, and, also, on his description of the algorism, par-
ticularly of the third book on algebra, based on the Vienna
codex 4770 and published in the BIBLIOTHECA Mathemati, XIII,
pp. 99-114.
"Joannes de Muris (Jean de Meurs, Murs, Muria) was
born in Normandy, c. 1310; died after 1360. He wrote on
arithmetic, astronomy and music." (Smith, RARA ARITHMETICA,
p. 117. )
xxitſ. THE ALGORISM OF JOHN KILLINGWORTH.
This English arithmetic, written in 1444, is des-
cribed by Dr. L. G. Karpinski in the ENGLISH HISTORICAL
REVIEW, vol. XXIX, pp. 707-717. According to this article
a unique copy of this algorism is preserved in is rifteenth
century manuscript in the Cambridge University Library, and
consists of 23 pages, divided into three parts, treating
integers and sexagesimals, and giving five sets of tables
for facilitating arithmetical operation.
In addition to the description just mentioned, Dr.
Karpinski has permitted the use of results of a further
study of the algorism, particularly with reference to the
treatment of fractions.
John Killingworth (d. May 15, 1445) is "one of a
long list of distinguished mathematicians and astronomers
connected with Merton College, Oxford."

(13)
.
XXIV. A 15th CENTURY ALGORISM.
This anonymous work was probably written shortly before
1483. Dr. E. Rath has described” its contents and compared it
- || ther EIN DEUTsches RECHRNBUCH Aus DEM 15 JAHRHUNDERT,
* Mathematºes, XIII.3, pp. 17-22 ||
with the Bamberg Arithmetic, a German arithmetic published in
1482+, and with the ALGORISMUS RATTS_PONENSIS, an algorism
| Smith, RARA ARITHMETICA, p. 12 ||
copied and perhaps written by Frater Fredricus of the Cloister
Emerans in Rºsensºrs. The work is perhaps based directly, or
indirectly on the other two e It, falls into three parts: chape-
ters I-VI, dealing with whole numbers, and resembling the work
of Sacrobosco ; #hapters VII-XI, repeating the algorism of frac-
tions of John of Lineriis (c. 1500-1350); and #hapters XII-XXI,
treating "practical" problems, after the fashion of Leonard of
Pisa's
XXV. ALGORITHMUS MAGISTRI JOANNIS DE GMUNDEN DE MINUCI TS
PHISIGIS. *
|| From the library of Geo. A. Plimpton,
New York | | s
This algorism is the last work in a volume of five
Works printed in Vienna in 1515. The title page of this
volume is as follows:
- CONTENTA IN HOC LIBELLOe
ARITHMETICA COMMUNIS.
PROPORTIONES BREurs.
DELATITUDINIBUS FORMARUM.

{14)
ALGORITH MUS. Me GEORGIJ PEURBACHIJ IN INTEGRIS.
ALGORITHMUS MAGISTRI JOANNIS DE GMUNDEN DE MINUCIJS
PHISTCTS. *
| | fr. 44-53. ||
The first three Works are by Joannes de Muris, Thomas Bradwardin,
and Nicolaus Horem respectively. All five works are well-known
and it is expressly stated in a manuscript of that period (1515)
that lectures then given on arithmetic were based on them (Smith's
Rara Arithmetica, p. 118).
Joannes de Gmunden (Johann von Gmunden, Johann Wiss-
bier? Nyden? Schindel? Johannes de Gamundia)* was born c. 1380,
| Bibliotheca MATHEMATICA, 1896, vol x. p. 4; Smith's
RARA ARITHMETICA, p. 117 ||
and died 1442. He studied and taught at vienna.
This work treats of sexagesimal (physical) fractions, in-
cluding their square and cube roots. It begins: QUAMVIS ARS
NUMERANDI IN MINUCI JS TAM vulgariºusºphistors ºrios: SIT
TRADITA IN ALGORITHMO DE MINUCIIS." -
And ends:
. . . . ET ERIT MINUTIA SEQUENS ALIAS SERUATAS:
ET STC FAC TOTTENS QUOCIENS VOLUERIS: UT HABEAS PRECISE RADICEM
IN GRADIBUS MINUTIS SECUNDIS ET CETERA: QUOUSQ3 TIBI SUFFICIAT.
XXVI. ELEMENTA ARITHMETICES AUCTORE GEORGIo PEURBACHIo. *
- - Professor -
|| From the library of) L. G. Karpinski. ||
This arithmetic , known by various names, is a brief
treatise on integers, fractions, roots, proportion and applica-
º
(15)
tions, written probably for students preparing to study astronomy,
in which Peurbach himself Was primarily interested. The work went
through several editions, the first bearing the date 1492.
The author, Georgé von Peurbach (1423–1461), who, also,
studied and taught at Vienna, was a pupil under Vºn Gmundern
and a teacher of Regiomontanus. *
| Smith, RARA ARITHMETICA, p. 53.)
XXVII. THE CALCULATION WITH DEGREES BY SIBT EL-MARIDINI.
- 7
This work is probably the one entitled RAQ (IQ EL-HAQAIQ,
on the calculation of degrees and minutes." There are several
º
copies of it, of which one is in Cairo (247, Trans. 167. )
sº-
-- -
--- ---
Sibt El-Märidini (Muh. b. Muh. b. Ahmed, 'Abū'Abdallah,
Bedr ed-din or Šems ed-din el Migr? el-Dimišqí) was born in 1423
and died 1494/95. He lived first at Damascus, and later at Cairo
Where he Was a mathematician and astronomer at the mosque EL-
AZHAR. He wrote many treatises, but none of them are very im-
portants (ABHANDLUNGEN ZUR GESCHICHTE DER MATHEMATISCHEN WISSEN-
SCHAFTEN, 1900, x, pp. 132-185 by H. Suter; and BIBLIOTHECA MATH-
EMATICA, 1899, New Series XIII, pp. 33-36 by Carra de Vaux. )
XXVII. THE ARTTHMERIC OF AL-KALSADT.
This is an arithmetic by an Arabic mathematician of
Spain, Aboul. Hagºn Ali Ben Mohammed Al-Kaijadi, who died in 1486,
ń
It treats of integers, fractions, roots and Algebra. (TRADUCTION
J -
DU TRAITE D'ARITHMBTIQUE D'ABAEL HAGON ALI BEN MOHAMED AL-
KALGADI par M.F.Woepcke, ATTT DELL'ACCADEMIA PONTIFICIA DE'NUOVI
LINCET, 1859, Vol. XII, pp. 230-275, 399-438. )

CHAPTER I
DEVELOPMENT OF FRACTIONS IN ANCIENT
AND MEDIEVAL TIMES e
- Operations with common fractions are definitely traced back
to the ancient civilization upon the banks of the Nile. For the
text proper of the oldest (ſirithmetic in existence, the Ahmes or
Rhind papyrus, estimated to have been Written more than thirty-five
hundred years ago and probably based on other texts still older
by many centuries, begins with the study of common fractions • In=
deed, the calculations with fractions recorded by Ahmes appear to
represent a high point in the mathematical development of that
age a
One prominent characteristic of Egyptian arithmetic is the
use of unit fractions, or fractions with (a constant) numerator
(of 1. It is by these fractions that Egyptian influence is trace=
able through Greek, Roman, Arabic, and middle-age European math-
ematics • Concretely, the Egyptians knew only this form, a single
exception being the fraction 2 . Their notation likewise was a
3
were expressed by special symbols,
single notation: 1, l ; l; and
2 3
º 4.
While other unit fractions were expressed by writing their res-
2
3
º
pective denominators with a dot above, as 7 for 1. Abstractly,
7
however, the Egyptians had other fractions, expressed always in
terms of unit fractions. For 2, they wrote l 4: 15 for 3, 1 + 1.
5 3 15 13 8 ' 52
+ #: To this end tables" were given by Ahmes for converting
| | Eisenlohr, pp. 28-30 || -
fractions of the form 2 (ne 1,2,... . . 49), into unit fractions.
- 2n + 1
These tables are probably the results of the accumulated expert-
Žence of earlier mathematicians and not of any general formula,

(17)
except for the special case when the 2n+1 is a multiple of 3e
Then a rule” is given to multiply the denominator by 2 and by 6
| | Ibid, pp. 13-14 ||
respectively; that is ,
2 = (lall) l = –1 a 1 . Accordingly 2 = -2 = 4-1 4- 1 - 1 =
# T \; -ā’; # 4 a. # * : * #’; + # #
####
& TT5
As a technical term, a common denominator is not speci-
fically mentioned; but, as will be seen later, solutions of ex-
amples reveal the conception of such in its practical applications
With unit fractions, numerators of fractions reduced to a common
denominator could be found very readily.
Ahmes solves examples involving all four of the funda-
mental operations with fractions • Subtraction is treated as ad-
ditive completion of the subtrahend to the minuend, and division
as multiplicative completion of the divisór to the dividends
Numerous examples are solved, but no formal rules are stateds
Before turning to the solution of any examples, the reader will
ºt, ſº …
recall the Egyptian method of multiplying integers, by repeated
duplication and addition of multiples required to give the de-
sired products To multiply a given number by 13, the given num-
ber is doubled, redoubled and then again redoubled, ºne, the
sum of the multiples 1, 4 and 8 give the product soughts The
same method is used in the multiplication of fractions o
A problem involving both multiplication and addition is
the following one in which the student is required to add to
l, and 1 their l and l ; that is, solve (1.1. 1) + (14.1) (1 x_1).
A 25 2 4 #4 #'+ 4 ++’% *s;
The text presents the solution in the following form:
| | Ibid, p. 36, problem 7.|| * -º


l —l. 1/4 1. l
4. 28 16 11.2
7 l 1 1/2 1/4 l/4
1/2 1 [1]
75 T55 together l
2
3 1/2 1/2
1 and 1 reduced to a common denominator of 28 give as numerators
4. 28
7 and 1 respectively. Dealing only with these *****, * of
1 and of -1 (= 1 + 1), becomes 3 l and l- 1 of , and of
4. 28 5 T55 2 2 4. 4. 28
(* 1 + l_) become l l l and l- Adding 7, 1, 3 l; l. l l l ; and
( IS IT2) 2 4 4 2 2 2 4
l, the sum is 14; that is, 14 or 1, as shown in the solution above •
4. 28 2
In the original text, the numbers 7 and 1 and the fractions accent-
ed with slanting linesſ, that is, the terms to be added, are Written
in a red color While the other terms of the solution are written
in black, - -
An example in subtraction is the following one in Which
the author finds the difference between 1 and the sum of 2 and 1. *
| | Ibid, p. 39, problem 21. || 3 15 -
"You are told to complete
2 —l. to 1.
3 5
10 l together ll remainder 4 multiply 15 to find
4 * t
15 *_l 1.
15
—l. 1 l together 4.
10 2
#1 3 Hence 1 and 1 are to be added to obtain 4.'
B 5 15 15
+\,.6%
Observe, the % is used to indicate which numbers are to be
added. Observe, also, the use of 1 of 15. From the Egyptian
10

(19)
manner of writing lo and 15 it was easy to see that l of 15 is
10
1 1. The author checks his result by addition: 2 , l ;
75 - 3 5
1 - 1,
--
- 15 1.
When added, give 1 -
If the civilization along the valley of the Nile can claim
greatness by virtue of its development of a particular system of
common fractions, then the civilization along the valley of the
º - -
- ºr
/Euphrates and Tigris likewise can claim greatness by virtue of
! º, º, º
--
its development of another and decidedly distinct system of fraces
tions, a system so convenient that, in succeeding centuries, it
became the choice of mathematicians and astronomers, and remained
in active use until replaced by the decimal fractions of modern
times. Evidences of it remain in our use of days, hours, minutes,
and seconds, and in astronomical calculations.
The basis of this system is 60, applicable both to inte-
gers, and to. fractions • According to it 64 * 60 ****
10° - 60 +40 F 1.40. A unit is divided into 60 parts, each of
these into 60 others, and so on. In fractions, the constant
numerator of 1 of the Egyptians gives place to a constant de-
nominator of 60, Which is never Wiitten, but borne in mind,
1 and l are expressed as 30 and 20 respectively.
2 3 The origin of this system is a matter of conjectures
Existing Baylonian tablets show that it Was in use about 2000
B.C. It came into use among the Greeks through Hipparchus, the
great Greek astronomer of Rhodes, who lived between 200 and
100 B.C.
The Romans showed themselves awkward in mathematical
form. But they developed independently a system of fractions
(20)
which governed the practice of calculations of the early middle
ages until the 12th century. The date of its first appearance
is not known, but the fractional unit of the system, the UNCTA,
Whi, ch is r; of the integral unit, the AS, is from o ºrº of
the sº and Etruscan languages and appears in the Roman
literature as early as the time of T. Maccius Plautus (254-184
B.C. )* At first the subdivisions of the AS were called MINUTIAE.
|| Menaechmi III, iii, 3:
Huc ut addas auri pondo unician;
Rudens IV, ii., 8:
* Neque piscium ullam unciam hodie pondo cepi.
The fractional parts were as follows:
_l = UNCTA, | 2. -
12 º tº º 3 ºr " ºº º: º º
ſiz = SEXTANs (a 'sixth),
QUADRANS (a fourth),
TRIENS (a third),
QUINCUNX, (quinque unciae, five unciae),
SEMIS (a half),
-
SEPTUNX (septem unciae, seven unciae),
BES (2 parts of an as ),
DODRANS (de quadrans, an as less a quadrans),
º
1.
10 - DEXTANS (de sextans, an as less a sextans),
12 -
ll. - DEUNX (de uncia, an as less an uncia),
12
12 = AS.
l2
Subdivisions of the uncia are given as follows:
_l = sºug|A (1 uncia),
24 - 2
_l = SICTTTCUS (l uncia),
48 4.

- - - - º - - -
- º 4- A. 4. - ſ º º º rº-
(21)
-
1 = SEXTULA % uncia), - -
7
2
Gradually these fractional parts of a unit assumed the re-
lation of pure fractional notation. Other fractional forms appear-
1 l
ed; as, 1 = SESCUNCIA (= 2 - 1 l uncia); l. TRIENS. l QUANDRANS =
8 l2 2
1 UNCTA ( = 1 - 1 = 1). To obtain products readily tables were
3 T Tº
formed , which remained in use as late as 450 A.D., as the CALCULUS
4 -
of victor.IUs ves º shows. *
|| Tropfke I, p. 77.
This system of fractions was incapable of much developments
* * * º ºwº,
Its difficulties and inconveniences in ºre. • Hence scientific
7" --- -
ºthºat ºf ,-, cºº
A. 3.
º º - º *...* - -
mathematics among the Romans lagged u e introduction of the
... " º … -
º ºf ko, ſº
| ºr position system. - º
The Greeks represented numbers by letters” and, in common-
\ –
|The following table from Cajori's History of Mathematics
(1901), p. 64, shows the Greek alphabetic numerals and
their respective values: -
vs ſº Y à e g ! ) 4 a / X. Z. 2/
1 2 3 4 5 6 7 8 9 10 20 30 40 50
§ 2 - 2 /2 / / / 2 X A & J 22- A.
€6 mo 80 90 100 200 300 400 500 éoo 700 goo goo looo good
, Y etc.
3OOO
%3 }~
AZ AZ AZ
10,000 20,000 30,000
e?c
->
The coefficient for M was some times placed before or
behind instead of over the M. The Greeks had no zero. ||




(22)
fractions, had notations for both unit fractions and non-unit
fractions. The numerator Was marked with one accent, and the
denominator, written twice, was marked with two accénts. #. W3.5
expressed thus: ºikº • The denominator was sometinº
placed above the numerator: es 2 or "º , or . . º In Writing
unit fractions, the numerator was omitted and the denominator
written only once, as / " for #" l and 2 had special symbols;
C and a character similar to a . º * of the
Egyptian forms for them. Ptolemy, Who flourished at Alexandria
|| 1 Delamºrgastronours Ancienne, II, p. ll.
about 137 A.D., expressed 2 as "2 parts', 3 as '3 parts', . . . . . .
n_ as 'n parts" - 3 4.
17 -º- -
Two definitions of a fraction were probably given during
the Greek period, or earlier. Euclid (c. 300 B.C.) defined 2.
reaction as being e Smaller number divided by a larger, if it
contains part of the larger without measuring it exactly. * An
| | Heath's EUCLID: VII, 3 and 4. ||
older definition is this: "a fraction is one of the subdivisions
of a unit, or as an aliquot part of a whole number different from
1. * *
| * Tropfke, I, p. 80. ||
Sexagesimal fractions, as noted above, were introduced
into Greece by Hipparchus (between 200 and 100 B.C.). * But Theon
|| The sexagesimal calculations of the Scholia of the Elements
of Euclid (c. 300 B.C.) were added by commentators at a
much later date. For a discussion of this point see
BTBLIOTHEGA MATHEMATICA, y, pp. 225-233.


of Alexandria (c. 370 A.D.) is the first Greek writer whose methods
of operating with sexagesimal fractions are known. * He dealt with
| * Delambre ASTRONOMIE ANCIENNE, II, pp. 3-31. ||
them in quite a modern fashion. Addition and subtraction were made
from left to right, although it is probable that they were also made
from right to left. According to Delambre, all of Theon's rules &
for the different cases of multiplication can be expressed by one
general formula:
Represent a, b, c. . . . by a' , b’, c''', . . . . . . as the Greeks did.
60' 59. 603 - - (m) , (n)
Then p(m) and q(n) represent and q and p" "' q^* =
gº)
- 60 m)
- : paſtºn). Let m = 0 and n = 3, then p(m) g(n) =
#ºy - . . .
11 ſº
e
ºccº
pſ. = p(;
Reciprocally pºº) = ()".
q\n \ºiſ.
This theorem is in erroot the same as that which Archi-
me des deomonstrated for the progression l: 10 : 100: 1000: e.t.c s
The on follows his rules With examples. In multiplication,
he chooses the side of an inscribed decagon, which is 37° 4' 55",
and squares it. Writing the multiplier above the multiplicand,
he multiplies the terms of the latter successively by those of
the former, beginning with the highest orders and going from left
to right. The partial products are then added and simplifiedo
Division will be sufficiently clear from the following-
example:
Dividend. 1515° 20 ! 15" |25° 12' lo” divisor.
30° 1st quotient.
25 x 60 . . . . .500 º -
-
Remainder . . . . . . . . 1.5 = 900
Total minutes = 9.30
- 12" x 60° = 720
Remainder . . . . . . . . . . . . 2CO

(24)
10" x 600 10
Remainder . . . . . . . . . . . Tº 250 12 * 10"
250 x 7" 175 7' 2nd quotient
I5' = 900". -
Combining with 15" 915"
etc.
Ubserve the procedure from left to right. The labor of
multiplication and division Was reduced by the use of tables •
- Among the Hindus, a common fraction is written with the
numerator placed above the denominator Without any intervening
line; * thus 4 for 4. An integer is written as a fraction with a
- 5
| p. 36 ||
denominator of l. In the case of a mixed number the integer is
2
placed above the fraction, thus 4 for 2 4, No definitions are
5
given, nor is any definite order observed, although all four
fundamental operations, squaring, cubing and extraction of roots
are treated. The common denominator is used, but, the lowest
COIſºſºl Orºl multiple is not particularly emphasized a The methods
employed are quite modern; for example , all four of the Hindu
Works examined multiply fractions by dividing the product of
- their numerators by the product of their denominators, and
divide fractions by inverting the divisor and multiplying.
Cancellation is used also. º, ), dº
Sexagesimal fractions appear to have been employed, only
in certain localities, for only Brahmagupta (c. 628 A.D.) refers
to them. *
id Ibia, p. 31 ||
The Arabs learned the fractional methods of the Egyptians,
of the Greeks and of the Hindus, and, improving them, gave to the

(25)
East, and to the West, the medieval forms from which the World has
derived its present systems. Two of the most important avenues
through which these fractional forms have reached Europe are
through the works of the learned Arab mathematician and astron-
omer Mohammed ibn Musa Al-Khowarizmi (first half of the 9th
century), and the work of Leonard of Pisa who, after extensive
travels, wrote his celebrated work, the LIBER AB/ACI (1202).
COMMON FRACTION.
For the term reaction, the Arabs use AL-KASR (or KESR,
from KASARA, to break into pieces). The translator of Al-Khow-
arizmi employs FRACTIO; Jordanus (d. 1237), Gernardus (13th
century), Liverius (c. 1zoo-lagoºndon (c. 1380–1442),
Killingworth (d. 1445) and Peurbach (1423-1461), FRACTIO and
MINUTIA; Leonard of Pisa MINUTUM RUPTUs and, occasionally,
FRACTIO; Joannes de Muris (b. 1310- d. after 1360), RUPTUS,
FRACTUM and FRACTIo. 1
| see lists of technical terms given in this chapter
at the close of remarks on respective treatises. ||
The two conceptions of a fraction held by the Gheek
mathematicians” are held by the Arabic and medieval authors.
|| This Fºren, p. 22 ||
But Gernardus is the first, to attempt, a formal definition. *
| | Ibid, p. 49. ||
The Arabs use the Hindu form of writing a fraction. But in the
Case of a mixed number they Write the integer both above the
fraction and to the left of it. in Europe, three forms appear:
(l) the Hindu form; * (2) the Hindu form modified by the in-
Described on p. 24 : Aſsed by Al-Khowarizmi p. 103

(26)
Germadus p. 9 o' and other’s . ||
sertion of a line between the numerator and the denominator; and
(3) a rare form in which the denominator is placed immediately
after the numerator and distinguished by a certain mark above it;
as 3 s for 3. The LIBER ABACI is the first work found contain-
ing a *...i. written with a line, VIRGULA, between its terms,
º very probable that Leonard found it already in use in the
Arabic translations With Which he certainly was familiar, for he
claims no originality for his work. The line gained general use
very slowly. gernardus (13th century) writes a thus: a or -ab. ,
b b
and in the Bamberg Rechenbuch of 1483 the line is missings. The
third form appears only in an anonymous 14th century treatise en-
titled BREVIS ARS MINUCIARUM. * -
|| It begins, cum MINOR QUANTITAs ALIQUOCIENs sumpta MAIOREM
COMPONIT and ends, ST MULTO MAIOR FUERIT. It is found in
the cod. Vatic. Ottob. 309. It contains this passage,
MINUCIAM VULGAREM SCRIBES SUPERIUS NUMERATOREM INFERIUS
DENOMINATOREM PONENIDO. . . . . . . EST ENTM AT, TUS MODUS SCRT-
BENDT NON PEIOR PREDICTO, UIDELICET SCRIBENDO NUMERATOREM
ET DENOMINATOREM DEXTRORSUM CURTELLA LINEUNCULA RECTE
IPSI DENOMINATORI SUPERPOSITA UT - 3. QUINTAS SIC 3 5,
SIMILITER .4. 7 "** A 7. G. Enestrém, BTELIOTHELA MATH-
EMATICA, VII, pp. 308-309.
Leonard of Pisa does not speak of the origin of his
fractions, but he uses the Arabic forms, influenced frequently
by the Egyptian unit fractions and the methods both of the
Greeks and of the Arabs. Indirectly this is clear in many
(27)
places in his work, and very directly in the whole of the sixth
part of chapter *a, pp. 77-83, which he devotes to the re-
solution of fractions into unit fractions, * and on page 69 where
| This research, pp. 43-45,144 | | -
he specifically mentions Euclid's way of dividing 83 by * 3: The
Arabic and early Latin translators and Ben Ezra derive common
fractions from the numbers 2 to 10 inclusive and from the prime
numbers 11, 13, 17. . . . . .Thus they obtain l; i. . . . 1, l, 1. . . . . .”
- 2 3 10 li 13
- 33-39, |
| | Ibid., pp. 32,35-36,/ and elsewhere.
All of these forms reveal the Egyptian influence immediately.
other forms are obtained from these, for example, l is express-
ed by Leonard of Pisa as 1 of 1 written l 0 °, by Al-Hajjar
6 2 6 2
* ,
/ cº-
written 1 *, Alj-Karkhi (after Hoch-
; 2 /
| Ibid. p. 99.
º
Suter ) 3.S
of l
6
l
2 -
|| 4. Ibid, p. 36. ||
heim) Writes # for l of l of l. Leonard Would have written
8 || 7 || 3
t 3 7 8
- it, hus, Q-l l_l - Leonard Writes 3 and l of l thus l 3; but Al-
, , 3 7 8 5 3 5 3 5
Hassar (ałe, Suter) reverses the Arabic order, which is from
right to left, and writes 3 1. other translators frequently
5 3
make similar changes. Leonard uses a great variëty of forms,
5
Some of Which appear to be original with him.
| | Ibid., p. 42, (b), (c) and (f). ||
No signs of operations are employed. Hence, in many
instances, the meaning of the Arabic is doubtful.
Al-Khowarizmi and those who follow him arrange given
numbers *...**ś f solutions in columns; 6 later authors
Ibid., p. 412-1 #




(28)
-
generally in rows. Leonard is original in his arrangements. *
| | Ibid, p. 121 and elsewhere. -
The Hindu process of reducing to a common denominator fractions
of different denominators is familiar to all Arabic authors. The
lowest common multiple, short cuts, *
and checks are frequently
employed. Al-Karkhi uses Euclid's method of obtaining the high-
|| * Ibid., pp. 120,128 , 137 for example. ||
est common factor”; Al-Hassar"checks by casting out 7; Leonard
| | * Ibid., p. 35 || || Tbid, p. 108, |
by casting out 7, 9, ll, or 13s 4.
| For an explanation of his manner of checking Arabic
fractional forms, see ibid, pe 42. ||
SEXAGESIMAL FRACTIONS.
Babylonian fractions are not treated by all Arabic
writers nor by all later non-Arabic writers. Al-Hassar (12th
century), Leonard of Pisa (1202), Al-Banna (13th century), Al-
Kalsadi (d. 1486), and Peurbach (1423–1461) do not treat them
at all. A few authors treat them without mentioning common
tractions.” Al-Khowarizmi (c. 628 A.D.), Al-Nasawi (c. 1030),
|| For example, Fifthuaes (c. 1337), Von Gmunden (c. 1380–
1442 and Killingworth (d. 1445). ||
Ben Ezra (d. 1167), Joannes de Miuris (c. 1310-d. after 1360),
Ben Gerson (c. 1321) and Liverius (c. 1300-1350) treat both
common fractions and sexagesimalsº
The Writers call them by various names: Al-Khowarizmi's
translator calls them 'sexagesimals'; Ben Ezra, "astronomical”
fractions; Gernardus, "philosophical" fractions and "physical”
fractions; Liverius, "physical" fractions.

(29)
In his astronomical tables, * Al-Khowarizmi divides a
|| DIE ASTRONOMISCHEN TAFFELN DES MAHAMMED IBN MUSA AL-
KHWARIZMI IN DER LATEIN, UEBERSETZUNG DES ATHELHARD
von BATH - suter (1914), p. 7. ||
revolution (ROTA; Arabic, FELEK) into 12 signs (SIGNUM), a sign
into 30 degeees (GRADUS ), a degree into 60 parts (PARS; Arabic,
DAKAICA), a part into 60 seconds (SECUNDA), a second into 60
thirds (TERTIA) and so on in this fashion to infinity. But in
his arithmetic *, he does not mention the sign. Also, he calls
|| trattati, I, p. 17. ||
subdivisions of a degree "minutes." Few subsequent writers treat
the sign, and, with rare exceptions, all call 60ths of a degree
minutes. A very interesting exception is noted in some early
European astronomical treatises • Walcher, prior of Malvern, in
an astronomical treatise of ll?0 uses PUNCTUS for minute, MIN-
UTIA for second, and MINUTIA MINUTIAE for third. *
1| ZoDIACUM TOTUM SICUT ET Nos IN XII°3FM SIGNA UNUMQUODQUE
ta
Nº.
SIGNUM IN -K×xº~. GRADUS UNUM QUENGUE GRADU/M In . Lx**.
PUNCTOS UNUM.UENQUE PUNCTUM IN .LXta. MINUTIAS MINUTIARUM
DIVIDEBAT . . . . . . THE RECEIPTIon OF ARABIC scrence in
ENGLAND by Charles H. Haskins • THE ENGLISH HISTORICAL
REVIEW, XXX, pp. 56-69. ||
PUNCTUS for minute appears also in an anonymous 12th century
- º, Cº -
º, º
arithmetic. Ben Ezra is rather conspicuous by his careful
# * + (*) **** º º ºf wº. Nº. ºutwº
intrºduction to his treatment of sexagesimals. Von Gmunden makes
a futile attempt to introduce a 'more convenient System of parts
(30)
by making 1 revolution = 6 signs, and l sign = 60°, and, also, by
making 1 day = 60 hours.
- The early authors arrange their work in columns, while
the general practice of later authors is to arrange it in rowse
sibt El-Mariäinichecks sexagesimals by casting out 7’s or 8's;
Killingworth, by casting out 59.
Throughout the whole period treated, the operations of
addition, subtraction, duplation, mediation, multiplication,
division and extraction of roots, both of common fractions and
of sexagesimals, are familiar operations. With the exception of
the extraction of roots, all of these operations are discussed
in detail in the following pages •
II. * BRAHMAGUPTA. *
wº
º!" No. T would include the remarks | | * Colebrooke , pp.
- |
* already given on Egyptian frac- 277-289.
tons. pp. 16-19.
Brahmagupta gives no explanation of terms, nor any for-
mal introduction to fractions. He merely states, "He , who dis-
tinctly and severally knows addition and the rest of the twenty
logisties, and the eight determinations including measurement by
shadow, is a mathematician. *
|| Chaturveda, "A celebrated scholist of Brahmagupta," adds
that the twenty logisties are: "Addition, subtraction,
multiplication, division, square, square-root, cube, cube-
root, five (should be six) rules of reduction of fractions,
rule of three terms (direct and inverse), of five terms,
a ſº.
seven terms, eleven terms and barter. The eight deter-



(31)
minations are: mixture, progression, plane figure, excavation,
stock, saw, mound and shadow. By mathematician is meant a cal-
culator; a proficient competent to the study of the sphere." ||
Illustrative examples are given, but they are added by
Commentators, probably by Chaturveda, and not by Brahmāgupta
himself. Addition, subtraction, multiplication, division, square s
square-root, cube, and cube-root of common fractions are treatedo,
Sexagesimal fractions are not treated with the * *
that a method of squaring them is given in the supplement. *
|| Par. 62, p. 322. |.
III. AL-KHOWARIZMI. *
| Trattai I, pp. 17-23. I
The author begins the subject of fractions immediately
upon concluding his treatment of integers.”
|| Trattati I, p. 17.
No Chapter division is indicated further than this brief
remark, "And now we begin to treat concerning multiplication of
fractions, and concerning their division, and concerning the ex-
traction of roots, if God wills."
However, extraction of roots is not found in the Work,
indicating that portions of it are probably lost. Moreoever,
º, ſº
after "multiplication and division of sexagesimal fractions,
there follow in order arrangement, addition, subtraction, dup-
lation, and mediation of sexagesimal fractions, and then multi-
plication of common fractions. Roman numerals appear frequent-
iy, no divisions are headed, error’s occur, and the treatment is
brief and abrupt.

(32)
The author says, 'I know that fractions are called by
many names, numerous and infinite, as a half, a third, a fourth,
a ninth, a tenth, and one part of XIII., and a part of .x.VIII, *
| * Probably XVII is intended since xvi. II is not prime and
would be expressed as l of 1s or l of l- ||
2 9 3 6
and so on. But the Hindus . . . . . divided one into • LX. parts, which
they called minutes, then each minute into • LX. parts, which they
called seconds a , . and so on to infinity." It is then observed that
one is equal to 60 minutes, equal to 3600 seconds, equal to 21,600
thirds, and so on to infinity. "Therefore the first is the order
of degrees, in which is an integral number, and in the second
place, MANSIO, are minutes. In the third, also, are seconds; and
in the fourth thirds, and so up to the ninth and tenth place."
Technical terms: CIRCULUS, COLLIGERE ( to add), CON-
STITUERE (to arrange), DIFFERENTIA (order), DIVIDERE, DUPLARE,
FRACTIO, GRADUS, (also integer), MANSIO (order, place), MEDIARE,
MINUERE (to subtract), MINUTUM, MULTIPLICARE, NUMERUS INTEGER,
SECUNDUM, SEXAGINTA,
IV. MAHAVTRACARYA. *
| chapter III, pp. 38-69.
No definitions of terms are given. One dedicatory para-
graph constitues the introduction to fractions. Then the author
SayS:
"Hereafter, we shall expound the second subject of treat-
ment, known as KALASAVARNA (i.e. fractions)." *
|| A footnote adds that "Kalasavarna literally means parts

(33)
resembling 1." ||
ºnatiºn, division, squaring, square-root, cubing,
and cube-root of fractions, summation of fractional series in
progression, fractions in series, and six varieties of fractions*
| These six varieties are given under addition. |
follow in ordere
v. SRIDHARACARYA. *
|| Bretromeca MATHEMATICA XIII.3, pp. 209-210. ||
Less than two pages are devoted to fractions • No intro-
ductory remarks and no topical headings are given. Neither is any
definite order observed, but each fractional operation is set off
in a separate paragraph-
VI. THE ARITHMETIC OF AL-NASAWI. *
|| Woepcke-JOURNAL ASTATIQUE, Ie, pp. 492-500; Suter -
BIBLIOTHEGA MATHEMATICA, VII.3, pp. 113-119.
This arithmetic is divided into four books, treating res=
pectively integers, common fractions, mixed numbers, and sexages-
imal fractions. The last, three books are subdivided into seven
chapters each, treating in order addition, subtraction, multi-
plication, division, square root and cube root of their res-
pective subjects. The title THE SATISFACTORY ONE arises from
the fact that the author purposes to present a simple and easy
treatment by avoiding the unnecessary processes and tedious
examples of his predecessors. The complicated fractional forms
of Al-Karkhi, Al-Haggar and Al-Kalsadi are wanting. His opera-
tions are much simpler than those of other authors, and are
(34)
quite modern. To multiply two fractions, multiply their numer-
ators and then their denominators, and divide the first product
by the second. To divide two fractions, multiply the first by
the second inverted. Also, bring the two given fractions to the
same denominator and divide the numerator of the first by that
of the seconds
Tn extracting square root and cube root, examples are
selected so that their roots are exact.
Al-Nasawi Wrote fractions after the manner of the Hindus ;
O
thus l for 1, the zero being replaced by the integer in case of
11. 11
a mixed number. *
|| Tropfke I, p. 79. ||
However, Al-Nasawi appears to have been considered radical
by Arabic calculators, who did not look upon his work with as much
favor as did the Persians,
VII. AL-KARKHT. *
| Chapters x-x111, xxv. ||
Both common fractions and sexagesimal fractions are treat-
ed in this arithmetic, multiplication, addition and division of
common fractions preceding multiplication and division of sexage-
simal reactions.
(a) Common fractions.
There are many fractions, but of those expressed in
Arabic notation there are nine : 1, 1, . . . ._ls. These are called
2 3 LO
simple fractions, and from them an infinite number of others may
be constructed; as 111, 111 || 1
# #. that, * } * #" -



(35)
are other fractions, also , such as one part of eleven, one part of
seventeen, and those formed from these and from simple fractions. *
º
|| The translator calls such fractions "die stummen Braeſe."
compare With Al-Haggar, Ben Ezra, and Al-Banna. pp. 36,39,51.
Unity can be divided into an infinite number of parts.
This division is different in different countries. The definition
and illustration of the denominator of a fraction, the definition and
the use of the Euclidian method in obtaining the common factor of any
two numbers, and the method of obtaining the lowest common denomina-
toº of several different fractions are all modern.
(b) Sexagesimal fractions.
"The degree contains 60 minutes, the minute 60 seconds, the
second 60 thirds, the third 60 fourths, and so on to fifths, sixths,
sevenths, eighths, ninths, tenths, elevenths, and . . . to infinity.
Therefore the minute is 111 of a degree, likewise the second of a
minute, and the third sº *ena. In general, each division of an
order may be placed into a ratio to a division of the next higher
order by means of ill.' Thus 20 seconds is 1 of a minute, or
LO || 6 - -
ll lll of a degree, or lllll of a degrees
O || 6 || 3 #
VIIT, AT-HASSAR. *
| Bibliotheca Mathematica, II.3, pp. 23-40. |
The seven chapters into which this work is divided treat
respectively the following topics: whole numbers, multiplication of
fractions, transformation of fractions, addition of fractions, sub-
traction of fractions, division of fractions, and extraction of roots
of whole numbers and fractions.
Each of these chapters is further subdivided. Chapter II
has 72 parts, of which part 1 gives the various kinds of tractions
and the manner of Writing them. Simple fractions include the first
(36)
nine fractions 1, 1. . . . l, and the fractions 1, l; 2, 5, etce
2 3 T5 11 13 11, 17
or those not derived from the first nine . These would be expressed
in the Arabic as "one part of eleven,' 'one part of thirteen,' etc.
If one wishes to write 2, one does so by writing two in the place of
3 -
the one of l and three in the place of the 2. “
2 Ben EZ ºa. - - E-
|| Compare with Al-Karkhi,ſand Al-Banna, ºrd–F#–. pp.35, 39,51. ||
Another class of fractions are those whose denominators are
composite, as 'one twelfth' written l, that is l of l;' three fifths
6 2 2 6
and one third of a fifth," written 3. 1", and so on.
5 3.
| The Arabic order is 1 3 | | -
3 5 -
IX. BHASKARA. *
|colebrooke, pp. 13-18, 42.
- Cl
Fractions are treated in this work in Section TTT of ſhap-
C.
tler TT and section II of ghapter TV, covering only seven pages • The
- º
order of the operations with fractions is not definite, but the
following are treated: addition and subtraction of fractions, multi-
plication and division of fractions, and square, square-root, cube,
and cube-root of fractions. These operations are the eight opera-
tions of arithmetic sometimes referred to . Some of the rules given
are later repeated in different words. In this translation frac-
tions are written in the form a , While the original Work has the form
b -
a • An integer may be written as a fraction with unity as a denom-
º šićhara and Brahmagupta followed * custom. * Several
|| Colebrooke p. 14, footnote. ||
fractions are read from right to left,
X: ANONYMOUS-LIBER ALGORISMI DE PRATIca ARISMETRICE. 4
| | pp. 49-72 ||
With the exception of the excerpts mentioned, pp. 93-133,
this algorism shows more finish than the Numero Indorum. The style


(37)
is more fluent, no Roman numerals appear’s topics are indicated by
proper headings and the work is enlarged both by more detailed
exposition and by the addition of examples, particularly in com-
mon fractions. Multiplication and division of sexagesimal fraces
tions are slightly confused in their order of arrangement, 'but
the general order of topics is , sexegesimals: multiplication,
division, arrangement, addition, subtraction, duplation, and me-
diation; common fractions: multiplication and division. Extract-
ion of square root follows fractions •
- The first sentences of the introduction to fractions read,
"Although the denomination of arly number of parts can be done in
infinite ways following infinite numbers, still it pleases the
Hindus to make the denomination of their fractions from sixty, For
they divided one degree into sixty parts, which they called min-
utes. * * And then the author explains the divisions and sub-
||frattata, p. 49. || -
divisions of sexagesimals and gives the names corresponding to
each ordero See ITT.
Technical terms: aggregare, giffra, (also, circulus),
denominatio, differentia, diminuere, dividere, duplares extrahera,
reactic, fractiones alterius denominatić his (common fractions),
gradus, mediare, minusre (to subtract), minutum, multiplicare,
numerus collectionis,” numerus communis,” numerus denominationis, *
|| See p. 112 for explanation of this terºme || -
numerºus dividers, numerºus dividendus, numerºus fractionis, numerºus
multiplicans, numerºus multiplicanăus, ordinare (to array), secun-
dum, tertium, vacuus.
XI. ANONYMOUS - A 12th CENTURY ALGORTSM.
The treatment of fractions in this work is for the most

(38)
part, clear and concise. There is no confusion of material. Its
divisions and subdivisions are all properly headed, its expositions
are brief - too brief at times - and its examples are given in Con-
densed forms and set off in figures • The order of topics is the
same as in III and X.
"A degree is an integer, a part of which is 60 minutes , .
2600 seconds, 21600 thirds. * *
| Trattata II, p. 21. II
Technical terms: addere, ciffra, circulus, colligere,
constituere, differentia, diminure, dividere, duplare, extrahere's
fraction gradus, mediare, minuere, minutum, minutiae diversorum
generum (common fraction) multiplicare, (also, ducere) partire
(to divide), retrahere (to reduce), secundum, summa-
XIII. ABRAHAM BEN EZR.A. *
| Ezra, pp. 31-36. I
Both common fractions and sexagesimal fractions are treat-
ed in Part W. of this Work. The treatment of the former includes
introductory remarks, multiplication, division, addition and sub-
traction of fractions; that of the latter, introductory remarks
and multiplication of fractions. No headings indicate the sub-
divisions of the chapter, but there is no confusion of material,
and the treatment, is clear a Many examples are given, and occas-
ionally the Work is set off by a marginal figures
(a) Common Fractions,
"It is known that 1, as it were, is the mid-point of a
Circle; therefore 1. cannot really be broken." Arithmeticians take
their ractions from Whole numbers; as , 1, #’’ ‘’’ ‘‘ſº These are
the first series of fractions and are from the 9 numbers 2 to lo
inclusive. The respective numbers from Which they are formed are

(39)
called denominators. 'some times the denominator is a number which
has no part which one can express, * for it is * Prime number, as
| compare with v1.1, VIII and XVII, pp. 35,36/M. Silberberg,
the translator of Ezra, explains (note 73) that these are
called fractions "which one cannot express" because the
Hebrew contains no word for the fractional part l, l;
11 13
*
. . . . Instead one says, "one part of eleven. . . . . "
11 or 13, etc.” On the one hand l is no number; on the other hand
it is a number - an odd number's
If one uses two fractions, which are not of one kind,
also not equal to each other, then orie determines the number from
which each of the fractions is derived and multiplies them together-
Their product is the common denominator, Tf a third fraction is
given, then find the number from which this third fraction is gº-
rived, and multiply it into the common denominator of the first, two
fractions. This last product is the common denominator of all three
fractions. Similarly one may find the common denominator of any
number of fractions,
(b) Sexagesimal fractions. * -
| Ezra, pp. 43-44. || -
Ezra calls sexagesimal fractions 'astronomical tractions,
and explains that, that astronomers divided the celestial sphere
into 360 parts, because this number is nearly equal to the days of
a solar year, and there is no number, smaller than this, which has
all parts (as integers) which one can express, except the seventh . . . .
If We divide the celestial sphere into 12 parts, then in 1 con-
stellation there are 30 degrees, and there is no number smaller than
this which has as many (integral) parts, for it has a half, a third,

(40)
a fifth, a sixth and a tenth. Since it has no fourth, one has
doubled this number, giving 60. Therefore one divides each de-
gree of it (by 60), then into sixtieths, and calls each part
* minute" (first, ) and divides each minute into 60 parts and calls
these 'seconds'. Likewise, one divides each second into 60 parts
and calls the result "thirds. " Thus one proceeds to 10, each
divided into sixtieths, and still further, if it is necessary •
The study of 'astronomical' fractions is regarded as
of special importance because, with them, Ptolemy calculated the º
1. - -
cords of a circular arc, and extracted the roots of irrational -
numbers; indeed, he showed that Archimedes' value of 77 , 3 12, a to
7Cº.
º
* * * * * 30 seconds. :
|| Neither is quite correct. ||
XTV. Leonard of Pisa. *
| Liber Abaci, pp. 47-83.
of the fifteen chapters of the Liber Abaci, chapters VI
and VII, are devoted to common fractions, sexagesimal fractions
not being treated. Multiplication of fractions is treated in
chapter VT, and addition, subtraction and division of fractions,
together with incidental methods of reductiºn to a common denomin-
ator and of simplification, are treated in chapter VIT. Some
simple cases of addition are found in chapter VI, pp. 53-55, in- -
cidental to multiplication. Chapters and, excepting a very few
omissions, sub-divisions of chapters, are all carefully headed,
Multiplication is presented in eight cases, or parts, which will
be given later. Chapter VII is sub-divided into six parts:
(1) Addition, subtraction and division of fractions of one line.
(2) Addition, subtraction and division of fractions of two liness

(41) .
(3) Division of a whole number by a Whole number and a frac-
tion or fractions, and conversely •
(4) Addition, subtraction and division of whole numbers With
fractions •,
(5) Addition, subtraction and division of parts of numbers
with fractions •
(6) Resolution of fractions into unit fractions • 1.
| | The wording of the text is: Septimum itaque capitulum in
}y. -
.-
partes sex dividifus • In prima guarum ad àictionem unius
virgule cum alia » nec non extractionem unius virgule de
alia demonstrabimus et divisionem unius virgule per aliam-
fā€¢
in secunda addictionem et extractionem duarum virgulam
- - - *T
cum duabus » et de divisione earum ad invicem •
- $t.
In tertia divisionem integrorèm numerorum per integros
- - T-
et ruptos et eorum contrarium •
In quarta additionem et extractionem et divisionem
integrorum numerorum cum ruptis cum integris et ruptis e.
in quinta autem addictationes extractiones seu divisiones.
partium numerorum cum ruptis edocebimus • -
In ultima quoque reductiones plurium partium in singulis
partibus ostendemus • Liber Abaci, p. 63. | |
No formal introduction to fractiqns is given. Neither is
a fract.ion. defined. Denominators great.er than 10 are, as far as
possible , reso1ved into factors less than l0 , The lo West, common
denominator is frequently used. References are made to Euclid,
particularly in finding the ratio or t, Wo expressione.* Numbers
| | Instances aae , Ibid, pp. 51 and 69. || -
are read from right to *•*** ••* • #* is read 5 l. Fractions are
2
classified according to the form of their expression. Tiiustaa-
tions of some of these expressions are the following:
(a) 2 4 3, meaning 3.4 of #. of 1 of i\;
3 5 7 #45 7t. 5 7.
(b) 2 4 3o meaning ### of 3. É of 4 of 3;
3 5 7 7 ' 5 '' 3 5 7
(c) og 4 3, meaning 2 of 4 of 3 ;
3 5 7 3 5 7 . -
| Leonard appears to be the author of the forms (b) aſ ā (c.). ||
(d) 4 29 3, meaning 3 of 29 4;
7 5 B 7
(e) & 5 33 13, meaning (3 + 1) (334.343 of 1.)
7. 9 5 4 4 ' 5 9 7 9
(f) 1 1 0°, meaning l of l l l of 1
--- 3. T; + + ---
#Tº Tº 4. 3
|| Probably used only by Leonard. |
The Be are other simpler forms, but they have been observed previous-
ly in Al-Hajjar (VIII) and elsewhere, and are undoubtedly from
Arabic sources •
An abundance of illustrative examples are proposed and
solved, and, in most instances, their results are set off in mar-
ginal figures. Results are checked by numbers, usually by 9 or ll.
7 and 13, also , are used. The check on a Whole number with a frac-
tion may be obtained before, or after its simplication; for example,
- 44% O = -
2 8 11. 11 8 11) 8 11)
obtained as the expression stands as follows: "multiply the check
the check by 7 on 1 & 3 14 (= 14.4 fºr 4 of , of 4) may be
2
of 14, which is 0, by the check of 11, which is 4, and add the 3
which are over the 11, there are 3; multiply that by the check of
8, which is 1, and add 3, which are above 8, there are 6; multiply
that by 2 which are under the line and add 1, which is above 2,
there are lă, of which the check, which is 6, is the check of
l_3_3 14- "
2 8 11. -
| | Ibid, p. 52. ||
Reducing the given number to an improper fraction, one obtains
#H. the check on the numerator of which is 6, as befores


(43)
Much skill is shown in the reduction and simplification of
fractions:
-* ---
3 9 9
of a ninth; " and so on. Part six of chapter VII, pp. 77-83, is
"For 3 you say l; for 1 3 (~ 3 - 1 of l) you say 1 and a third
º 3 + # 5’ → ~ :
devoted to the resolution of fractions into unit fractions. This
part, , therefore, reflects the continued influence of the Egyptian
fractions. It is sub-divided into seven cases: -
(1) When a larger number below the line is to be divided by a
smaller number, for example by itself, which is above the line :
(a) Simple fractions, as 3 - 1, where the aerºsis
divisible by the numerator; 12 4.
(b) Composite fractions, as 2 0 (= 2 of 1 = 1 of l) = 1 0 or
Tº T 5 ºf 5 2 9
—l;
18 - -
(c) Reversible composite fractions, as 3 0 (~ 3 of 1 = 3 of 1 -
- 5 9 5 9 9 5
of 1) = 1 0 or l-
5 3 E TE
(2) This case applies when the larger number can not be divided
l
3
by the smaller, but such parts can be made of the smaller that the
larger is divisible by some of them, as in case l; for example,
5 = 3 2 - 1 l
6 6 6 2 3
Work in this case is facilitated by the use of tables to
Which the Whole of page 79 is given. For the number 6 as a de-
nominator, one has the table º Parties de 6
- TTÜGT5 Test. Tº
in figure la Similar tables -
2 1.
are given for 8, 12, 20, 24, 3
- 3 1.
60 and 100. º
4. 1. 1.
6 º
5 l 1.
- 3 2 |
Fig. 1. (L.A., p. 70)

(44)
(3) The third case includes fractions of which the larger de-
nominator is increased one in order to make it divisible by the
smaller numerator; as ,
1. 2 12 11 12 12 11 11.
l, l 0.
6 6 Ll y
(b) _8 = 6. 2. But from (a) _6 = 1 1 0, and -2 =
- IT TT TT TT 2, 3 II 11.
1 1 0. Hence 8 =l 0 l 0 l = l l l
& GTT II GTTT 2 TTT - 66 35 ;
In cases (4) to (7) the fractions are resolved into other
fractions to which cases (1) to (3) apply. They will be sufficient-
ly clear from illustrative examples, although the text treats them
in detail:
(4) — 5 = 1 - 4. But from (3), _4 = 1. l. Hence 5 = l l l
- TT II II 11 33 3 11 33 11 .
(5) 11 = 2 9. But from (l), 2 = 1, and, from (3) 9 = 1 1.
26, 26 26 36 Tó 36 75 3
Hence ll = 1 - 1. l.
26 78 L3 3 -
(6) ** 14 into unit fractions • Dividing 13 by 4, the
3
quotient lies between 3 and 4. Hence 4 lies between l and l.
3 4.
#3 is one whole unit 1 of one unit is 1-3 (= 3+l of ºl, which
13 4. - 4 l; 13 * 4 13
subtracted from 4 (= +3+...+1 = x3 + 4 of , 1) leaves 3
T&T Tº T3 13
- T 4.
which from (2) reduces to 1 1 0 (= 1 of 1 + 1 of ºl)
- ? T3 Zi
—l —l."
13 2 26
Therefore a 4 = 1. l. l. 4 2 is
3 53 2.5 Å
In conclusion, a general rule is given. * Stated briefly
| | Ibid, p. 82. ||
it is ; Multiply numerator and denominator of the fraction by 12,
24, 36, 48, 60, or some number such that it is more than one half
and less than twice the denominator, and then divide the resulting
numerator successively by the factors of the denominator, and re-
duce the results as in the cases given above. For example,
17 - 17 - 24 = 408 - 2 14 (= 14, 2 of 1) = 14 2 o. But
29 29 . 24, 29.34 35 T24 #*#5 º 24, 35 3.

14 = 7 = L 1, and 2 O (- 2 of l = 2 of 1 = 1 of l) =
757. Tº TE 2 29 24 29 24, 24 29 12 29
1 O = 1 = Therefore 17 = 1 l l -
12 29 348 29 48 12 2
3
The same result Would have been obtained had 36, 48, or
any multiple of 12 been used.
Technical terms; addere, circulus, denominans, denomin-
º
atus, differentia (order, class'), dimidium, disgregati; (dissolu-
tion), dividere, extrahere (to subtract), fractio (also, minutum,
ruptus), multiplicare, numerus integer (also, sanus), pensaſ test,
check), residuus, summa (sum, also, product), virgula (line).
XV. JORDANUS NEMORARIUS. *
|| Bibliotheca Mathematica, XIV2, pp. 41-54. ||
For this discussion Eneström employs the text of the
Tractatus minutiarum, as found in the Vaticanus Ottobonianus 309,
and the text of the Demonstratio de minutiis, as found in the
codex Latinus Berolinensis 4° 510. He makes parallel outlines of
them. The former is probably the older. It includes 26 pro-
positions, while the latter includes 35, propositions 6, 26-29,
31-34 being additions. Eneström gives the text of the introduc-
tion of each together with discussions of them. The first Writes
3 thus: "3,4", or "3 quarte"; the second writes 2 thus 3 *. Both
- . 3
|| As in x1 p. 116|| , .2 : º, . . . .
10 … "
write 2° 10' 32" 57" thus 32. * A vacant space is filled with a
67-68 57 -
|| As in IIIpp. / ; x, p. 76 ; XI, p. 77 ||
a ciphere The first contains some discussion, at times very ob-
scure, of the questions whether by multiplication the product is
smaller, by division the quotient is larger than the given frac-
tion, and why a minute multiplied into itself gives a Second,
while a degree multiplied into itself gives a degrees

(46)
Both introductions use many technical terms:
Tractatus Minutiarum: minutia (fraction), numerus denºsinans,
numerus numerans, vaga sumptio partium (common fractions),
ordinaria sumptio partium (fraction with subidivisons, or the
Roman fraction or sexagesimal fraction), consimilis sumptio
partium (fraction with a mixed denominator of the form a", or the
sexegesimal fraction), and dissimilis sumptio partium fractions
with a fixed denominator but not of the form aº, as the Roman
fractions).
Demonstratio de minutilsº fractio or minutia, numerºus
denominans, vaga sumptio partium, ordinaria sumptio partium, con-
similis sumptio partium, and dissimilis sumptio partium, all of
which have just been defined in the Tractatus minutiarum. Further
technical terms defined are : fractiones eiusdem generis (fractions
of the same kind, as different parts of a length), tractiones
diversorum generum (fractions which can not become equal, as parts
of a unit of length and of a unit of area); similes fractiones
(fractions of the same name), and then the familiar terms: addere,
dimidiare, dividere, duplare a multiplicares_pars multiplicans,
pars multiplicataa and radicem extrahere, etherometra and ysometra
are two terms whose meaning is not clear, but from what is said
one concludes that they refer to extraction of roots of sexage-
simals, and that ysome tra are such fractions that can have a root,
as seconds, and that etherome tra are those that have no direct root,
as minutes • Of these works themselves, Eneström gives only à.
brief synopsis. He considers the Demonstratio de minutiis (here-
in after noted as number I), section by section, in regular order,
and then states the number of the corresponding section of the
(47)
Tractatus minutiarum (here-in-after noted as number II.)
These works are probably the basis of the algorism of
Gernardus (this research, XVI), which Eneström has published and
commented upon and compared with these. Hence it Will be con-
venient to have in one table * a synopsis of all three works in
| see table p. 48 ||
so far as they relate to fractions • Number I treats of addition,
subtraction, aviation, mediation, multiplication, division and
square root of common fractions, and multiplication, division and
square root of sexagesimal fractions; number II lacks the square
root of common fractions and a few unimportant special cases of T :
the third contains 42 propositions, and adds to I cube root of
common fractions, and addition, subtraction, duplation, mediation
and cube root of sexagesimal fractions. *
- | | All three treat square root of integers. See Biblio-
theca Mathematica, VIII.3, 24–37, 135-153; XIII.3, 289-
332. ||

|Sectio
n Numbe
º -
º
º
º
*†
43
Synopsis of sections treating the fundamental
º TT TTT operations with fractions
7 # 6 7 Reduction of common fractions to a common denomina-
toº : the rule is moderne - – -
8 7 23 Reduction of common fractions to the lowest terms:
the rule is moderºn.
Q 8 9 Addition of common fractions: a c ad -- bc.
+ + i = −3
TIO 9 Il T Subtraction of common fractions: a - c – ad - bc
- b d d
Duplation of common fractions: 2. a = 2a: … a
11 10 | 12 5 - T5 ºr E-
2
- 3.
12 11 12 || Mediation of common fractions: 1 * – 8 2
2 * B = 2E or TE-
Multiplication of common fractions : Timited to forms
13 12 13 l - 1 = L. minutia here means unit fractions.
- b di Ed
- l
14 13 14 Finding a part of a part : b = 1 -
C bc.
23 22 T3 || Multiplication of a whole number by a fraction
15 ab,
3. * - ºn TE
34 T33 13 || Multi Tication of a fraction Ey a fraction: a 3 ac.
- - - . b º d bd
Multiplication of a mixed numbers: (a1 b)(d e) - ad H.
25 24 |13, 17 - C. : -
& -- bd + º- Multiplication of sexagesimals: 2 methods
------ | f c ºf (Cf. Al-Karkhi, this research, p. 107- )
3.
26 Division of common fractions : E - ad -
c be
27 19 Division of whole number by a larger; a , . . a . -
- - b
| | Division of a fraction Ey a who Te Rūmīº a T -
28 b - —º e
--- - c be
Division of a Whole number by a fraction: a ac
29 - - r TE E
Ç
Division of common fractions: (1) When a - Ho, and E- iºd,
& 3. 3. acd
30 25 18, 19 || 5 || C = h; (2), in general, E - Edd - ad .
3 TE : Ta- C *Sº
d d d. º
Division of proper fractions and mixed numbers: Reduce
30 25 the latter to improper fractions and proceed as
above. *-
Division of sexagesimals: Cf. Al-Karkhi, this research,
30 25 27 | p - 138; in I and TI the dividend is taken greater than
the divisors

(49)
zºvt. MEISTERS GERNARDUS. *
|| Bibliotheca Mathematica, XIV3, pp. 99-149.
Gernardus treats both common and sexagesimal fractions, the one
immediately rollowing the other in each of the following operations:
addition, subtraction, duplation, mediation, multiplestion, di-
vision, square root, and cube root. “
+
|| This is the only work mentioned in which both common and
sexagesimal fractions are found in all of the eight opera-
tions stated above, unless an exception is Liverius whose
complete work was not accessible • -
A fraction (minucia, reactio) is the quantity enumerated and
denominated;" as, one fifth, three fourths, three thirds, five
|| minucia sive fractio est quantitas numerata et denominata-
Ibid. p. 100.
(4)
thirds. Formal definitions are given of many terms ; as denomination,
(2) - ut . 3. ut secundum - 4.
| Cum minor quantitas secundum aliquem numerum multiplicata
ut - 3 - ut - 12-
maiorem perficit dicitur quantitas minor maioris par’s multi-
plicativa sive pars quota sive pars simpliciter nomine res-
ut, - 4. ut . 3.
tric to. Numerºus autem secundum quem minor quantitaš multi-
pligata maiorem componit est denominacio partis ad totum- | |
5 6
denominator, numerator, multiplication, division, squaring, square
ut - 4 - guia quater.
| | 5 Est numerºus denominans in quo to clens est unitas quogiens
ut, , 3. -
par’s denominata in to toe | |
ut. • 3.
-
|| 6 Numerus numerans est in
quo toeiens est unitas guociems pars
in minucia • Compare with Liverius, Von Gmunden, and
Peurbach, pp. 54,60–61,61-62, respectively. ||
root, cubing, cube root.
Fractions are divided into two classes: common fractions and
sexagesimal fractions. The latter are called 'philosophical"

(50)
fractions, since philosophers use them. In a printed edition of this
manuscript they are also called "physical' fractions minucie
phisice , a term not noticed before this time." In them the de-
| Dr. L. C. Karpinski explains the probable origin of this
1ast, term as follows: "sexagesimal fractions came to be
called 'physical' fractions through mistaking the abbrevia-
tion of the word 'philosophical', another name for sexa-
gesimal fractions, for the abbreviation of the word 'phy-
sical', 'philosophical' and 'physical' having the same, or
very similar abbreviations. |
gree is the integer and its reactions are the minute, the second,
the third, and so on, with a ratio of 60, as in previous works.
They are arranged in order from right to left. 2° 10' 32" 57"' are
written 2 10 32 57, the denominators never being written.
Other fractions are called common fractions. The fraction
is written a, or more frequently, - a b.
b
The author reduces fractions to a common denominator, or to
#
their lowest terms, and a given integer or a fraction to a fraction
With a given denominator in the same manner as one does today.
| Throughout the whole work, letters and lines are used to represent
numbers. Frequent references to Euclid are made.
Technical terms: addere (to add), cifra, deduplatio (halving),
delere (to delete), denominatio, dividens (dividend), dividere ( to
7
divide), divise (divisor), duplatio, fraction (also minucia minutum)
minucie - philosophicie (philosophical fractions, sexagesimal frac-
tions), minucie phisice (physical fractions, sexagesimal fractions.)
minuole vulgares (common fractions, ) multiplicare (also, ducere),
numerºus denominans (denominator), numerus numerans (numerator), 98.I.'s
mul - - - º -
tiplicara - pars Quote = pars (a fraction whose numerator is 1),
subtrahere (to subtract.)






(51) º
XVTT. TBN AL-BANNA. *
| Atti dell'Accademia Pontificia de' nuovi Linesi, XVII,
pp. 307-309. || -
section 2 of this work is sub-divided into six chapters respect-
ively treating the following topics on fractions: names and classi-
fication, addition and subtraction, multiplication, division (includ-
ing denomination), restoration and abasement, and transformation. 2
| The treatment, of restoration, of abasement, and of transforma-
tion is wholly rhetorical kākā and is, in effect, the same as
given by Al-Hassar and Al-Kalsadi. see Al-Hassar pp. 139,74,
Definition of a fraction: "A fraction is the relation which
exists between two numbers when it expresses one or many parts (of a
unit). The relation between the numerator and its denominator, you
call a fraction."
- There are ten” simple names for fractions: a half, a third. .
| Al-Banna names only nine here, including the word "a part", but
Marre, his translator, includes 'a tenth". This statement
then agrees with Al-Karkhi (VII), Al-Haggar (VIII), Ben Ezra
(Xtſ I), and Al-Kalsadi (XXVIII, this study, pp.34, 36,38,64 ||
a ninth, then a part (not articulated). You repeat these fractions
in addition, ending with a value of the numerator less than that of
the denominator, as 3. If you join these simple names to each other,
there results a *-isºla of two, or more names, as a half-
fourth for an eighth, written 1. The numerator differs according
to the variety of the reasº." Classified according to their num-
erators, there are five classes of fractions: (1) Simple reactions,
(2) relative fractions, (3) heterogeneous fractions, (4) fractions
- - - 4.
divided into parts, and (5) subtractive fractions.
|| Al-Kalsadi gives the same classification. ||
_
(52)
T - -- - - - -
These classes and the manner of reducing them Will be clear
rom illustrative examples: *
| These examples are added by Marre; Al-Banna, himself, gives
no examples • |
(l) 3. Evidently the numerator N = 3.
4.
(2) 4 3 (~ 3 + 4 of 1.): N = 3.5 + 4 = 19.
5 4 4 ' 5 4
(3) 4 3 (= 3 + 4). N = 3.5 + 4 = 7 as 43.
5 7 7 " E
(*# (= 1 of 4 of 6). N =l - 4 - 6 = 24.
- 7 || 5 || 3 3 5. 7
(5) (a) disjunctives 1 7 - 4 (- Z + 1 of 1 - 1). N = (7.2+ 1)/
2 8 4 3 2 8 4
4 - 1.16 - 44.
(b) conjunctive”: i. 3 5 - 1 i.
3 4 7 3 4.
| How this is obtained is not very clear from Al-Banna's direc-
tions. Al-Kalsadi is more intelligible: "4 l moins 1 5" =
5 8 3 7
( 1, 5) less l 5 = l (5 + 1 of 1) - (3 + 1 of 4)
3 7 3
)
(37) 37 IIA GT I ºf 7 7
+ #
1.
l
4.
}
-
5
8
(
(4 I) (3 l
( ) 8 8
Whence N = (3.5 + 1) 5.8 - (3.5 + 1)(5.1 + 4) = 496. (Ibid,
XII, p. 267. ) Now if one follows Al-Kalsadi's directions,
then to obtain Al-Banna's result from his (Al-Banna's) ex-
ample, one would have to interchange the terms of his ex-
ample, writing l l moins l 3 5. Then N = 560, as shown. |
3 4. 3 4 7
N = [(5.4+ 3) 3 + 1.] (3.4) - [(5.4 + 3) 3 +1] (1.3 + 1) = 560.
These five elementary types may be involved in many differ-
ent combinations. Also, whole numbers may be present.
Thus:
(a) l 5 5 (* 5 + 2 + 1 of 1). N = (5.6 —- 5) 2 + 1 = 71.
2 6 Ö 2 --- -
(b) 8 # |- 8(3 + 3 of 1j. N= ( 3.6 + 5)8 = 184.
6 6 4. -
(c) * } # |- 5(5 A 3)]. N = (5.4 +3.6).5 = 190.
- 6 4
(d) 1 3 l 3 ſ. 3 iſ 3, 1 of 1)]. N- (3.5 + 1)(3.3
E 3-4 ##4-# * 40 (3.5 + 1)(3.3 + 1) = 160.


(53)
other examples are given, but the meaning of many of the
longer ones is doubtful owing to the absence of symbols of addition
and multiplication among the Arabs •
xvi.II. LEWI BEN GERSON. *
| pp. 89-194. ||
Both common and sexagesimal fractions are treated in the second
part of this work, but neither very completely. No formal introduction
to fractions is given, nor are the terms employed defined. The arrange"
ment of material is confused. Addition of sexagesimal fractions, and
subtraction of both common fractions and sexagesimal fractions are
missing. Multiplication is treated first - An air of scholarship and
of originality pervades the Works
XIX. PLANUDES.”
| pp. 31-33. || -
The subject matter of this work is well arranged and presented,
and is very modern in its operations. The treatment of fractions in-
cludes only sexagesimal fractions, whose sub-divisions appear in regu-
lar order: introductory, addition, subtraction, multiplication, and
division. The introduction suggests the introduction to sexagesimal
fractions by Ben Ezra, * He says tº whaenº), "I must now speak
| Abraham ben Ezra, Das Buch der: Zahl., pp. 43-44. ||
of the constellations, degrees, minutes and seconds. Notice therefore
that the sun in its orbit goes through 12 signs, each of which falls
into 30 degrees, each degree into 60 minutes. . . . . , and so on to in-
finity; however, astronomers leave all others except eight and take
only four; viz. , signs, degree, minute. . . . . , and seconds. . . . . ."
XX. NICOLAS RHABDAS.
Rhabdas gives no systematic treatment of fractions, but he
does give illustrative examples in their multiplication, division -



(54)
and extraction of square root. He gives, also a table, arranged in
rectangular form, for noting the products of the fractions 3, 2, l
- ?
2 3 2
l, . . . . 1 by the integers 1, 2, 3, . . . . 10 respectively a * The examples
3 Tö
| p. 169.
employed in multiplying and dividing fractions involve a series of
unit fractions, which he first reduces to a common denominator, and
then multiplies or divides as required. The result is reduced back
to unit fractions. * Sexages imal fractions are not mentioned-
| For example, pp. 177-183. ||
xxt. Johannes DE LIVERİs. *
| | fr. 21- 25. || -
Certain folios treating the introduction, addition and sub-
**º-da º Paºlº ºwſ ºf: d (a º *A* * º
used, awe-eeen other sources,” it is learned that Liverius furnished
|| From a 1540 edjation and from the Rara Arithmetica by Smith,
traction of fractions were missing from the copy of the 1483 edition
p. 13. | |
the treatment-of-fractions-given-in-Beldamandi's arithmeties. The
following operations of fractions follow in order: addition, sub-
traction, dupāation, mediation, multiplication, and division. each tº
treats both common fractions, vulgares fractiones, and sexagesimal
(a.e.) a tº a ,
fractions, phisies/fractiones, the latter following immediately after
the former. The methods employed are quite modern.
Two numbers are necessary to express a fraction: a numbrator,
numerºus numerans, and a denominator, humºus aenominans." Fractions
| Numerºus numerans ast ille in quo totiens est unitas quod parties
integri Volumus representares Sed numerºus denominans est ille
in quo totiens est unitas quotiens pars denominate est in suo
Compare these definitions with the definitions of numerator

(55)
and denominator given by Gernardus, p- 49, Von Gmunden,
60-61
pp. / and Peurbach, pp. 61-62.
are called common which are enumerated from common numbers, as a half
from two, a third from three, and so on. Sexagesimal fractions are
called physical fractions. They are fractions that are represented by
divisions of sixtieths. A degree is called an integer, and is divided
into 60 parts, which are called minutes , and a minute is divided into
60 parts which are called seconds, and so one
In common fractions the denominator is written below and the
numerator above, as four sevenths 4. In sexagesimal reactions, only
the numerator is written and the ºis, for the denominator is helds
They are written in the order of their places ; as signs, degrees •
minutes, seconds, and so on, the first place being for signs, the se”
cond place for degrees, the third place for minutes, and so one
Common reactions are reduced to a common denominator,” and in-
tegers are reduced to fractions, or vice versa, just as one reduces
them to days
Technical terms adderes cifra, denomination denominators
denominator communis, dividere, duplare, frangere, gradus, hora,
mediare, minutia (also minuta, fractio), minutia vulgares (common
fractions; also, minuta simplices, tractiones vulgares), multi-
- f - -
pligº, numerator, numerºus denominans (denominator), numerus numer-
ans (numerator), pºſioie minucii (physical fractions), signum (sign, )
subtrahere-
KXTT. JOANNES DE MURIS.
DE MURTS defines a reaction as "a part or parts of an integer:
7
|| Fractio est integri in pars uél partes. Pars ugl partes here *
refers both to unit fractions and to non-unit fractions. ||
He calls its terms the numerator and denominator, both of which he
(56)
definese
In fractions, there are nine species: representation, num-
eration, addition, subtraction, duplation, mediation, multiplica-
tion, division and extraction of roots, * The treatment of duplation
| | Representacio, numeracio, addiclo a subtractio, duplagio.”
mediacio, multiplacio, divisio, radicum extraction |
and mediation does not appear to be given in the prose section of the
work. The given operations deal with common fractions only, except
the last which treats extraction of roots with sexagesimal approxi-
mations. In another place rules are given for transforming common
fractions into sexagesimals, Fractions are classified into "simple
or sterile fractions, fractiones s_implices aut steriles, as l; and
'composite or pregnant fractions", fractiones composite aut *.
as 2, 3- These classes respectively correspond to unit and non-unit
rºle. A second classification is into 'primary fractions, *
fractiones principales, as lº 2, 3; and 'secondary fractions', fracº
tiones secundarie, which º: rººtiºns of fractions, as l of 2, written
2 2 2 3
l. 3, or 1 -e-, or 1 g • Incidental ly, the author remarks that others
º
*... *..., * . Primary and secondary fractions are also called
reactions in sº and in obliquo. Before operating with them, se-
condary fractions are always reduced to primary fractions.
In general, de Muris dotains a common denominator by taking
the product of the denominators of all of the given fractions. When
the denominators are not prime to each other, he uses their least
common multiple some times, but not usually. No tables are given, but
reference is made to them,
The fact that the manuscript is partly in prose and partly in
verse partially accounts for his great variety of technical terms: addi-
C Co Nº Nº adiungere (to add) demere (to subtract; also,
- 2 - -


(57) *
- &
minuere), denominatio, denominatio communis, denominatio dissimális,
denominator (also, numerus denominans), dividere (to divide; also a
º - -
partere, scindere), divisio, duplacio, fractio (also, fractum, minucia,
ruptus), fractiones composite aut pregnantes," fractiones simplices.
|| This and the five immediately following are defined on p. 56 ||
aut steriles, fractiones principales, fractiones secundaries fraction-
es in obliquo, fractiones in recto, fractiones philosophice (sexage-
- -
simal fractions; also, numeri circulorum), mediacio, multiplicacio,
ſaultipliacio) de cruce (multiplication in the cross), multiplicare,
(aiso, augēre, ducere, extendere, ferre), numeracio, numerator, (also,
numerºus numerans), radicum extraction representatio, subtraction
summa multiplicationis, virgao”
t -
| Leonard, p. 44-3 has virgula; Peurbach, a century later, also
uses virgula. Leonard uses ruptus and minutum; de Muris,
ruptus and fractum. | |
XXIII. JOHN KILLINGWORTH,
In this work only sexagesimal fractions appear. They are
treated in the second part, 2* para de abbreviantibus opus calculand.
in minucijs seu fraccionibus physicis, in three chapters: chapter 1,
including introductory remarks, addition, subtraction, duplation, and
mediation; chapter II, multiplication, division and extraction of
square root : and #hapter III, discussion of checking operations. The
check is by 59, the use of which the author explains is the same for
sexagesimals as 9 is for integers,
Killingworth gives signs, degrees, minutes, seconds, and so on,
as divisions and subdivisions of the zodiac. He makes a sign equal
to 30 degrees, and performs the fundamental operations after the cus-
tomary manner of his time. The slate (lapis calculatorius) is just
coming into use , and Killingworth evidently uses two of them at a time a

(58)
But a more interesting peculiarity of his work is his return to column
reckoning, now obsolete. This peculiarity is particularly prominent
in his operations with sexagesimals.” This is practically the last
| Primo tamen sciendum est quod in scribendo minucias seu
fracciones physicas quodlibet genus per se separate in sellula.
sibi appropriata scribendum est- Cellulas autém distinct as
sic habebis prºne in magno lapide tº saleſ toºls lineas
equidistantes secundum distancias signorum- ||
appearance of column reckoning.
Of the five tables, the first gives the products of all inte-
gers from 1 to 99 by the respective numbers 1 to 9; the second, the
Tabula uniformis addicionis, and the third the Tabula difformis
addicionis, give sexagesimally the sum of any two numbers from 1 to 59
inclusive; the fourth; the Tabula multiplicationis fraccionum et
autºnia, is unfinished but is intended to give all sexagesimal
products from Ix I to 59 x 59 inclusive; and the fifth, the Tabula Re-
duccionis Integrorum ad minucias physicas, gives the number of degrees,
minutes, seconds, thirds, fourths and fifths in the integers from 1
to 59 inclusive a With the exception of the secºnd table , the arrange-
ment in each of them is simple and ordinary • The Writing surface is
ruled with vertical and horizontal lines, making little cells,
cellula, one for each number. To find the sexagesimal sum of any
two given numbers, the reader merely notes in table three one of the
numbers in the first row and the other number in the first, column
(counting from top to bottom, and from left to right) and then notes
the sum sought at the intersection of the column and row in which the
two given numbers lie - Tables four and five are similarly arranged,
But the second table, sexagesimally written in column form, presents

(59)
rithmetical series with common differences of 1, 2, 3. . . . .30 With-
hout repetition; the sexagesimal complements placed at the bottom
of the columns enables one to use the same table for numbers with
differences 30 to 60; thus the fifth column is 5, 10, 15. . . . 50, 55,
of 6,11, 16,..... 51, 56, 117, 12, 17,.....52, 57,218, 13,....58,319,
14, . . . . . . 59,4. The heading at the top is 5; at the bottom 55s, Hence
read down the column for numbers with common differences of 5, and
up the column for numbers with common differences of 55.
Technical terms : addicio, aggregatum, cellula, ºntº,
º
denominatio, denominator, divisio, duplacio, fraccio (minucia seu
fraccio), fraccio phisica, gradus, mediacio, multiplicandus, multi-
plicare, multiplicatio, signum, subtractio (remanens)-
xxTV. ANONYMOUS – A 15th CENTURY ALGORISM. *
|| Bibliothese Mathematica, 133, pp. 19-20. II
As has already been noted, hapters VIT-XI of this Work
treat; of common fractions, treating in order multiplication, addi-
tion, subtraction, mediation and division. The Bamberg algorism has
this same order’s But the Algorisms. Ratis ponensis has this order:
(\ddition, subtraction, duplation, mediation, multiplication, divi-
sion and extraction of roots s”
|| gompare With the order given by Jºnny. as wº, p > 54 ||
The numerator of a fraction is termed the "figure above",
and the denominator, the 'figure below. " A line is written between
thems
xv. Johannis won GMUNDEN. *
| fif, 44–53. ||
Johannis von Gmunden treats sexagesimal fractions only. His
work differs little from similar works of his immediate predecessors,
or of authors today. But it is of more than ordinary interest
(60)
because he has attempted to introduce an "easier and more useful"
division of parts. Instead of dividing a circle of 360 degrees into
12 signs of 30 degrees each, he divides a circle into 6 signs of so
degrees each, thus making the division of signs uniform with the
division of degrees, minutes, seconds and so one. The usual divisions
he entitles 'common' fractions, fractiones communes, and refers to
its sign as the common: sign; his own divisions he entitles, "phy-
gical" fractions, minucie phisice or fractiones phisice, and its signs
the 'physical' sign. He observes that a "physical" sign may be re-
duced to a common sign, or conversely, since one hysical" sign is
3. cº
equal to two 'common' signs. -
Tikewise, instead of dividing a natural day into 24 hours,
each hour into 60 minutes (called “hour - minutes' #, each minute into
6O seconds, and so on, he divides a day into 60 hours, each houjº in-
to 60 minutes (called 'day-minutes" ), each minute into 60 seconds, and
so on as before.
The treatment of fractions is divided into ten cases: (1)
representation of fractions, (2 ) reduction of integers to fractions
and conversely, and reduction of fractions of different denominations
to the same denomination, and conversely, (3) addition, (4) subtract-
ion, (5) mediation, (6) duplation, (7) multiplication, (8) division,
(9) extraction of square root, and (10) extraction of cube root, a In
each case, "physical' fractions are treated first, and then are noted
the variations necessary to secure corresponding results in * common.”
fractions, -
To express a "physical" fraction two principal numbers are
necessary: a numerator's numerºus numerans, and a denominator, ſluſſle Lºuis
denominans. *
º-º-º-
- - O
|| Numerator est ille: in quo to cies est. unitas guſt parties

(61)
integri vel alicuius fractiones volumus representage. Sed denominans
--- - T.
seu denominator: est ille a quo aliqua fractio denominature f * 45°
compare With the corresponding definitions given by Gernardus,
Liverſus and Peurºach. pp. 49,84,61-62. II
For example, in 32 seconds, 32 is the numerator and seconds, as a
two , is the denominator, 2 signs, 24 degrees, 36 minutes and 45
seconds, may be Written thus: 2.24. 36.45 - or thus:
signa TGradus T minu sed" a ºn
*g 24 36 45 (f. 46. )
Similarly, one may Write 18 days, 16 hours, 42 minutes, 35 seconds a
The method of reducing 'physical' fractions, or * common"
fractions, or days, hours and their fractions to the same denominator,
or vice versa, is the method used today, remembering, of course, that
a 'physical' sign is equal to 60 degrees, and a common sign 30 degrees,
Technical terms: aggregare (to add, also, addere), additio
Qaddition), cifra (also, circulus), denominatio, denominator, divisio,
duplare, dies, fractic, fractio communis (common fraction), fractic
phyisica (physical fraction; also, minucie phisice) gradus, genus
(order), hora, integra (used synonymously with hora, and gradus),
mediare, minuta, multiplicatio, numerus denominans (denominator),
numerus dividendus (dividend), numerºus numerans (numerator), signum,
signum commune (common sign), signum phisicum (physical sign), sub-
trahere (to subtract), subtrétic (subtraction).
xxvt. PEURBACH. i.
| re. 43-47.
A fraction is a part, or parts of an integer. Two characters
are necessary for denoting a fraction: a numerator, numerator, and a
denominator, “ denominator. -
| T ºr
| | Numerator est in quo suf tot unitates, quot partes integri
signific are volumus. Denominator vero qui totiens habet.


(62)
unitateſ, quotiens pars denominata est in suo to toe Peurbach,
f. 43. Compare Gernardus, Liverius, and Von Gmunden pp - 49, 54,
co-ei -
This work treats only common fractions. A common fraction is
one which is enumerated from numbers uniformly in the following order:
as, a half from two , a third from three, and so on. They are divided
into two classes: primary fractions and secondary fractions. * A primary
|| Compare With Joannes de Muris, p. 56 ||. -
fraction is one which is expressed by two characters, as lº 23 a.
secondary fraction cannot be expressed by less than four tº sº,
as l l (~ 3 of l). A secondary fraction is always reduced to a
º rººf before operating With it. This reduction is effect-
ed by taking as a numerator and denominator respectively the products
of the numerators and denominators of the secondary Praction.
The methods employed for performing the following operations
are the same methods that one employs today: To reduce several frac-
tions, or two fractions to a common denominator; to reduce a whole
number to a fraction with a given denominator; and to reduce a frac-
tion whose numerator is greater than its denominator to a whole num-
ber or to a Whole number and a fraction. The common multiple of the
denominators of the given fractions is usually used as the CO miſſion.
denominator’s
º
Examples: 2, 3, 4 reduced to a common denominator are 40,
4.25 60
45, 48; 4 integers are equal to 20; 37 is equal to 5 integer's and
O 60 5 7 : -
|
of an integers
The treatment of fractions is subdivided into the following
topics: addition, subtraction, mediation, duplation, multiplication,
division, square root, and cube root. Each topic is appropriately
headed. The general style of the author is shootº and lucids
_
(63)
Technical terms: addere, aggregatum (sum) coniungere (to
add), denominator, denominator communis, dimidiare, duplicº duplum,
fractic (minucia) fractio primaria (primary fraction), fractio se—
/ - - -
cundaria (secondary fraction), fractio vulgaris (common fraction),
+===ee, multiplicare, numerator, productum, subtrahere (to subtract),
virgula (line . ) -
XXVII. SIRT EL-MARTDTNT. *
|| Bibliotheca mathematica, 1899, new series XIII, pp. 33–36.
In his treatise on the calculation of degrees, Sibt El-Mariain?
gives two interesting examples of sexagesimal division where the quo-
tients are periodic , each period occupying eight places: 47° 51' divi-
wv/ v vſ. viſ
ded by 1° 25' gives the quotient 33° 45' 52" 56" 2é 14 7" 3" 3/
ºf f/
45' 52*... ... the first period extends from minutes to eighths in-
clusive. 1° 18' divided by 19 25' gives the same period.
Sibt, El-Maridini tests his results in sexagesimals by casting
out 7 's C. Jº 8's . To test 15° g 24” On 1 40"
by 7, he begins with 15°.
The check on 15° is 1. 9" is equal * gº of a degree, and the check
on 60 is 4. Hence multiply 1 by 4 and add 9, giving 13, the check on
which is 6. Similarly for the next place multiply the remainder 6
by 4 and add 24 to the product, obtaining 48, the check on which is
6. Continue in this way with succeeding fractions until the last one
is reached. The result is the check sought. In the same manner the
author obtains the check With 8, -
XXVIII. AL-KALSADT. *
|| Atti dell'Accademia Pontificia de' Nuovi Lincei, 1859, XII,
pp. 264-275. ||
The second part of this work is subdivided into an intro-
duction and eight chapters a treating the following topics on fractions:
addition, subtraction, multiplication, division, denomination, restore-

(64)
tion, abasement and transformation.
Although each topic is set off into a separate chapter,
the topics themselves and their order of treatment are the same as
given by Al-Banna p - 51 . Al-Banna is Wholly rhetorical, whereas
Al-Kalsadi includes illustrative examples, many of Which have al-
ready been included in the present study of Al-Banna. Otherwise,
Al-Banna and Al-Kalsadi do not differ from each other in arly € SSGn-
tial point. Therefore nothing further need be said relative to
Al-Kalsadi • -


CHAPTER TI.
ADDITION OF FRACTIONS. !
In the Hindu arithmetics, addition seems to have no définite
position with reference to the other fundamental operations with
fractions. But of the Works examined, in the Arabic arithmetics and
in their translations, up to and including the work by Leonard of Pisa,
addition follows multiplication, or multiplication and division,
whether the fractions treated be common fractions or sexagesimal frac-
tions or both. Ben Gerson (c. 1321) places addition in the midst of
multiplication, while a work as late as the 15th century” places it
|| A anonymous 15th century Algorism (XXIV). || -
after multiplication. In the Works examined, Jordanus is the first to
use decisively the modern order: addition, subtraction, duplation,
mediation, multiplication, division and extraction of square root of
fractions. With the exceptións mentioned, this is the order followed
by subsequent Writerse
- Usually only two fractions are added at a time . To add two
common fractions, if their denominators are alike , add their numera-
tors for a new numerator, but retain the same denominator. If the
denominators are unlike , reduce the fractions to a common denomina-
tor, and then proceed as before • When several common fractions are
to be added, two of them are added, then to their sum a third is added,
and so on, until all are added. However, Brahmagupta (c. 628), the
very first author treated, teaches addition of several common frac-
tions, both by this method, and by reducing them to fractions with a
common denominator, and then dividing the total sum of their numera-
tors by their common denominator's Leonard, also, uses both methods.
Planudes (c. 1337) is the first who undoubtedly adds several sexa -
gesimal fractions at a time, for he includes a figure * showing the
| | Fig. 12, p. 83 -

(66)
addition of four fractions at a time *
Mixed numbers are added "in two ways": in common fractions,
integers and fractions are added separately and their results com-
bined, or the given numbers are reduced to improper fractions and
then added as proper fractions; * in sexagesimal fractions, the inte-
| pp. 75, so, 82. . . .
gers and their fractions are added as given, corresponding orders
being added to corresponding orders; or they are reduced to frºac-
tions of the same denomination, added, and their sum reduced back
to fractions of higher denominations. Usually they are added as
given. . Some authors give both methods.
Common reactions are arranged for addition differently by
different authors, The sum usually appears below. In the earlier º
Works, sexagesimal fractions are arranged in order in columns with
integers at the top; but later they are arranged in rows, at first
from Eight to left, and then from left to right. Gernardus (13th
century) adds two fractions at a time, arranging the given integers
and fractions in order from right to left and putting the sum in
place of the top row deleted. Planudes (c. 1337) puts the sum above.
Addition is begun with the highest, with the lowest, or with an inter-
mediate order. Later Works begin with the lowest order. The process
of adding is modern. -
Leonard of Pisa. is the only author Who checks his results in
ºłºś. - -
addition, and even he does so only sometimes a Two instan ces are
cited in which he checks by ll and by 13 respectively.
D
II. BRAHMAGUPTA.”
| p. 277, Par. 2; p. 281, Par. 8, together with their foot-
notes. ||
"Quantities, as well numerators as denominators, being
(67)
multiplied by the opposite denominator, are reduced to a common de-
nominator. In addition, the numerators are to be united."
"Example. : What is the sum of one and a third, one and a
half, one and a sixth part, and the integer three, added together?
Statement : 1 1, l l l l 3• Or reduced 4 3 7 3.
3 2 6 3 2 6 1.
The numerator and denominator of the first term being multi-
plied by the denominator of the second, 2, and those of the second by
that of the first, 3, they are reduced to the same denominator
(8 23 and, uniting the numerators, 17). With the third no such
cºlon can be , since the aeºntº is the same: union of the
numerators is alone to be neae, 24. which a bridged is) 4. So with
the fourth term: And the addition being completed, the sum is 7."
- Another rule is : "The sum of numerators which have like de-
nominators, being divided by the [common.] denominator, is the re-
suit in the first reduction to homogeneousness."
"Example: Half of unity, a sixth part of the same, a twelfth
part of it, and a quarter, being added together, What is the amount :
Statement : l l l l Reduced to like denominators the
2, 3 Tº Zi
numerators become _6 2 l 3. Added together and divided by the de-
nominator, the res
The supplement closes, with these words: "[The end] of chap-
ter, t, Welve [comprising] sixty-six couplets on addition, etc. *
-
| p. 324. Par. 66. ||
III. AL-KHOWARTZMI. *
|| Trattati 1, pp. 21-22. ||
Before treating addition the author tells how to arrange
(constituere) integers and fractions: The method is clear from the
following example :
*But if we wish to array - XIT. degrees and exº~K. minutes,

(68)
.x1,.. and . VI. * seconds and . L. fourths, we place XII, then we put
|| 1 should be v ||
under them • KXX. in the place of minutes ; and, under • KXX. XLV. in
the place of seconds. In the place of thirds, however, We put ciphers
hºrs "circulii" since it is without thirds, ......next we put under
the ciphers fifty in place of fourths; and this is their figure.”
|| Both X and XI pp. 76, 77 have this same array; also, each
adds a figure. This work gives no figure • ||
The rule for addition is too briers "And similarly [we] put
the different orders of fractions under themselves in turn; and as
often as • LX. , or more, are added in any place (mansie * We put in
their order. Whatever remains above • LX. 3 and make of every . x. Cºló e
We put that in the order above. ”
No examples are givene
IV. MAHAVIRACARYA.
| pp. 50-69. ||
After treating multiplication and division of fractions,
squaring and square-root, cubing and cube-root of fractions, and
fractional series, the author says, "Hereafter we shall expound the
six varieties of fractions: Bhaga (or simple fractions). Prabhaga
(or fractions of fractions), then Bhagabhaga (or complex fractions ),
then Bhaganubandha (or fractions in association), Bhagapavaha (or
fractions consisting of two or more of the above mentioned fractions.")
(1) simple fractions: (addition and subtraction.) Rule: "If, in
the operations relating to simple fractions, the numerator and the
denominator (of each of two given simple fractions) are multiplied in
alternation by the quotients obtained by dividing the denominators by
means of a common factor thereof, ( the quotient derived from the de-
nominator of either of the fractions being used in the multiplication


(69)
of the numerator and the denominator of the other fraction), those
(fractions) become so reduced as to have equal denominators. (Then)
removing one of these (equal) denominators, the numerators are to
be added ( to one another) or to be subtracted (from one another, so
that the result, may be the numerator in relation to the other equal
denominator);"
This means that we have a pair of fractions such that
a , b = az, bx = az, E. b. *
#### * : + £ = *::
- # Foot-note , p < 50. # -
"Another rule for arriving at the common denominator in
another manner: -
The Niruddha (or the least common multiple ) is obtained by
means of the continued multiplication of (all) the (possible) common
factors of the denominators and (all) their lultimate) quotients. In
the case of (all) such multiples of the denominators and the numera- -
tors (of the given fractions), as are obtained by multiplying those
(denominators and numerators) by means of the quotients derived flºom
the division of the Niruddha by the (respective) denominators, the de-
nominators become equal (in value): " -
º
Example: Add together 43 , , -z, i. and l:
2O 36 63 21.
Several other examples are stated, but none are solved a
Following these rules and examples, there are tedious and obscure
rules covering each of a variety of examples, differing but little,
but yet unrelated to each other, and imprºactical in their natures
The following are illustrations, *none of which are solved:
|| See respectively the following sections pp. 54-60; sections
74, 76, 79, 81, 84, 92. ||
"The sum of (certain) numbers which are divided (respective-
ly) by 9, 10, and 11 is 877 as divided by 990. Give out what the
numerators are (in this operation of adding fractions. }"

(70)
"The sum of five or six or seven (different fractional)
quantities, having 1 for teach of) their numerators, is 1 (in each
case). O you, Who know Arithmetic, say What the * required) denom-
inators are."
"The sums ( of certain intended fractions) having for their
numerators 7, 9, 3, and l3 (respectively) are (firstly) (secondly)
1 and ( thirdly) 1 . Say What the denominators (of those frac-
4. 6
tional quantities l are."
"of three (different) fractional quantities having 1 for
each of their numerators, the sum is l; and of 4 (such other quan-
4.
tities, the sum is) 3 Say what the denominators are."
7
"Say What the denominators are of three (fractional quan-
tities) which have 3, 7, and 9(respectively) for their numerators,
when the sum (of those quantities) is 73."
48
"The sum of 1, l, l and l is 1 - If
- † 4, 5 20 -
What two (fractions) having 7 and 11 for their numerators should be
_l is left out here,
20
added (instead so as to give the same total)?"
"Here end simple fractions."
(2), (3) Compound and Complex Fractions.
| | pp. 61-63.
"The complex fraction here dealt with has an integer for a
numerator and a fraction for the denominator."
The author does not definitely state what variety of frac-
tions these are, but they appear to belong to the second and third
varieties; i.e., the Prabhaga (or fraction of fractions) and the
Bhagabhaga ( or compáex fractions) varieities.
Rule: "In (simplifying) compound fractions, the multiplication
of the numerators (among themselves) as well as of the denominators


(71) -
(among themselves) shall be \ the operation). In the operation ( of
simplification) relating to complex fractions, the denominator of
(the fraction forming) the denominator (becomes) the multiplier of
the number forming the numerator (of the given fraction)."
Example: "A certain person gave (to a vendor) l of 1, 1 of
º 2
- º 2
l of 1, 1 of 7, 2 of 2, and 2 of 1, (of a pana) out of the 2 panas
3 2 4 9 3 7 7 9 -
(in his possession), and brought fresh ghee for (lighting) the lamps
in a Jina temple. J friend, give out what the remaining balance is."
# 1 l l -
"Given)-4-3 4 and 3 ; say What the sum is When these
8 9 4 -
are added."
Other examples are given, but none are solveda
Further rules are given to cover cases represented by each
of the following example:
"The sum of 1, 1 of 1, 1 of 1, 1 of 3
8 4 5 2 5 4.
quantity is 1. What is this unknown (quantity?" *
l of L, of a certain
3 -
| | Ibid, see 108, p. 62. ||
"(The following partially known compound fraction, viz.,)
# of a certain quantity, # * } of another (quantity), * , of
2 of (yet) another (quantity give rise to # as (their ) sum. What
*... the unknown (elements here in respect of these compound frac-
tions?”
"Thus end Compound and Complex Fractions."
(4) Associated fractions: *
| pp. 63-66. ||
The rule involves two kinds of associated fractions: a frac-
tion associated With an integer; i.e., a mixed number, and a frac-
tion associated with a fraction; i.e., association in "additive con-
secution", as "ä associated with 1" means l l of 1.
2 3 2 3 2
Rule: "In the operation concerning (the simplification of )
(72)
the Bhaganubandha” class (of fractions), add the numerator to the
| Associated tractions. ||
(product of the associated) whole number multiplied by the denomina-
tor. (When, however, the associated quantity is not integral, but is
fractional), multiply (respectively) the numerator and denominators
of the first (fraction, to which the other fraction is attached) by
the denominator combined with the numerator, and by the denominator
(itself, of this other fraction).
Examples: "Lotuses were purchased fore 13 **pher for 10 l;
and saffrom for 2 1 (niskas). What is (their total) value when added?"
2
"O friend, subtract from 20 t the following) :- 8 l; 6 l; 2. l.
8 6 12
and 3 E."
3
"O friend, you, who have a thorough knowledge of Bhaganu-
bandha, give out (the result ) after adding l associated with l of it-
- 4.
*
associated with l of * , associated with l of itself,
4. 2
iated with l of itself, * , associated with , of itself."
3 -
There are other illustrative examples • None of them are
self,
#
3.5 SO
#
Ç
solved a
- Then immediately following are rules and illustrative exam-
ples for finding out one unknown (element. ), at the beginning, or any
unknown fraction in other *Jºguired places (in each of a number of
associated fractions, their sum being given):
"Thus ends the Bhaganubandha class [of fractions)."
(6) Bhagapavaha Fractions. *
| pp. 66–68. ||
Bhagapavaha literally means fractional dissociation, and is merely
the converse of Bhaganu banda.
(6) Bhagamatr Fractions.
| p. 69.


(73)
This is the class of fractions containing all of the foregoing var-
ieties and "the respective rules pertaining to the different varie-
ties hold good. It is of twenty-six varieities, formed from taking
the preceding five varieties in combinations of two, three, four or
five at a time e
"Thus ends the Bhagamatr variety of fractions."
"Thus ends the second subject of treatment known as frac-
tions in Sarasongraha which is a work on arithmetic by Mahaviracarya."
v. SRIDHARACARYA. * -
| Bibliotheca Mathematica, XIII., pp. 209-210.
3 -
"Reduce to a common denominator and add the numerators. Of
a whole number the denominator is unity." No examples are givene
A little later, the author says, "To reduce (two) fractions
to a common denominator multiply the numerator and denominator re-
ciprocally."
VTI. AL-KARKHT. &
|| Chapter xxxi. || -
* Tif you wish to add different fractions to the same and
determine their sum, then you seek the common denominat, or of the
same , multiply them with it, set the sum of the products of the
common denominator into the batio, and shorten the expression?
Example: If you wish to *### and 3 + 1 ||, take the
8 8 2
general denominator, which is 224, and * } ++ # to this de-
nomination, obtaining 103; and *#4 # obtaining 98. Then
you take the sum of 103 and 93, and divide the sum by 224. What
you obtain is the solution. - -
WTTT. AT- i." R. *
|| Bibliotheca Mathematica Trz, pp. 28-34. ||
- C.
Addition of fractions is described in #hapter TV of this work;
(74)
but, in chapter III, prefatory to addition and subtraction of reac-
tions, 'transformation' of fractions is describeds
Example: If 5 are to be changed into sevenths Write it thus:
6 -
. Then multiply 5 with 7 and divide the product 35 by both denomin-
6 5
7
tors, first by 6 and then by 7 obtaining 35 (= 3 + .3) written
* 42 7 " 42
º-5
#
Chapter TV contains 22 subdivisions most of which are devoted
to algebra. * The translator gives only two examples on addition:
| An algebra resembling that of the Egyptians. | |
Part 3. ''Addition of a fraction with a fraction to two
single fractions'. - -
- 5 i -
Example 6 2. This means 94.
6 :
9
a) = 2 49 (= 2 + 3 + 1 or 6),
+; 72 + # * 8 9
#
#
#
#
#
written 2 6 i.
9 8 - - - -
Parºt, 3. Addition of a fraction. With a Whole number to a sim-
ilar expression."
- 7
Example: , that is (7+ 2)(94- 7). * The result, is
8
9 3
:
12 # (=*4 # + # of #” Written 12+ e
TX. BHASKARA. *
| celebrooke, section 29–30, 33-34, 36–37, . 13-17. ||
(a) Rule for "Simple Reduction of fractions": "The numerator
and denominator being multiplied see precally by the denominators of
the two quantities, they are thus reduced to the same denomination,
or both numerator and denominator may be multiplied by the intelligent
calculator into the reciprocal denominators abridged by a common
measure. "
"Example: Tell me the fractions reduced to a common denomina-
tor, which answer to three and a fifth , and one-third, proposed for


(75)
addition; and those which correspond to a sixty-third and a fourteen-
th offered for subtraction.
"statement: 3 l l .
1 5 3.
"Aswer. Reduced to a common denominator 45 3 5, Sum 5
15 15 5
*
3
5
5
"Statement of the 2nd example: _l l.
63. 14 -
"Answer. The denominators being abridged, or reduced to
least terms, by the common measure seven, the fractions become l l =
- 9 2
"Numerator and denominator, multiplied by the abridged de-
nominators, give respectively 2 and 9 .
- - L26 126
"Subtraction being made , the difference is 7_."
1.33
(b) Rule for "reduction of quantities increased or decreased
by a fraction: " "The integer being multiplied by the denominator,
the numerator is made positive or negative, provided parts of 3.11
unit be added or be subtractive. But, if indeed the quantity be
increased or diminished by a part of itself, then, in addition and
subtraction of fractions, multiply the denominator by the denomina-
tor, standing underneath, and the numerator by the same augmented or
lessened by its own numerator. “
"Example: Say, how much two and a quarter, and three less a
quarter, are , when reduced to uniformity, if thou be acquainted
With fractional increas: and secrease.
*Statement f # *
4, .4/.
"Answer. Reduced to homogeneousness, then become 9 and ll."
4. 4.
(c) "Rule for addition and subtraction of fractions : *
l
|| This rule repeats (a) in effect and states two of the eight
rules of arithmetic" applying to the "eight operations of
arithmetic" already mentioned. ||
The sum of ſin case of subtraction] the difference of fractions
having a common denominator of a quantity which has no divisor. “
(78)
"Example: Tell me, dear woman, quickly, how much a fifth,
a quarter, a third, a half, and a sixth, make, when added together.
Say instantly what is the residue of three, subtracting those frac-
tions?"
"Statement:
l l l l l-
5 4. 3 2 6
"Answer: added together the sum is 29.
- 20
"|Statement 3 l l l l l
l. 5 4 3 2 6
"Subtracting those fractions from three, the remainder is
31."
º
x. ANONYMOUs – LTEER ALGORTSMT DE PRATTCA ARISMETRICE. *
|| Trattati II, pp. 54-55, and last Par. of p. 71.
(a) Sexagesimal fractions •
Before adding, subtracting, doubling, or taking half of sexa-
gesimal fractions, they must be arrayed in proper gradus • 12.
order. See page 68, where the same numerical ill- |minuta • 30.
ustrations given in this Work have already been Secunda .45.
noted. Note here, however, the insertion of a Tert, ia. AO0.
figure in vártical form. (Fig. 2. ). Quarta - . 50.
Fig.2 . (Liber, Algorismi
p. 54) -
Then in adding, like are added to like , and so on as has been
seen in ITI. -
No examples are given.
(b. Common Fractions •
In the text, addition immediately follows division. Hence,
in addition, common fractions, reduced to common denominators for
division, merely add their final numerators instead of dividing them,
(See division p. 141). No examples are given.


(77)
XT. ANONYMOUS – A 12th CENTURY ALGORTSM. *
| Abh'i zur Gesch. der Math., VIII, p. 22.
The order of arrangement of sexagesimal fractions is as given
in TTT and X. pp. 68, 76 . Also, the same illustration 1224
- - - 3O
is used, except that 12 degrees is replaced by 1224 45
- - OO
degrees. This work add a figure (Fig. 3) 50
Fig. 3. (Abh'1
VII, p. 22)
"But When we wish to add we begin at the lower
2999
place ", in which iſ 60 accrue, one is required for it 59
- - - 59
|| TTI and x begin to add at the higher place. 59
- - . 59
in the higher place in this manner' (the manner shown in Fig. 4 s
- - (Tbid,
Fig. 4 ). No example is given. p. 22)
XIII. ABRAHAM BEN EZR.A. *
| | pp. 39–42.
Ben Ezra gives no rules for the addition of fractions and,
- 4. - -
strictly speaking, but two examples •
| other examples are given under that heading, but they are
really exercises in the solution of first degree equations.
They are Egyptian in character and in the manner of solu- -
tion a Foº example, 'We have added to a number its ninth
and its tenth and obtain 50. ” compare with it, this exam-
ple from the Ahmes manual 'Ahau, ** * its Whole, it makes
19" (Eisenlohr, pp. 43-44. ). Both examples are solved in
Sºme | -
they manner. |
Example (1): "We add 2 to 5; how great is the sum?'
- - 5 7
The common denominator is 35. Taking
and of it, and add-
2 £
" 5 7
ing, one obtains 39. For 35 one takes 1 Whole number.
4. -
namely 5 , or 4 -
Tº 35
4 remains,

(78)
Example (2): "We take l of a number, its What
5
l, its l-
7 9
part is that of the number: *
Two solutions are given: (a)* In modern notation, the first
| See Ben Ezra, p. 105. ||
is as follows: -
1 l l l = i. i. 1 + 14(l − 1) + (1 - 1)= 3 + 4 + ·2 = 3 tº .
5 + # 5, '''.i. 5 * * * * * * : * * * * * (#3
( 8 - 4)|- 2 = 3 + 1 + 1 - ( 8 - A] 2 = 4 + 2 - # - 5 =
# - #'J 4 gº 5 # 4 |} +2= 634, 45 * 3: # , ºft ST
4. 3 – 8 - 28 - - 4 3 - 2 # = 4a: . - 4 "F a
#4 g; 5 *** - * ~ *- : * + 3
63 º
(b) * The second solution is similar to that just given for
*
| see Ben Ezra, pp. 40-41. ||
example (1). In modern notation the solution is as follows:
3 3
1 + 1 + 1 = 63 45 39 = 143 = 4 a. 35 = 4 L 5, as before.
#4 # + 3 + i + i +* * * * * + -ā- #4 #
- 9
xv. T.EONARD OF PISA.
Addition of fractions is treated primarily in subdivisions
1, 2, 4, and 5, * chapter VII, of this work, but the first case is
|| See this thesis, pp. 40–41.
also found previous to this :
| Liber Abaci, pp. 53-55.
(a) Addition of fractions of "one line." -
Rule: * 'Multiply the number which is under the first line by
| | Ibid, p. 53. ||
the number which is under the second ; and What results, place under
some line: next multiply the number which is above the first line by
the number which is under the second; and the number which is alo ve
the second you multiply by the number which is under the first ; and
unite these two multiplications ; and What results you place above the
line, and you have the desired (result ).

(79)
Example: add 1. to 2. Applying the rule above, their sum is
2 5
9. *
T5 -
|| The text gives the solution in detail. ||
Later” this statement is found: 'If you wish to add l to lº
- 3 4.
| | Ibid., pp. 63-64. ||
we teach you to do this in two ways; ' Then the author proceeds to
add l and l in two Ways, the first of which follows the rule above ;
3 4. -
the second is clear from the solution: "Multiply 1 which is above
1–
the 3 by 4, there are 4; which place above la and 3 4.
- 3
1 which is abová 4, multiply by 3, there are 3 ; l l
- 4. 3
which place above 1, and add these together, there
4. -> 7 additio
are 7, which divide by 3 and by 4 which are under 12
Fig. 5. (L.A. p. 64- )
the lines, this is 12, 7 result. . . . . " (See figure 5. ).
12 - -
Fract, ions whose denominators have a common factor are reduced
to the lowest common denominator before adding; as, 5 and 3 are re-
duced to 10 * †. respectively, and then added. * ºr. 1. given here
| | Ibid, p. º || -
a table of values for the sum of any two proper fractions whose denom-
inators are equal to or less than 10-
When more than tWo reactions of the above type are to be
added, add two of them; then to their sum add a third, and so on un-
tºil all are added,
(b) Addition of two simple fractions to two simple ºne."
|| De Additione duorum ruptorum adiunctione - Ibid., p. 65 • |
No formal rule is given. But illustrative examples are
proposed and solved for each case a
Example (1): To add "1 l and l 1'; that is, to add (L_1)
T ºf 7 E - ###
and (1 + 1), one may reduce the fractions to a common denominator
5 7
and add as though one had four fractions of 'one line', as in (a)
(30)
1 and l, and then L and 1 by (a)
3 4. - 5 7 -
ast, results. The common denominat, or
above, or one may first combine
above, and then combine these l
is 3 times 4 times 5 times 7, or 420. Whence, in modern symbols »
one has -
l l 1. l = 140 105 34 60 = 389.3 or (la l) (1 + 1) =
#4 + + š + # #33-### 7:3 + 7.5 - #5 #4 #' + \# * *
(4 + 3) 7.2 + (7+ 5) 2-4 = 342 + 4. is b389 - 339 * -34 of -1.
420 420 42O 42O 420. 6 - 7 - T.C 10 7 LO
+ 3 of 1 of 1, Written 5 l 9, as shown
6 7 1. 6 7 10
in Figure 6. 144 245
l l l l
7 5 4 3
5 l 9 additio
6 7 10
Fig. 6. (T.A. p. 65.)
Example (2): Similarly one may add '2' 3 and 2 3'- Figure 7
- 7 5 9 3
Shows the result, , Observe that 5 - 7 - 8.9 has
been replaced in the last step, by 1505 2232
- 2 3 2 3
4, 7,9. 10 9 8 7 5
l. 3 7 4 1 additio
4. 7 9 10
Fig. 7 - |L.A. p. 66)
(c) Addition of whole numbers with fractions. -
|| De additione integrorum numerorum cum ruptis. Ibid, p. 71- |
Rule : The text gives the rule very much in detail, Stated brief-
ly it is: Reduce mixed numbers to improper fractions with common de-
nominators and treat as in (a) and (b). or add whole numbers and
fractions separately, and then combine the results obtained.
Example (1): ' If you wish to add 1521 as TT
3
l 12 and 3 126, Write the numbers as here #126 #12. º
3 4. - -
- Süſºlº jº
shown (Figure 8.) . . . . . . ' , and so on as direct- l 3. Functioni *
- T. 139
ed by the rule. The text solves this Fig. 8. (L.A. p. 7T)
example in detail by both methods • Figure
8 shows the steps of the first methods


(91)
Example (2): Similarly one may add
'l l 15 and l 3 322, ' that is (15 1. l)
7 : ** 7 E *** + #4 +
and (322–t 3. + A*- Figure 9 shows the steps
5 7 -
of the first method. *
| solved in detail, Ibid, p. 74. ||
(d) Addition of parts of whole numbers with
13552 6545
1 3322 * 115
7 5 4 3
5 l 3 +
3-#-F# 338
Fig. 9 . (The text
gives no Figure . )
fract, ions. *
| --- - - -
|De Additione partium numerorum integrorum cum ruptis,
Ibid, p. 75. ||
No formal rule is given.
Example (1): Add '2 29 3 and 2 128
- 5 4. 9
In modern symbols, the
5; "
7
29 2 and 5 of 12s 2 .
5- 7 9
solution of the text is as follows: 3 of 292 +
4. 5
5 of 2 - 3 of 1.47 l. 5 of 1154 =
# * 128 # = # * ++ # * ==
lAZ. 3. 7.9 + 1154.5. 5.4 = 27783 - 1154.00 =
- 4. 5.7. 9 4 - 5.7. 9
1431.33 = -lºlº - 1134 3 + 3 of 14 3 of
T. 5.7. 9 2.7. G.T.C 10 9 LO 7
l of 1 + 1 of l of l of , 1, written
9 10 2 7 9 1O
1 2 3 & 113, as shown in figure 10, The
2 7 9, 10
check is by ll.
that is , add 3 of
4.
10) (s). T
115400 27753
21285 229 3
9 7 5 4.
addictio
1 2 3 6
#–4–3–1; 113
ext, ractic
1 2 3 5 , ,
- 6
ā-ā-ā-Hå 69
Figure 10. (L.A. p. 75.)
The lower result in figure 10 is the difference of the given
expression; this is, llā400 – 27783 - 87617
4. 5, 7, 9 4, 5.7. 9
Example (2): Similarly one may add
"2 5 33 l 3 and 1 & 244 l 3 * : that is, add
#–3 & # # * † : 244 +
(l, 30 (33 + 3 + 2 of 1) and (3 + 1 of 1){
#t # - + 9 * 7 9 #t 4. 7
(244 – 5 1)." The check is by 13.
+ 3 + Ti
(Figure 11).
= 1 2 3 5 2-,
++, 69.
(O) (7)-
31521.75 334438
—l 5244 lºg 3 2331 &
TT 6 4 7 7 9 5 4
check is by 13. (7)
3 & 4 & 6 145
TSTOTTI
Fig. 11. (L.A. p. 76 . )
The text gives many other examples, some of which involve
very complicated combinations of the four cases just treated.
XV. JORDANUS NEMORARTUS.
(See Section 9 of the synopsis of this algorism, p. 4°. ).


XVI. GERNARDU.S. *
| sections g and 10. |
(a) Addition of common fractions.
To add two fractions the author proceeds as one does today;
for example, in modern symbols, #4. % : **** º
To add more than two fractions, add any two of them, and then
to their sum add a third, and so on, until all are added .
To add whole numbers, or whole numbers with fractions to whole
numbers or to Whole numbers with fractions, reduce the fractions to one
denomination; then reduce the whole numbers into fractions of the same
denomination. Next add as above a
(b) Addition of sexagesimal fractions.
Arrange given fractions in order in rows from right to left,
the one to be added being under the one to which it is added, so that
corresponding denominations fall under each other. Beginning at any
place, add the lower number to the one above, writing the result in
the place of the number above, that number having been 'deleted.' If
the result is 60 in any place not integer's Write a cipher in the de-
leted place, and then add one in the next place toward the right. If
there should be no number above a lower number, then Write the sum
above itself' s
XVII. IBN AL-BANNA. *
|| Atti dell'Accademia Pontificia de' Nuovi Lincei, Vol. XVII,
º
pp. 209-210.
Rule: "The operation in addition consists in this that you.
multiply the numerator of each fraction by the denominator of the
others and that you divide the sum by the denominators. In subtract-
ion you subtract the less from the more before dividing by the de-
nominators.”
-

(83)
No comments nor examples are given. *
3 5
T6
|| Al-Kalsadi, ibid, Vol. XII, p. 269, gives this: ; that
- 1. 3
57
is , \; + 3 of 1) + (3 + 1 of 1). The 'total numerator' is
6 4. 6 7 5 7 -
llê9 • Dividing 1189 by the successive denominators 4, 5,6
and 7, one obtains, 1 + 2 + 5 of 1 + 2 of l of 1 + 1 of
+ # + # *# # + # *
6 6
l of l of lº written l 2 5 2. l. ||
5 6 7 4 5 6 7
XVIII. LEWT BEN GERSON. *
| p. 92. ||
only one paragraph, together with an illustrative example,
appears on addition, and this only incidental to multiplication:
"Addition of fractions of different denominations: seek the
smallest common multiple for the tractions, that is the denominator,
and from it take these fractions in their totality, and, divide the
result by the denominator, which is the result, sought."
Example: In modern symbols, 2 L 4 L E L 3 l l = 8 96 a
# * : + š + i + # * Tā; + rj +
100 + 15 = 291 = 2 and 51.
120 l2O l20 120 a
XIX. PLANUDES.”
|| p. 32. ||
To add 'astronomical' fractions, arrange them as shown in
r Sº a lº º º Sekºund
figure 12, and add as one would add them today, Billier Grade Minut. ”
0 22 57 35
placing the sum at the top and the respective 3 18 44 52
4. 23 54 49
amounts carried forward at the bottom, and 2 25 32 25
- l 14 45 29
remembering that a sign is equal to 30 degreese 2– 22 12 2
Fig. I2. (FIanudes , p. 32. )
xx. LIVERTUs.”
| The folios containing addition and subtraction were missing
from the text used, but from an analysis of the parts not
missing it may be surmised that the treatment of these

(84) º
operations includes both common fractions and Sexagesimal. ---
fractions, and, in methods employed, differs little from
modern addition and subtraction of fractions. |
XXII. JOHANNES DE MURIS • -
Addition is defined as the union into one fraction of sever-
all fractions of the same denominations
Rule: If you wish to add two fractions then multiply the
numbers below; the denominator is made. After that multiply that
which is above by the lower, multiplying in a cross, add together
what results. [That is] collected put finally for a nuºrator and
collect your fractions with the fractions [and] integers, if there
are any a
Observe the use of the phrase "multiplying is a cross,"
(fer...ad crucis). De Muris is the only author examined who uses it,
but this phrase is quite common later a
Another rule is given in prose: If fractions are of the same
denominator, add their numerators, retaining the same denominatore
But if they are of different denominators, reduce them to a common
denominator, and proceed as befores
Addition of fractions in recto and in gºals is divided
into four cases according to Whether one adds one or many fractions
in recto to one or many fractions in obliquo.

Example T1) . In modern notation, the author adds 2, 3 and
3 4.
4 as follows: 2 #4 4 = 2 - 4 + 3-3 4- # * +7 4 + = 17. 5 + 4. 12 *
5 #+ i + š 3.4 5 12 5 12.5
--------
6O 60 60
3 and 4 are all primary fractions or fractions in rectoº
5
Example (2): To add fractions in recto or fractions in obliquo, add
G -H 48 = 1.33 = 2 integers and 13 of an integer.' Observe that 2,
#
the fractions in recto, multiply those in obliguà, and then add the
results thus obtained. For example, to add 1. § and 1 #, the author
3. º 4

(85)
obtains first l and
6
XXV. VON GMUNDEN. *
, which he adds as above •
TE
#
| F. 47. I
No examples are given, nor does the author make it clear as
to the position of the sum. But otherwise detailed instructions are
given for adding both 'physical' and 'common' fractions. The methods
are modern in arrangement and in addition, except in the valuation
of a º 60 degrees already explained on pages (, o – (, | .
- Like Wise days, hours and their fractions are arranged and
added as one does today.
A. second method of addition is given as follows: Reducing
the 'physical fractions of different denominations to the same de-
nomination, add them and then reduce the sum back to fractions of
higher denomination.
XXVI, PEURBACH.
| f. 45. ||
The rule given for adding common fractions is precisely the
Sºſſlé 2 S the rule given today for adding them.
Examples: Employing modern symbols, l
5
É
5
l l = 20 lb lº = 4.7-
5 6 60 60 60
Several fractions of different denominations may be added in
"another way" ; add any two fractions, to their sum add a third; to
the sum of these three, add a fourth; and so on. Thus one needs to
know only how to add two fractions. The rule is modern.
Example: 3 + 3 = 3.4 p. 3-3 = 8 tº * : * 'one integer and
3 4. 3 * 4 3 & 4 l2 12
-
–8
12
XXVIII. AL-KAISADT. *
|| Atti dell’Accademia Pontificia de Nuovi Lincsi, 1859, vol.
TI, pp. 265-369. TGSee Al-Banna, pp. . )
CHAPTER TTT .
SUBTRACTION OF FRACTIONS.
Except in the Works of Al-Karkhi and LeWi ben Geron, wherever
addition of fractions is given, subtraction of fractions is also given,
º
--
and given either in connection. With addition, or immediately after it c
The smaller fraction or mixed number, or sum of fractions or mixed
numbers is taken from the larger. In general, subtraction is regard-
ed as the inverse of addition; and hence the arrangement of fractions
for subtraction, the place at which subtraction begins, the disposi-
tion of the remainder, the "two Ways" of subtracting integers with
fractions are as in additione
- Of the authors examined, only Planudes and Leonard of Pisa
check their results after subtracting a Planudes checks by adding
the remainder to the subtrahend to obtain the minuend; When dealing
, Leonard checks by casting out 7's, 9's, ll's or
with large numbers
IT. BRAHMAGUPTA.
The same rules and examples given in addition are employed
in subtraction, except that the difference is taken instead of the
---
º -
---
TTI. AL-KHOWARIZMI. *
|| Trattati T., p. 22. | |
Arrange the given integers and fractions as in addition. We
begin subtracting from the top; " and subtract any place from that
Which is greater. But should there be in that place less than what
-
you. Wish, . . . . . . . . . . . , subtract from that ; or should there be in it
a cipher, subtract one from the place Which is above it ; and make that
one . TX. parts of the fraction which you are 'operating', and take
9 2



º (87)
from it, what you are 'operating'; and add what should be the remain-
der besides the unfinished place ; and if there should be a cipher
above that place subtract one from the place which is above that,
and reduce it to . IX. parts in the place Which is just below: then
in turn subtract, one from it also, and make it. [XL] parts as above
in the place Which you wish. After this subtract from it what y Oll
wish; and put What remains in that place , which ends What, is sub-
tracted from it, ".
| The tedious style and obscure structure of this passage
suggests that the translator is unfamiliar With his subject,
Moreover, the language is that of the 6A5acus. Notice parti-
cularly the Roman numerals and the phrases de fractione guam
operaria and quod operaris. ||
Tv. MAHAVTRACARYA.
Subtraction is the inverse of addition and has been treated
along with it. (see p. 68 ff.)
W. SHTTYHARACARYA. * -
|| Bibliotheca Mathematica, VIII.3, p. 209.
"Reduce the first and the other to a common denominator and
take the difference of their numerators."
No examples are givene
VIII. AL-HASSAR. *
|
Chapter V of this Work is devoted partly to a description of
|| Bibliotheca Mathematica, TT3, p. 34.
fractions, partly to a description of algebraic solutions of first
degree equations." Subtraction is treated briefly because one may
u ºn at ..… ººs ---
| "...", º ths tº of the Egyptians. ||
*
know that all cases, which have appeared in the chapter an addition
are repeated for subtraction.

(88)
1.
Example (1) : Given 3 . Which means 5 - l. This is equal to
6 6 4
3-4 - 1.6 = +7 (= 3 + 1-1) written 3 1. Observe that the minuend is
4 e6 12 6 * 2 2 6 2 -
Writ, ten below the subtrahend.
Example (2) : Given 4 la which means lo – 4 l. This is equal
5 2 - ll 5 2
to 10.10 - (2.4 l) = 1 (= 1 - 1) written l.
10. 11. 111 10 11 11 LO
IX. BHASKARA.
Subtraction has been treated in connection. With addition.
(See p. 74 ff. ) * - |
X. ANONYMOUS - LIBER ALGORISMI DE PRATICA ARISMETRICE. * -
|| Trattati II, p. 55, and first Par. p. 72.
(a) Sexagesimal fractions are subtracted as in III, pp. 86-87 -
No examples are given.
(b) Three lines are devoted to the subtraction of common fractions,
* for it is merely the inverse of addition' . No examples are givene
XI. ANONYMOUS - A 12th CENTURY AT GORISM.
The same as in III and X above. • No examples are given.
XIII. ABRAHAM BEN EZR.A. *
| One paragraph ibid. pp. 42-43. |
en Ezra says, "Subtraction is easy, if you have a denomina-
top for both fractions."
Example: ' If ye wish to subtract 4 from 5. The denominator
9 7
is 63, 5 of it = 45, and 4 of it 28. We 4 5 63
7 9 2 3 5 4
subtract. 28 from 45, and 17 remain". See 17 7. 9
- fºr:
Figure 13. (Ezra,
xty. LEONARD OF PISA.
Subtraction being the inverse of addition, all the cases
and rules given for addition apply to subtraction, except that ,
_

(89)
of course, after fractions have been reduced to a common denomina-
tor, the smaller numerator is always to be taken from the larger
numerator instead of added to it. It has been convenient to use
as illustrative examples those that have already been used in addi-
tion.
(a) Subtraction of fractions of 'one line."
Rule: See addition (a) p. 73 , With corresponding changes
for subtraction.
Example (1): Subtract 1 from l. Common denominator one
4. 3
has 3 from 4, or l = l of 1, Written l Q -
12 12 12 6 2 6 2
(b) Subtraction of the sum of two fractions of 'one line' from
the sum of two fractions of 'one line'.
Example : Subtract ''1 1 from l l ; that is, subtract l i
7 5 4. 3. 5 7
from 1 1. See this esample in addition
3. 4.
(b) pp.79–80 for detail steps in reducing to 144 245
1 1. 1. l.
a common denominator. The remainder is 7 5 4 3
10l or 101 = 2 2 of l 3 of 1. 5 2. 2 Extractio.
3 - 4 - 5 - 7 6.7. LO 10 7 T.O 6 7 6 7 10
of l, written 5 & 2. (Figure 14- ) -
1.O 6 7 10 Figure 14. (L.A. p. 66.
(c) Subtraction of mixed numbers:
Example (i); subtract 'l 12 from 3 126'. Two methods of
solving this type of examples wº given 1. addition. * The same two
| p.80 - ||
methods may be applied in subtraction. By the first method, the re-
mainder is 1373 twelfths, or 114 5, written -
Tº 1521. 148
5 . (Figure 15). The second method leads 3.126 1.
+ 114 tº -" ZF #12
• * - Residuum extractioni
to the same result. 5 ul. Cºll. S
- – 11.4
| 12
Figure 15. (L.A. p. 72's

(90)
Example (2): Similarly,
one may subtract 'l 1 is from 1 & 322'; and may 135552 6645
4 3 7 5 - l 3322 l llā
that is , subtract **** ++ from 322 – # F#- 7 5 4 3
By the first method the result is 129007 or l 4 l 307 Ex- |
3 - 4 - 5 - 7 6 7 LO *
129 OO'7 * 1. 4. l 307 - (Figure 16. )
6 - 7 - 10 6 7 10 -
Fig. 16 (The text
gives no Figure. )
(d) Subtraction of parts of whole numbers with Tractions.
5' ; that is, subtract
-
Example: Subtract '2 as 4 from 2 l28
- 9
5 4.
3 of 29 2 from 5 of 128 2. See Figure 10, page 8 l ; and a sketch
4. 5 7 9 - -
of the 4++ solution of this example •
The text gives many other examples, some of them involving
very complicated combinations of the four cases just given.
xv. Jordanus NEMORARTUs. -
See Section 10 of the synopsis of this algorism, page 4/8 -
XVT. GERNARDUS. *
| section ii. ||
(a) subtraction of common fractions.
If the given fractions are not of one denomination make them
go. Next, subtract the numerator of the one from the numerator of the
other, and the operation is done.
(b) Subtraction of sexagesimal fractions,
Arrange the given fractions as in addition, placing those
of the minuend above. Then prodeed as one does today, placing the
remainders above in the piace of the numbers of the mausºld deleted,
XVII. TBN Al-sama. *:
|| Atti aeli Academia Pontificia de 'Nuovi Lincoi, XVII, pp.
309-310. ||
The rule for subtraction of fraction was given together with
(91)
the rule for addition of fractions. Hence see addition page 32 °
No examples are given. * - 4 3
|| Al-Kaigadi (ibid, Vol. XII, p. 270) gives this:
). The total numerato
sive denominators 3, 5, 7 and 9,
of 1, Written 0 3 6 1 . |
9 3 5. 7 9
XIX. PLANUDºS. *
| pp. 33-34. ||
In the subtraction of sexagesimal fractions the arrangement
is as shown in figure 17. The top row is the remainder, the second
and third rows are respectively the minuend and O 27 3O 41.
- - 1.0 24 58 25
the subtrahend. The columns passing from left 9 27 27 44
-
to right are respectively signs, , degrees, min- Fig. 17, (Planudes,
p. 33.)
utes and seconds. The manner of subtracting is the same as that
used today. 'But observe that if you borrow one from minutes for
seconds, or from degrees for minutes, ſhe one is equal to 60, but
- º, ,
from Signs equal, 30.
Test. Add the remainder to the smaller series, and one should
receive back the larger series.
Tf signs, degrees, minutes and seconds are subtracted from
signs, and degrees, arrange as shown in figure 18, l 24, 5 lb
- - 10 24 O O
supplying the minutes and seconds of the minuend S 29 54 45
with zeros. Then borrow from degrees, calling it Fig. 18. (Planudes,
p. 34. )
first 60 minutes, and then 59 minutes and 60 seconds, Whence proceed
º;
-
as in the first example . .
XXT. LTVERIUS. *
| see Footnote page. C3 ||



(92)
XXT T. J. OHANNES DE MURIS.
The rule of subtraction corresponds to that for addition.
Example (1). Subtract 3 * from 5 * • These are frac-
tions in obliquo, or secondary **. º them to frac-
tions in recto, or primary fractions, the author obtains 3 and 5.
The common denominator is 40.42 or 1680. Multiplying * ºna
subtracting, the author has 74 for a numerator. The result, then is
74 , which is incorrectly reduced to 9 -
1680 ". l 210.
Example (2). To subtract '3 e from 1 and l', the author
4 5 - 11.
obtains as a common denominator 4.5-ll, or 220. Whence 1 and l
- - 1. - - ll.
give 240 and 3 * or 3 give 33. 33 from 240 leaves 207. Therefore
- 4. 5 20
207 is the result sought -
220
Using modern notation, a second solution is given as
follows: 1 - 1 - 3 of 1 = 1 - 1 - 3 = 1 -
-- - –3 + 4 + · · · · = 3.9%
ll. 4. 5 11. 2O 2O 11. 2O 11 220
as before a -
Example (3). An interesting example is one in Which the
author shows that 'if you subtract 2 from 10 o will remain. '
XXV. VON GMUNDEN. - 3 15
|| Fr. 49-50.
No examples are given. Th 'common' fractions” the given
| See 9. 60 for his meaning of a 'common' fraction and
of a 'physical' fraction. || -
integers, and fractions are arranged and subtracted as they are to-
day. Hoºſever? the remainder is placed above in place of the minuend.
"Physical' fractions and days, hours and their fractions are sub-
tracted in the same manner, except for the variation of the value
of a 'physical' sign, and of a day, already noted in the intro-
duction.
(93)
A secºnd method is given as follows: Reduce the given
fractions of the minuend and subtrahend to the same aenomination,
then subtract the one from the other, and then reduce the remain-
ing fraction to fractions of higher denominations.
XXVT. PEURBACH. 1
| Fr. 45-46.
There is no difference between Peurbach's method of sub-
tracting 'primary' fractions and the method used today -
Examples: Using modern symbols - 3 - 2 = 5 - 2 = 3; 7 - 2 =
- 7 7 7 7 3 3
3.7 - 2.8 – 21 - 16 : 2.
3 = 3 3. 8 24 24
To subtract many fractions from one fraction, or from many
fractions, combine the fractions of the minuend and of the subtra-
hend respectively into one fraction, and then subtract as usual.
* Secondary' fractions are converted into 'primary' frac-
tions, and then subtracted as above-
XXVITT. ATI-KAI.S.A.D.T. *
|| Atti dell'Accademia Pontificia de 'Nuovi Lincei, 1859, KTT,
pp. 269-270. ||
Soo Al-Banna, page 91 -


CHAPTER TV
DUPLATION AND MEDTATION OF FRACTIONS.
The operations of duplation and mediation of integers are
of Egyptian origin. Adopted by the Arabic mathematicians, they came
into Europe through the Work of Al-Khowarizmi. These operations alº 3.
not found in any works of the Hindus, nor of the Hebrews, nor in the
Liber Abaci, nor in the Arabic treatises not based on Al-Khowarizmi,
but only in the algorisms. They remained in use until the 16th
century, although they possessed no importance after the development
er the operations of multiplication and division by direct methods.
pupiation and mediation of fractions follow from duplation
and mediation of integers. Of the Works examined only nine include
them as operations of fractions, and , of these nine , it is signi-
ficant that not any give illustrative examples.
Duplation and mediation follow immediately after subtract-
ion. In the later algorigms mediation precedes duplation.
III. AL-KHOWARTZMI. *
|| Tratvati I, p. 22. II
Both in duplation and in mediation the given integer and
- fractions are supposed to be arrayed as in addition, although the
author does not specifically say so. The process of doubling and
halving is rather incomplete -
"When you double any number or fraction, begin at ths
higher place, then at that, which follows it. Whenever there should
be collected in any of the places that which is more than the
º
number of its parts, place the that that is over in that place,
and raise up one from the place which is just above . "
(95)
But in mediation, begin at the lower place, and halve
that; next the following; and if you find one there, do with it
after the manner T explained to you in the first of the book."
X. Anonymous - T,IBER AT GORISM DE PRATTCA ARTSM-TRICE. 1.
|| Prattati I, pp. 55–56.
See Al-Khowarizmi above - |
AT. ANONYMOUS – A 12th CENTURY ALGORTSM. *
|| Abh' 1 zerºeshºe der-Math. Wisconsenatºon, VIII.3, p. 23. |
In audiation we begin at the higher order, but in mediation
at the lower order, as above. ”
XV., JORDANU.S.
See sections ll and 12 of the synopsis of this Algorism,
p. 48 º'
XVT. GERNARDUS. *
| section 12.
(a) pulation and mediation of Common fractions,
You double a common fraction by doubling its numerator, or
by taking half (deduplacio) of its denominator if it be an even num-
ber. Forº four? thirds are the double of four sixthse
Like Wise you mediate a common fraction by doubling its
denominator. Two fourths are half of two halves. or by taking
half of the numerator if it be even. But if it be odd, subtract
one from it and take half of the remainder. The error would be
no thing.
(b) Duplation and mediation of sexagesimal fractions.
In duplation of sexagesimal ('philosophical' ) fractions,
in the place of integers, you write the double of the number of


(96)
integers. In other places, if the result is less than 60, write.
that ; if 60, Write a cipher and add one in the nearest place toward
the integers; if more than 60, Write the excess quantity and for 60
add one in the nearest place toward the right.
But in mediation, if any digit be odd, if it be the very
first, subtract one, halve the remainder and, for the half of the
unit, give 30 to the nearest place toward the left a But if the
digit be odd in another order of one place, subtract one, halve the
remainder and for the one give 5 to the digit figure of the preced-
ing. For example, to mediate 75 minutes, for 5 take half of 4 and
give 30 to the place of seconds. Next subtract one from 7, take
half of 6, and give 5 to the half of 4. Do likewise in others. The
reason is clear.
XXI. LIVERTUs.”
| Fr. 23.
(a) Duplation of fractions.
In common fractions, double the numerator, or add the num-
ſ:
erator to itself, the denominator not being changed.
In physical fractions that is done as to 1d. in addition.
(b) Mediation of fractions. -
In common fractions, the author merely says, 'Take half
of the numerator, the denominator not being changed.'
Trn sexagesimal fractions, if they are reduced to the same
denomination, do as in integers. But if they are not, reduce them
to the same denomination: now begin at the lowest fraction, and if
it be even do as in integers. If, however, it be odd put half of
the nearest, even number contained in it in that odd place and,

(97)
making so rºom the unit, ºut 30, which is hair or it, in the place
nearest toward the right. Do likewise With preceding fractions,
adding half of any odd number, that is 30, to the fraction follow-
ing. But in halving an odd number of signs or days, the one sign
is 30 degrees, half of which is 15 degrees, to be added to the
degrees following: and the one day is 24 hours, half of which is
12, to be added to the days followings
XXV. VON GMUNDEN. *
| | f : 52.
(a) Mediation of fractions.
Sexagesimal fractions and days 2 hours and their fractions
are mediated just as in Liverius above , remembering the respective
values of a 'physical' sign and of a 'common sign."
(b) Duplation of fractions •
Gmunden disposes of duplation in two lines: Duplation is
performed by adding two equal numbers •
XXVI. PEURBACH*
| f. 46. ||
(a) Mediation.
If the numerator of the fraction is even, take half of it
for a numerator, the denominator not being changed. If, however,
it is odd, form a denominator the double of the denominator, the
first numerator not being changed a
- (b) Duplation. -
Double the numerator of your fraction, the denominator being
kept ; or take half of the denominator, the numerator being kept,
Secondary fractions are first converted into primary frac-
tions, and then doubled or halved as above-
CHAPTER V.
MULTIPLtcºmron or ractions.
In fractions the operation or multiplication is recognized
as the simplest of the fundamental operations; for , or the authors
examined, except the Hindu authors, all of the early ones and,
later, Leonard of Pisa (1202), Ben Gerson (1321), and an anon-
ymous writer of the 15th century place multiplication first, a Jor-
danus (d. 1237) is the first definitely to present multiplication
after addition and subtraction. -
The processes of multiplication show few changes for the
period studied. Both Brahmagupta (c. 628) and centuries later,
Peurbach (1423–1461) find the product of two, or of several given
common fractions by multiplying together their numerators for a
new numerator and their denominators for a new denominator. Both
alike reduce mixed numbers to improper reactions and treat them as
proper fractions. The early Latin authors employ the same me-
thods, although they employ sexagesimals, also. l l by l l becomes
90 minutes by 90 minutes, while elsewhere in the sº ºz ºr a
become 12. 1 Ben Ezra reduces the fractions to a common asſºr,
| Ai-Khowarizmi, this study pp. 101,103 |
and divides the product of the numerators by the square of the
common denominator. He has 2 by 3 = 3 by 9 = 72 = 1. Yet he,
3. 4. l2 12 12 * 2
too, immediately observes that 'if you multiply 2 by 3, then like-
wise there results the half of the denominator, 12.' *
ºf ,
|pp.116-117. a ºf * * * *
The product of several common fractions by several CO mm On
fractions is obtained by reducing the fractions of the multipli-
cand and of the multiplier first to one fraction each , and then

(99)
multiplying these as usual. Al-Karkhi, in some instances, sim-
plies his work by introducing a convenient factor, as 60, into
the multiplicand before multiplying and later dividing the pro-
\
duct by 60. *
| p. 106.
|
In multiplying one sexagesimal fraction by another, the
roduct, is obtained as in ordinary integers. To obtain the order
of the product, Al-Karkhi (c. 1010) says, 'The result belongs in
that order, whose distance from the order of one of the two t'aic-
tor's is equal to the distance of the order of the other factor from
the order of the degree.'” And Liverius (c. 1300-1350) says,
| p. 107 ||
"Denominate the product by the number Which is the sum of the de-
nominators of the reactions multiplied, or give the product a de-
nominator such that its distance from one of the fractions is equal
tº ºthe distance of this fraction from integer. * Three centuries
* ******* -
| p. 131. || -
produced no essential change in the manner of obtaining the order
of the product of two sexagesimal fractions. The product of sever-
all sexagesimal fractions by several sexagesimal fractions is found
both by multiplication of the fractions separately, and, also, by
reduction of the given fractions of the multiplicand and mul—
tiplier to one order each and then multiplying as usual. Some
authors present both methods. *
|| Among these are Al-Karkhi, Ben Ezra, Gernardus and Liverius,
pp. 107,118-119,125-126/.
The relative positions of the given fractions and of
those of the solution vary with the authors. Many authors set
|

(100)
off their solutions in marginal figures, some of which are quite
unique. Only Leonard of Pisa and Al-Hassar check their results.
Short cut, 5 **** are more frequent than in the other .
operations of reactions.
II. BRAHMAGUPTA.-
| p. 278, Par. 3 and 4 and Footnotes; p. 281, Par. 8 and
footnotes. |
Integers are multiplied by the denominators and have the
numerators added. The product of the numerators, divided by the
product of the denominators, is multiplication of two or more
terms. *
"Example: Say quickly What is the area of an oblong, in
which the side is ten and a half, and the upright seventy sixths."
- "Statement: 10 1. 11 4." From the rule, the two quan-
tities become , after ºrigine," a and 35- "Then from the product
of the numerators 735, divided ºne º of the denominators
6, the quotient obtained is 122 l. It is the area of the oblong."
The method Of ºne oneous is that of reduction
to the form of an improper fraction, as given in the foregoing
rule.
The method of finding the result of the "second assimil-
at ion" consists in the multiplication separately of numerators by
numerators, and denominators by denominators, and then proceeding
as stated above ,
"Example : Half a quarter, a sixth part of a quarter, a
twelfth part of a quarter, and eighth part of ten quarters, a
fifth part of seven quarters: summing these and adding three
twentieths, let us quickly declare the amount. It is a sum, which


(101)
We must constantly pay to a learned astronmer.
Statement: l l l l l l 10 l 7 l. 3: or l l —l lo
4. 2 4 6 4 12 4 3 4 5 20 8 24 43 32
7 3.
Answer: the sum, is one."
III. AL-KHOWARTZMT. -
| Trattati I, pp. 18-20, 22-23.
A. Sexagesimal fractions.
(a) Multiplication of single sexagesimal fractions:
Every integral number by an integral number gives an inte-
gral number, and every integral number by any fraction gives a frac-
tion of the same order. 2 degrees into 2 minutes are 4 minutes; 3
degrees into 6 thirds are 18 thirds. Minutes by minutes are seconds;
6 minutes into 7 minutes are 42 Seconds. . . . . . and so on.
(b) Multiplication of mixed numbers by sexagesimals :
The method is given rhetorically and then illustrated by
examples. It is sufficiently clear from the exampless
Example (1) - "If you wish to multiply one and a half into
one and a half, * make jºx3xxxxx ſixmåka inskäää ºf its axiºm ºf gikºkº
|| The text reads in duo et dimidio instead of in unum et
-
dimidium as in the solution following. See pages 111 and
115 for this same examples • ||
one and a half minutes , and there are • XC - Again make one and a
half which you wish to multiply into those minutes, and there are
like Wise .x.C. : multiply the one into the other and there are . VIIIs
thousand and - C - seconds ; divide seconds by LX. , and there are . . . .
.C.. XXX. V. minutes . . . . . . . and divide by - LX. , and there are two
degrees and XVI* minutes, which are one fourth of one.”
| | Should be xv. ||
(102)
Example (2). Similarly, to multiply two integers, that is
two degrees, and XLV. minutes into three integers and X. minutes,
and. XXX. seconds," We have 165 minutes into llá30 seconds which
multiplied and reduced gives 8 degrees, and 43 minutes, and 52
seconds, and 30 thirds. *
|| See also on page 115. Compare with (2) page ill . . .
A general rule now follows: Similarly you do with all frac-
tions; obviously reduce each of those which you wish to multiply
into another to the lowest, order which is in each of them. After
this multiply one of them into the other; and keep that which re-
sults; and note in which of the places it should be next divide
by . LX. according to the methods I have told you."
The author states that there is another shorter method
which the Hindus used.* -
| | Et est ei alius modus brevior; set hio ordo est, quo usi
sunt indi, super quem figurare numerum suum. Trattati
I, pp. 19-20. ||
Be Common fractions •
The subject of multiplication is logically closed here; but
after treating in order division, arrangement, addition, subtraction
auplation and mediation, the text suddenly reverts to multiplication.
It reads, "And if you wish to multiply fractions and a number, and
fractions beyond minutes, or Seconds, as fourths and sevenths, and
other's similar to these, and divide them in turn, the work in these
will be as the Work of minutes and seconds; and I shall arrange for
you an example, if god wills."
The examples given are examples in the multiplication of

(103)
common fractions, solved without the aid of sexage simals, but after
their fashion.
Example (1) - "Should you wish to multiply . III. sevenths
by . IIII. ninths, the seventh and the ninth would be in the first
place of fractions as minutes; and you multiply them in turn, and
render in their place after the order of seconds. Whenever you.
wish to raise them to an integral number, divide them by each order,
which are sevenths by ninths a . . . . . . . And if it can not be divided,
they are parts of that class by which you divided. Three sevenths
by . IIII. ninths are .xII. parts of . IX. three parts of one."
Example (2). When, you Wish to multiply three and a half,
by .VIII. and three parts of . KI - , Write three, and place under
them one , and under one two. Thus you. Write three and a half; since
a half is one part of two , just as one minute is one part of LX.
parts of one. After this you write in another part .VIII. , and
under them three, and under three .x.I. and thus you arrange .VIII. "*
| | See both of these examples pages 114 and 115-116. -
Here the Work suddenly stops, evidently parts are missing.
TV. MAHAVTRACARYA. " -
| pp. 38-39. |
Rule: "In the multiplication of fractions the numerators
are to be multiplied by the numerators and the denominators by the
denominators, after carrying out the process of cross-reduction, if
that be possible in relation to them."
is reduced as l x lº
2 3
Tilustration from footnote: When 3 x
4.
the process of cross-reduction is appliede
O
+.
9
Example. "Tell me, friend, what a person will get for 3.
- 'E

(104)
of a pola of dried ginger'; if he gets 4 of a pona for 1 pola of
4.
9.
such ginger."
Four other examples follow. "one are solved, "Thus ends
multiplication of fractions."
v. SRIDHARACARYA. *
| Bibliotheca Mathematica, VIII.3, pp. 209-210. ||
"The product is obtained by dividing the product of the
numerators by the product of the denominators."
No examples are given.
"For the assimilation of sub-fractions multiply the denom-
inators and also the numerators."
A root-note gives this illustrative example: "How much
money is there when half a kakini, one-third of this and one fifth
of this are added together?
l l | H 1 l | 1 1 1 i
l. 2 || |l & 3 || || 1 2 3 5
Answer: Varatikas 14."
Statement,
l
2 3 3 : 3 TC)
20 varatikas - l kakini the answer is 14 varatikas.
This means 1 x l + 1 × 1 × 1 + 1 × 1 x l x l = Z, and since
o -
"For the assimilation of fractional increase add the numer-
ator to the product of the integer with the denominator."
A foot-note add: a |b = ac -i- b -
"Another rule : mutini, ºnominator by the denominator
standing underneath and the numerator by the same augmented by its
own numerator."
VI. AL-NASAWT.
See page 34 .
VII. AL-KARKHT. *
|| Chapters xxx-xxxrt; xxxvi.


(105)
A. Multiplication of common fractions.
(a) General case : If you wish to multiply a fraction by a
fraction, or a Whole number and a fraction by a Whole number and a
fraction, you turn first to the multiplier and seek the general de-
nominator of the fraction. When you have ſound it, then you multiply
it by the common multiplier and note the result as the denominators
Then you proceed with the multiplicand, as you have done with the
multiplier's The result, you multiply by the result noted and divide
that which results by the product of both common denominators. The
result, is the solution.
Example (1) : To multiply * | * + —l by 1 + + ++, the pro-
13
cedure is as follows : you seek a sº a. ** of multiplier and
multiplicand respectively. 44 is the common denominator of 4 and ll,
and 65 of 5 and L3, l. 3. multiplied by 44 gives 59, and 1 1
1 multiplied by 65 give s: The product of 59 and 33 is 4397. 5
Divide 4897 by 65 into 44, or 2860 and the result is the solution.
* * * * * # *** * * * | # •
(b) Multiplication of two fractions of different denomina-
tions: Again ir you wish to multiply a part of one denomination by
a part of another denomination, you multiply the number of the first,
fraction by that of the second. You divide the product by one of
the two denominators, then the result by the other denominator:
Example : 13 tenths multiplied by 5 thirteenths gives 65
to be divided by ten and then by thirteen.
(c) Multiplication of fractions by means of 60: If you
wish to multiply an expression which contains a fraction, or a Whole
number and a fraction, by another, then multiply the same by 60 and

(106)
r
then the product by the multiplicand. Whence you divide the product
by 60, or place it into ratio to 60. The result is the solution.
Example: To multiply la ll ill by 1 || 1 || 1, you multi-
e O y +++ + J ++++’ ”
LO 10 | 6
ply 60 into *** - #. and the product into 1-- 1: 1. The
LO LO | 6 4 5 -
product of 60 into *** + 1 Il is 67. If you multiply this number
LQ ---
LO | 6
by l l, l then it gives 97 – 1 # Dividing this by 60,
H- #-F. 5 + LO º 2 | | -
the final result is l l l l. 1 l 1 || 1 |l.
H # - Iā-F T###. 10|10|4
"This method likewise holds in general for all multiplica-
tion, but is practicable only in simple cases.
- (d) Multiplication of a mixed number by a mixed number: "If
you wish to multiply a large Whole number with a fraction into one
other large whole number (with a fraction), you multiply the whole
number of the multiplier into the Whole number of the multiplicand,
then the fraction of the multiplier into the Whole number of the
multiplicand, then the fraction of the multiplicand into the whole
number of the multiplier, finally you multiply the fraction of the
multiplier into the fraction of the multiplicand, and add all these,
which gives the solution.' * -
| Here follo Ws a rule and an example for adding common frac-
tions. See addition, p. 73 ||
(e) Another Way of multiplying mixed numbers: Tſ it be said
multiply l l by l 1 by 1 + k, then We have £ by 4 by 3, Whence
4. 3 2 4. 4. 3 4. -
multiplying 5 by 4. by 3 it gives 60, which we divide by the product
of tho denominators; that is , 4 by 3 by 4, or 48. The result is
* #: ‘ By this rule, all belonging here can be treated. But that
which I have said of the multiplication of Whole numbers in suffi-
ient. Praise be to Allah!”
(107)
B. Multiplication of Sexagesimal fractions.
(a) Multiplication of single sexagesimals: 'If you wish to
multiply a single expression Which contains degrees, minutes, Seconds,
and the like by a number of parts of a degree, then you multiply the
number of the part of the multiplier by the number of the part of the
multiplicand. The result belongs in that order, whose distance from
the order of one of the two factors is equal to the distance of the
order of the other factor from the order of the degree."
Example: ' (multiply) 5 minutes by 7 thirds. You multiply
5 by 7, it gives 35, which are fourthe, for the distance of fourths
from the thirds is equal to the distance of minutes from degrees."
The author adds that the order of the product of two single
numbers of a degree, or parts of a degree may also be gotten by add-
ing the orders of both factors. -
(b) Multiplication of other sexagesimal fractions: (l) The
multiplication of composite numbers consists in that you multiply
each single number of the multiplier into all numbers of the multi-
plicand. What results is the solution. -
(2) Or reduce the numbers of the multiplicand and multi-
plier to the respective lo West classes, multiply the results, deterº-
mine the order of the product as in (a) above, and then reduce the
product to higher orders by successive divisions by 60. What re-
sults is the Solution.
WITT. AL-HASSAR. *
| Bibliotheca Mathematica II, pp. 23-28.
Muliplication of fractions is described in the other 71 parts
C. -
of £hapter II, mentioned on page 3.5 - The forms appearing result
from combinations of Whole numbers, simple fractions, "fraction

(108)
fractions' , and fractions with 'fraction fractions.'
Part 2. 'Multiplication of a fraction by a whole number. "
B.ample: If you wish to multiply a by 10, write it thus
5
7 6
LU. Then multiply 5 by 10, and divide by 6, giving & 2. Check by
- 6
7.
Part 3. 'Multiplication of a fraction with a fraction frac-
tion by a whole number.'
Example: If you Wish to * * * a half of 1 by 12,
- - 5
Write them thus l l . Then multiply the l over the 5 by the 2 below
5 2 -
l2
the line and add to the product the 1 over the 2, giving 3. Mul-
tiplying this by 12, it. gives 36. Divide this by the product of the
denominators, i.e., 10, and the final result is 3 (=3| 3 || 0 of l),
3.
5 5 2 5
written 3 3 O. Check by 7.
- 5 : .
Paiº 4. 'Multiplication of a fraction fraction by a whole
number."
- —l.
Example : 7 3 . This is solved as in part 3.
º 25 -
Part 5. 'Multiplication of two different (simple) fractions
by a whole number. "
3.
Example : 4.
l6 -
is modern. This case is really a special case of part 3.
4
T;; that is ( 4)15. The method of solution
Part 9. 'Multiplication of two different simple fractions
by a whole number and a fraction."
3 4
Example - 4 5 - One has here (3.5-H 4.4) (5.6 + 5) =
5 # 6 - 5.4
#e 9 g (= 9 || 9 || | | | | }, written 99.1 !.
---ºu- 120 6 30 120 6 5 4
- - . . -
The original text gives here 'for the first time in this
º - - /
- - N - -
* . - \
chapter the general rule, which though repºated brº, ſº
Wanting in most cases. *


(109)
Part 13. "Multiplication of a fraction with a fraction
fraction by a whole number and a fraction fraction?
5 1.
Example: 6 2. Une has here (5.2 + 1)|(8.7 tº 5) 5 + ll-
5 l - 7.6 × 5.2 -
8 3 ( =9 || 0 || 0 7, 5 3 0 ), written s 3 * -
º' Tº H + · · · · ··· + £3 7 GT5 3
Part, 23. "
M
ultiplication of a (simple ) fraction by a (sim-
ple) fraction."
i
One has 63 ( = +4++ of 7), written 7 7.
T5 10 1. & - 1O 8
Multiplication of a fraction of a whole number
Example :
l
Part, 41.
º
O
and a fraction by a similar expression.'
... " Tt, is written
),
Example: ' Multiply 2 of 5 3 by 6
- 3 6 7
5 - The product is 28 56 (= 29 |
this : el 2 o
37 5 - 3: Tº ºf Tiji
2 5
º
6 3
7
Written 28 l 2, 2 9
--~~~~ *
re are given other examples, all containing different
combinations of the elementary forms mentioned in the beginning;
- 1 5 L -
; meaning (3 - 1 £5 1 ) (4 + 3 * 4)=
2 3 4 5
3
3
such as (Part 55)
- 4.
5
and (Part 59), a
º
51. 17, Written 51 3
6
3.
n
d
P
8.
rº
t
5
8
)
;
4.
2
2
l
2
l
7
4.
TÜ
*
still more complicated example, 5 l 3
6
4. 3 multiplied
9
Kact, notation for addi-
2
3.
4.
3
5
by a similar expression. The lack of an e
tion and multiplication makes doubtful the correct meaning of these
longer examples •
In part 60, another meaning is given to some of the shorter
forms 3 3 does not mean 3 and l of 3, but 3 of 3- Al-Hajjar calls
- 7 5 7. 7 5 5 -
this 'the fraction with the omission of the 'and' ', or 'the fraction
taken of a fraction." Al-Kalsadi (after Woepcke ) inserts a ver-
tical line between the respective fractions of the latter form; * as
- Sº, -
|| This is the 4th class of fractions given by Ai-Banna. See p.
- A
51. º ||

(110)
2 3 || 3 2
33, meaning thereby 3 of 3. Accordingly (part 67), 75 3 means
7 5 7 5 3 & 15 l
- - 4 6 2.
(2 2 + 2)(3 # - 4) = 12 L7, Written 12 O_3_2_0_0.
35 3 4. 2 23O 8 7 5 9 2 -
Part 70 contains this example, the answer to which is 5 + .
as one can readily see: l l - 1 1 - 1 l l l - i. 1 . l l l l .
2 3 4. 5 6 7 º
l l l l.
9 10
Parts 71 and 72 describe 'multiplication of fractions by
separation." -
----
| See (5), page 52 . || 3 ill? L.
- 4. 6 -
Example, part 71: 4 ill? L. ºrj or (3 - i) ) (4 – 1 ). The
result is 98, Written 2 3 3.
- 360 9 5
Thºlusion, AlêHa;;ar says (part 72), 'What we have now
said of multiplication of fractions shall be sufficient for him who
studies it carefully.
Tx. BHASKARA. 8
| celebrooke, sections 31–32, 38-39, pp. 14-15, 17.
(a) "Rule for reduction of Subdivided fractions: "The numera to ps
being multiplied by the numerators, and the denominators by the de-
nominators, the result is a reduction to homogenous form in sub-
division of tractions."
"Example: "The quarter of a sixteenth of the fifth of
º
three-quarters of two-thirds of a moiety of a dramma was given to
a beggar by a person, from whom he asked alms: tell me how many
cowry” shells the miser gave if thou be conversant, in arithmetic,
|| A cowry shell is the 1280th part of a dramma. ||
- - - - . -
With the reduction termed subj'division of fraction.”
(#) 1 2 3 4 5 16 4.
| observe the writing from right to left- | -
statement,”, l l 2 & 1 1 1.



(Ill.)
- - 5.
"Reduced to homogenousness or in least terms 1 -
- 7690 128
"Answer: A single cewry shełl was given.
(b) "Rule for, multiplication of fractions. *
|| This is another of the "eight rules of arithmetic."
See p. 36 . Tt repeats rule (a) above. }
"The product of the numerators, divided by the product of
denominators, [gives a quotient, which] is the result, of multi-
plication of fractions."
"Example: hat is the product of two and a seventh, multi-
plied by two and a third: And of a moiety multiplied by a third?
Tell, if thou be skilled in the method of multiplication of frac-
tions."
"Statement: 2 2 (or reduced Z - 15) and l . l.
l l 3 7 2 3
- 3 7 -
"Answer: the products are 5 and l."
- l 6
x. ANONYMOUS - LIBER ALGORISM LE PRATTCA ARISMETRICE. *
|| Trattati II, pp. 50-53, 56-66, 68-69. -
A. sexagesimal reactions.
(a) Multiplication of single sexagesimal fractions.
see Al-Khowarizmi, page 101 -
(b) Multiplication of mixed numbers by sexagesimals.
See Al-Khowarizmi, pages 101-102 .
Example (1): To multiply one and a half by one and a
half, . . . . . . . . See p. 101 example (l), where it is solved in exact-
ly the same manner. See p. 115 , also.
Example (2): Similarly to multiply "two degrees and . 10.
minutes by one degree, and two minutes and .30. seconds," one has
130 minutes into 3750 seconds, or " :Gºogſ thirds," which reduces
(112)
to two degrees, and 15 minutes and 25 seconds. *
| compare with this example (2), p. 102.
B. Multiplication of common fractions.
- The author refers to these as 'fractions of another denomine-
tion', and says that, if you wish to denominate fractions by any other?
number, as halves by two , thirds by three , and so on, another rule
must be observed ; that the numbers of the multiplicand and multiplier,
and of the dividend and divisor must be arranged in rows or columns
' opposite each other'. Moreover, careful note must be made of cer-
tain technical terms, numerºus fractionis, denominatio, numerºus de-
nominationis; numerus communis, and numerus collectionis. The num-
- - - ſin - - - +
ofºus fractionis is the numerator. #he denominatio is the denomina-
tor – 'What part of one integer it is , as a half, or a third, and so
on'. The numerºus denominationis applies to the common denominator of
the fractions of one column only, and is obtained by taking the pro-
dućt of the denominators of the fractions of a column; for example,
the numerºus denominationis of one half, one fourth and one fifth is
40. The numerus communis is the numerºus denominationis of both
columns taken together; for example, if the numerºus denominationis
of one column is 40 and of the other 20, then the numerºus communis
is 20 times 40, The numerºus collectionis is the sum of the numera-
tors of the integers and fractions of a column after they have been
reduced to a common denominators
Example (1) : " .. 8. integers, and one half, and one fourth, and
one fifth are to te multiplied into - 3 - integer's A and one ºira, and
Cºle ninth."
Arrange the multiplicand and multiplier as shown in figure
19. The common denominators numeri denominationium, of the columns
are 40 and 27 respectively; the common denominator of both columns,

(113)
numerºus communis, is the product of 40 and 8 3 +
• 1. 1.
27, or 1080. Reducing the first column to 2 ; -
1.
40ths and the second column to 27ths, one º 4. 9
- - 1.
obtains, 358 40ths, and 93 28ths, respective- 5 —#
40 27
ly. 358 and 27 then are the numeri |1080
º - - 353 33294. 93
collectilonium- Their product is 33294, a 3.0
- º - . 3.9.4
which divided by 1080, the numerºus communis, •l - 0. 6-0
Fig. 19. (Trattati II,
gives for a result '30 integers and 894 p. 63. )
thousand eightieths' written as shown in Fig. 19. t
- - tºº
Observe that, having obtained the numeri collection, 358
and 93, one may add, subtract or divide them as well as multiply
them.
Example (2): When integers only are to be multiplied by
fractions only, or vice versa, the solution is T.E.T.
similar to that of the example just given. 5. –
- - 2. |-
Figure 20 gives the solution for finding the l: 0. -
5.0 - l. 5. i
product of 2 by 15. In symbols, one has 5. 0.
5 - |30. 15.
2 + 2).” = (£3 F. 10)15 – 30 x 15 = 450 = 9. 4 50
5 TT6 Eð I 50 55 I5 - 9
Example (3) following immediately Fig.20. (Trattati
- II, p. 65. )
is of a slightly different type: " .. 3. eighths
- 4.
of one sevenths of one tenth are given to be 3
- 3
multiplied by . 4. " One obtains for a new 7
- T.O.
numerator 3 times 4, or 12, and for a denomina– 560
- 2240
tor & times 7 times 10, or 560, Now 12 should 672O
- Figure 2.I. (Tratta Ei
be divided by 560, 'but since that which you II, p. 66. )
divide is less, therefore denominate that by the larger thus. '
However, it is not clear just what conclusion is to be drawn from

(114)
the figure. (Fig. 21. )
Example (4): ' But in all these', says the author, 'when
one fraction only is to be multiplied by an 1. • 2 -
- --- 2. 3
integer, or by another such fraction, the 2 3
- - - • 6 -
numerºus fractionis Will be the numerºus collect- l 3
5 - 2 :
ionii. " For example, 'one half is given to be Fig. 21a. (Trat-
- tati II . 66.
multiplied by two thirds in this manner." (Fig. 21a. ) { * 9 )
A page and a half of the text is now devoted to division,
---- - - - - - - Cº.,
when multiplication is again taken up, and two more examples, pro-
posed and solved. Their solutions are similar to those already
given, but they are stated here because they occur in III and XI +:
|
the first “ , ” (one) proposes for multiplication . S. integers and three
| pp. 103, 115 -
|| Preliminary to these , two examples the author remarks, Hoc
i
dem est illud 9 tiam quod de multiplicatione, et divisione
integrorum, et fraction m alcorismus dicere vide tºur, et slº.
re-
a liter, this is the same then that Alcorismus is seen to
teach concerning multiplication, and division of integers,
-
and fractions, and so another. ||
º . . - a -
elevents by three integers, and one half by this mode"; the second,
'. 3. sevenths are multiplied by . 4. ninths. ' "
|| Trattaui II pp. 68-69; Trattati I, p. 23; Abn'l zur gesch.
der Math. VIII, p. 23. ||
- ſº
jºr. Anonymous - a 12th CENTURY AIgorism.”
| Abhis zur Gesch. der Math. , vol. VIII, pp. 21-22, 22-24. ||
(a) Multiplication of sexagesimal fractions.
"And so every fraction multiplied by itself or another
diminishes in the Whole of itself or of another whatever part of

(115)
an integer it is ; for example, 12 minutes taken into 24 matues de-
crease into 288 seconds, and 14 minutes into 15 seconds make 210
thirds. And so in others, of which this is the figure." (At this
point there is inserted a table for obtaining the order of sexage-
w - -
sºmal products up to eighteenths. It is written in a triangular
form; many of the characters used are Very curious. }
'But every whole number if taken into any fraction is of
the order of that fraction." 25 degrees into 5 minutes are lº 5 min-
utes, and 4 integers into 6 seconds are 24 seconds. And here the
author proposes and solves two examples whose solutions, similar to
the solutions given here , have already been noted: the first, to
multiply one and a half by one and a half"; and second, to multiply
| See pp. 101-102,111 ||
2 degrees and 45 minutes by 3 degrees and 10 minutes and 30 seconds. *
|| See p. 1.02 ; compare with p. lll. (2) . || 8 -
This work sets off the second result in the form shown :
in Figure 22. - º
(b) Multiplication of common fractions.
The authors speaks of common fractions as "fractions of
different orders.' * Two examples are proposed and solved.
|| De Multiplicatione minuttarum divorsorum generum. ||
Example (l), multiply 3 sevenths by 4 ninths, and example
A 4
(2), multiply 3 and a half by 8 and 3 elevents.
Figures are added
| These same examples and solutions have been given on pp. 103,
Whi. C. 3 & .
11.4 , which see.
for each of them. (Figures 23 and 24. )


(116)
7 -
12 . .63 - | 25.21. Tº
* 4. 5. 23
-ā-
-
&
Q 3.3.
(Abh' l. . p. 23.) (Abh' 1. . . . p. 23. )
Fig. 23. - Fig. 24.
A third example ; similar to these, is given: "Multiply 2 thirds by
3 fourth G. " +
| Aºi.... ------ p. 34. ||
As in multiplication of sexagesimal fractions so here , the
author inserts, a triangular table for obtaining the denominator of
the product of any two fractions up to ninths times ninths.
XIII. ABRAHAM BEN EZR.A.”
| ºp. 38-37. I
A. Common fractions.
* The multiplication of fractions is the converse of the multi-
plication of whole numbers. "
(a) Multiplication of a fraction by a fraction.
Examples: l by l is
2 2
l; is l; etc. These are fractions
4. -
whose numerators are unity. Uni
l
3 9 :
y is then divided by the square of the
denominator. But to multiply 3 by 3, calculate for each fraction 3
- 4. 4.
integers, whence their product is 9, which divide by 16, the square
of the denominators. It gives * + 2. of an eighth. Similarly, 3
- 2 2 - 5
by 4 gives lº divided by 5 squared. The result is expressed as ,
5
"2 of a square and 2 twenty-fifths."
&
º -
5
If the fractions are not of the same kind, they are first
reduced to a common denominator, “ and then multiplied as before. To
| | By the method explained on p. 39 - ||
multiply 2 by 3, multiply 8 by 9, or divide 72 by 12 squared. The
3 4. 3.2 12
result, reduces to 1 l. However, Ezra observes that "if you multiply
2
(117)
2 by 3, then like Wise there results the half of the denominator 12. "
From which he concludes that it is not necessary to use the square
of the common denominator; but , instead, the product of the de-
nominator's as given, such a product to be taken into the product,
of the numbers corresponding to the respective given numerators.
Similarly three fractions may be multiplied togethere
(b) Multiplication of a whole number by a fraction.
Example: To multiply 4 integers by 3 multiply 4 by 3 and
divide by 5, whence 2 2 result - 5
(c) ºne ºf of a whole number and a fraction by a whole
number and a fraction.
One may multiply fractions and Whole numbers separately by
(a) and (b) above, and combine the results; or one may reduce the
given expressions to improper fractions, and multiply them as in (a).
Example (1) : Multiply 4 2 by 5 3. The product is 24 16.
- 5
2
TE - TE
Example (2) : Multiply 6 7 by 3 4. The product is 26 l .
3 5 8
Each of these examples is solved completely by both of the
methods just stated-
(d) Multiplication of fractions 'which mankind cannot express.'"
--
| Fractions Whose denominators are prime numbers greater than
10, see p. 39 ||
Example (1): How much is 3 by 5?
7 ll
The solution follows readily from (a). In modern symbols,
it is : 3 x 5 = 33 x 35 = 1155 - le = 1 4 sevenths, or 2 l elevenths.
7 11. 77 77 77 = 77 11. 7
Example (2): Multiºſ; o' by lz. The solution is similar to
13 19
that of the preceding example. The result is 8 1 thirteenths, or
3 T.9
11. 10 nineteenths.
Tº 13 -

(iia)
Aº-
A.
(e) Multiplication of triple fractions*
Geº
| Ezra, pp. 38-39. | 2
A 3 6 -
Example : Multiply 4 by 7; that is, multiply l of l bf 2 by
- - 5 8 5 4. 3
l of 6, or, What is the same thing, multiply 2 of l of 1 by 6 of 1.
8 7 3 4. 5 7 8
Ezra "teaches a short method' of solving this. Since 3 and 4 are
found in 6 and 8 respectively, one needs only to find the multiple
of 5, 6, 7, and 8, which is 1680. Take this as the common denomin-
of l of 1680 is 56; and 6 of
5 7
y 180 and divide the product
ator of both fractions. Then 2 of
3
# of 1680 is lö0. Now multiply 56
by 1680 squared. The result is 1 .
280
(B) Multiplication of sexagesimal fractions.
(a) Multiplication of single sexagesimal fractions.
Consider the degree as an integer. Degrees multiplied by
degrees give degrees; any other order multiplied by degrees gives
the same order. Minutes, or seconds, or thirds, etc. by minutes,
or seconds, or thirds give results as explained on p. 101 .
(b) Multiplication of other sexagesimal fractions.
º ſº Ezra ºloº gives 'two correct methods' of multiplying sexage-
ºnal fractions: 'The one by means of writing'; the other " by word
of mouth'. * Th figure 25 is shown the solution of an example by
the first, method. The different orders are separated by perpendi-
cular lines • Observe the Writing of orders in an order reverse to
that of whole numbers . When any order is missing a zero is put in
its place. The multiplication is as in Whole numbers , and when we
obtain the last row 'We divide each place by 60, add the quotient
to the preceding place, and Write the remainder separately. Thus
We proceed with each place, until We come to the degrees, which are

(119)
equal to whole num-
Toers . Thus there re-
mains then 33 sixths, (ITITIT) (TTTTI) (TTTT) (III) (II) (I) (O)
3 3 4. 9 2
56 fifths, 31 fourths, 13. 44 18 3
º 22 38 36 6
24 thirds, 30 seconds 99 396 162 27
- 44 176 72 12
7 minutes, and 7 de- 33 4.32 54 9 |
33 176 32? 499 262 63 6
grees. ' w 2 5 8 4.
33 56 || 31|| 24 30 7 7
Fig. 25. (Ezra, p. 45. )
The second method is the same as that shown on p. 107 º
Applying it here, Ezra obtains 464.643 thirds multiplied by 71.5451.
A
£hirds, or 332429298.993 sixths, which reduces to the same result
as above. *
º, ºr e-le. ||
XTV. T.EONARD OF PISA.
- With the exception of some simple cases on addition,” the
| pp. 53–55, 57. || -
whole of chapter VI of this work” is devoted to the multiplication
| pp. 47-63 - | |
of fractions. One general rule is given at the opening of the chap-
ter, and is followed by many illustrative examples, presented under
eight heads, or cases. Mixed numbers precede proper fractions.
General rule: "But when you wish to multiply any number of
any aegree with any fraction, ruptum, or fractions by any number with
any fraction or fractions, Write down the larger number with its
fraction, or fractions under the smaller number with its fractions
(minutum.) , that is number under number, and fraction under frac-
tion, and consider the number above with its fractions. And then

(120)
make (of it) such reactions as are those Which are with the number
itself. And similarly for the lower (number) you make its frac-
tions. And you multiply the fractions made of the upper number by.
the fractions made of the lower. And divide the product (summa)
by the fractions of each number under one line, that is collected;
and you have the multiplication of any number with fractions. And
as this is shown more intelligently with the demonstration of num-
bers, we divide this chapter into eight parts: -
(a) Multiplication of Whole numbers With one denominator
under a line . *
|| De multiplicatione numerorum integrorum, cum uno rupto sub
una virgula, p. 47- | |
— -
| pensa est LA 7
Example : Multiply 3, 18 by 4 a.4- per 11 (4
S. # 84 3 is
One has (18 30 ( 24 4) - (147) 0) 3 T
3 9 8 220 (0.
(220) = 32340 = 449 l 4 of l; written 424
9 8.9 9 8 : 9 9
4 l 449. The check is by ll. (See 4 l 449
3 9 - º
figure 26 - ) Fig.26. (L.A. p. 51.)
The author points out that one may multiply "a third part
of 147, Which is 49, by a fourth part of 220, which is 55, and
divide the product by a third part of 9, this is 3, and by a fourth
part of 8, this is 2.' This is modern cancellation.
(b) Multiplication of numbers With many denominators under
one line. *
|| De multiplicatione numerorum cum pluribus ruptis sub una.
virgula, Ibid. p. 51.
1
pensa est, 215
Example (1) : Multiply l 3 13 by 3 per llll 313 (6
2 3 2 3
3 2 24; that is, multiply 13 -- 3 + 1. of l. - 375
4 9 5 ' 2 5 |3) 3 2 24 (6
4. 9
5 § 5 326 |
9 T

(121)
by 24 + 2 - 3 of l. The first number is reduced to sixteenths, and the
second º ºf-ty-ºxºns, giving 215 over the first number and 375
over the second number. (See figure 27. ) 215 multiplied by 375 gives
188125, which is to be divided by 2-8-4.9 or 8.9-9. The result is
* + 3 + 3 * #-F# of l of 1, Written 5 3. 5. 326. The check is by
8 9 8 9 - 8 8 9
ll.
Example (2): Similarly l 3 & 14 multiplied by 1 2 4 25
- 2 911 - 3 9 13
gives 4 6 was . * The check is by 7's
| For solution see the text, p. 52- |
(c) Multiplication of numbers with two denominators under
two lines. *
|| De multiplicatione numerorum cum dual us rupt is sub duahus
virgulis. This case begins on p. 52 of the text but the
title is omitted; see text, p. 47, for it . ||
Example : Multiply + 1 15 by l l 263 that is multiply 15 + 1.
T 5 3 E T #
3
1 by 26 l 1.
4. |--|--|---
5 '6 187
The solution of this example presents l l 15
- 4 3
nothing new. Figure 28 gives the result e 791.
1 l
Observe that, 3.4 ± 5.6 is replaced by S 5 26
1 7 & 410
4.9 - 10 in the last step- 4 9, 10
Fig. 28. (L.A. p. 52)
(d) Multiplication of numbers with two lines with many de-
nominators. *
| De multiplicatione numerorum cum duabus virgulis cum pluribus
ruptis . This case begins on 9 - 55, of the text, but omits
the title ; see text, p. 47, for it. ||
5
O
1.
Example: Multiply l 2 1 & 17by 2 1 & 4 es; that is multiply
2 5 5 8 ll.
17 - (5 1 of 1) + (2 l of 2) by 26 F ( 4 || 3 of 1) (1 ., 2 of 1).
+ 3 ++2° 5' + \# F# Ti ºf 3 “ Ti' Hº; + # * *
Except as to the form of their statement this class of examples does
8
:


not, differ from those of
preceding classes. Figure 29
shows the steps taken. Ob-
serve that 2.5-5.5-8.8-9-11 is
replaced by 2. 3.9.10.10.10.11.
The check is by 7 e
1, 2 9 T. 1. (3
l 2 l 2 i
5 9 2 3 7 -
6 3 O 9 L (O
2 + 2 + 28
5 5 GTII tº
1, 6 4 1. 2 7 2
ă ă ă Id Ið Iö Ii º
Fig. 29. ( ...A. p. 55.)

The text gives other examples containing still more denomina-
tors under two lines, but all are solved similarly.
(e) Multiplication of numbers with three lines. *
|| De multiplicatione
begins on p. 56 of
47 of the text, for
Example: Multiply
3 4. 5
before, reduce multiplicand and multiplier
to improper fractions, and divide the product
to the resulting numerators by all of the
numbers under the lines' , that is, by 3-4-5.7.
8.9, or 6.7.8.9.10. The check is by 11, Figure
30 shows the result, a
Figure 31 shows another
example under this case a
it. ||
1.
-
l
- - 5 4
21. + + + l + l hy 32 + # + # + #.
the text but omits the title,
l
3
As
__
numerorum cum tribus virgulis - This case-
= 21. by
See 9 e
1 2 3 32; that is multiply
8 9 Tº
5-0 7.
21. (9
19
32
(8)
7 5 9.
- 713
8 9 TO -
Ö. (L.A. p. 56.7
l
:
5
6
Fig
Ensa Test IO per II
3 8 5 L 3 l 3 O
(7
l 2 l l 2 3 l l 6 il
3 5 9 2 9 LO 2 7 11.
14328871 O (6
5 l l 2, 2 l 3 & 22
6 7 5 7 9 2 3 LO
pensa est (10)
2. l 0 l 3 9 & 0 4 274
7 7 8 9 9, 10 10 lo 17
g. 3i. (T.A. p. 53. )
(f) Multiplication of fractions without whole numbers, *
| | De multiplicatione ruptorum sine ganis. Ibid., p. 59. ||
º (123)
tº
º
This case treats the several preceding types of fractions,
º
their whole numbers being omitted. It is, therefore, a special
|
base of the preceding cases, and the solutions of examples present
nothing new .
Statements of some of the illustrative examples are: - (1)
!! Ibia, pp. 59-60. I
Multiply 3 by 33 (2) Multiply 1 % by 3-3 (= 4 + 1 of , by 3 + 3 of 1);
2 7 3 5 7 2 7 5 3 5
(3) Multiply l l l by l l 2 (= ++ 4 + + Py 2 + 1 + 1); (4) Multiply
5 4 3 7 6 5 * 4 ' 5 5 6 7
3 2 & 3 l 6 by 4 l l 3 l 7 (= 2 + 3 of l 3 + 3 of —l 6 - 1.
4 9 5 10 2 ll 5 & 3 7 2.13 # F# 5, # F# Făt Ti F =
of 1 by l - 4 of l ; 3 l of l 7 -- l of l) •
++ ° #-F# * + + #-F# * # + +, + = ** +
- - - 2
lines are terminated Lºl circles • *
+
|| De multiplicatione numerorum et ruptorum duº virge terminan-
ºur in circulo. Ibid, p. 61.
Example: Multiply 2 3 4...oil by 66 g .9 223 that *** multiply
3TST, 7
9 10
ll. 4. 5 of 4 2 of 5 of 4 by -
+ 3 + 3 + 3 + # * * * * 2 4 3 9 (3
7 9 10 - 3 8 9
Ubserve the difference between a 1 4 4 8 9 - (2
o 6 & 9. 22
circle" to the right of the line and one O 1'." 10
-- 270
left. Except as to the form of its || 3 5. 7 9 -
to the le Fig. 32. (L.A. p. 61. )
statement, this case offers nothing new -
Figure 32 shows the result, . The check is by 11-
Figures 33 and 34 show further applications of the circle :
º
--
- -
-
-

(124)
Ig 6 2 5 T 7 6 4 8
2 5 4 o 3 5 & 3 4. 7 oil (O
3 8 9 4. 6 11, 8 9, 10
4 3 2 4 2 5 7 9 3 6
o 6 & 9 o 3 & 2 o 3 + 7. 22 (4
7. 9 LO 4. 6 11 6 9. 10
3_5 probatur per 7 (0)
# 3 i 5 5. 7 0 0 8 7 296
FI. 33. (T.A. 6 8 9 10 10 ll ll
p. 61.) - Fig. 34. (L.A. p. 62.3}
(h) Multiplication of parts of numbers with fractions. *
|| De multiplicatione partium numerorum cum rupt is - Ibid., p. 62. ||
by l 5, 244 l 3; that is a multi-
Example : Multiply 2 5 33 l 3 -
º ; .. 5 4. º 4. Tº
ply (3 - 1)(33 + 3 2 of l) by (3 + l of l)
7 ºf + +" ºf #4 # * * #t # * # 4TO 3 OT (ST
(244 + 3 + 1). See figure 55. * The check is 2 & 33 l 3
6 11 º 7. 9 5 4.
$3 R33. || The text gives this figure only 2. l O L 4: 5 ( 0
in part, but gives the solution ii. 6 7
2 6 0 l 4 + 362s
in full. || - | 3 7 ? 8 9 ll
- Fig. 35. (L.A., p. 63.)
by 13. Figure 36 shows a more complicati d example. Many other
examples are given in the text a
g 2 & 5 i. 3 3
—l 2 3 42 2 & 5
13 ll 5 7 G 9
2 6 1 3 3 O 9 8 i
3 O & 331 l l 5
3 5 11 9 8 7
1 9 & 3 7 7 7 4 & 7 glia
2 7 7 9 9 10 lo ll: ll 13 -
Fig. 36. (T.A. p. 63. )
XV. JORDANUS NEMORARIUS.
see sections 13-14, 23-25, of the synopsis of this algor-
ism, p. 48 = - -
XVT. GERNARDUS. *
| sections is-in. ||
(a) Multiplication of common fractions.

(125)
A fraction is multiplied by a fraction if the numerator is
multiplied by the numerator and the denominator by the denominator-
An example is given in which lines instead of numbers are usea, and
which is demonstrated after the method of Euclid.
If many fractions are to be multiplied by many fractions,
the fractions of the multiplicand and multiplier are reduced to one
denominator respectively, and then the two resulting fractions mul-
tiplied as abo Ve =
Integers With fractions are reduced to fractions and then
treated as fractions.
- (b) Multiplication of sexagesimal fractions.
The several methods given for multiplying sexagesimal trac-
tions are as have already been given by Al-Karkhi p. 107 and Bºx
Ezra pp. 118-119,2xcept for the case When the multiplicand and multi-
plier consist of several fractions each. Then arrange the given
integers and their fractions in order from right to left, placing
the integers of the multiplier under the last fraction of the multi-
plicand, as shown in figure 37, Where f fourths, g thirds, h
minutes, and k integers are multiplied by a fourths and b minutes.
| lin-
te-
- - lº —/
- C - ... f. • C- th: | *k.
• 3 a. - b. locus |
inte .
Figure 37. (Figure 13, Gernardus . )
Place a cipher in each vacant place. Next multiply the
number in the last place of the multiplicand by that in the last
place of the multiplier, and place the product in the last space
over, the multiplier if such a product is less than 60. In the exam-
ple given, one has as fourths times ºf a fourths which gives - c.



eighths, which is placed over .a. fourths , . c. being less than 60-
If the product be greater than 60, subtract 60 from it as many times
as you can and place the remainder, if there be any, in the space
above the multiplier, and in the nearest place toward the integers
put as many units as is the number of times 60 was subtracted. Like-
Wise do with others. But notice if integers are given with minutes,
when you come to minutes, before multiplying them, reduce the in-
tegers and minutes to minutes v preferably yuu may do this before be-
ginning the operation), leaving the place of integers vacant. Now
proceed as before. The last term of the multiplicand having been
multiplied by each term of the multiplier, the terms of the multi-
plier 2 Tºe all shirted one place toward the integers, or until the
first place of the multiplier comes under the next to the last term
of the multiplicand, and the operation described above repeated with
this term of the multiplier, each product being added with the num-
belº already in the respective places and the respective/ºuses by
60 as above . -
Gernardus gives a shorter method which is the same as the
second method, given by Al-Karkhi, p. /*, and others.
XVII. TEN AL-BANNA.”
|| Atti dell'Accademia Pontificia de Nuevi Lincai, ºvir, p. 310.
Definition and rule : rais is taking of one of two fractions
plied -
to be multi, the one by the other, a part indicated by the value of
A
the other. The operation consists in this that you multiply the
numerator of one of two fractions by the numerator of the other,
and you divide the result by the denominators. *
No comments nor examples are given. *
|| Al-Kalsadi Ibid, vol. XII, p. 271. gives this example:
(127)
3 5 3
7 ° 4; that is, multiply (3 + 3 of 1 + 3 of 1 of 1) by (5 + 3 of 1-1.
l 3 5 # T â + + i = i # T : * :
3 4 7 - -
l of l of 1). The result is 11480 divided successively by 3, 4,4,
3 4. 7
6, 7-7, giving 2 + 4 of J. H. 3 of l of l = 3 of l of l of l of l.
7 7 6 7 7 4. 4. 6 7 7
2 of l of l of l of l of l. Written 2 & O 9 4 5.
3 4. 4. 6 7 7 3, 4 TG 77
Fractions with whole numbers are reduced to improper
fractions, and then treated as other fractions. |
XVIII. T.EWT BEN GERSON. *
| pp. 89-92, 94-97, 99-100.
(a) Multiplication of common fractions,
The author approaches multiplication of tº re- the
standpoint of ratio. He says that the ratio of any fraction to 1.
is equal to the ratio of 1 to the denominator of the fraction; thus,
l s 1 is equal to l ; 3.
3 And likewise the product of a given number of Whole num-
bers 2 or of one whole number by a given number of fractions, or by
one fraction, is as many parts of the parts determined by the de-
nominator as is the product of the multiplier and multiplicard.
For example, if one wishes to multiply 5 and 40 integers, one
asserts that the result is as many seventh parts as is 5 times 40,
Which is 200 a
- Similarly for the multiplication of a fraction by a frac-
tion: The product of a given number of fractions or of one fraction
by a given number of fractions or by one reaction is as many parts
of the parts named by both denominators as is the product of the
numbers of the multiplier and the multiplicand.
The denominator of a double fraction is the number, which is
composed of the product of the denominators of these fractions,

(
1.
2
8
)
After finding the denominator, one can effect from the
for going multiplication by Whole numbers as well as by fractions.
These rules are extended to finding the product of several
fractions. Each case is illustrated by an example -
Example (1) : If you * } * Z. 42 results, and if you
divide 42 by 7 the result contains eighths . . . . the result is 6. If
you divide 42 by 3, then the result contains sevenths. Then is the
result 5 and 2 of one sevenths.
7 8 - -
Example (2): .28 by 6 of a third – 168 of a third – 5 of
- 29 7 - 7. 29 7
a third and 23 of a seventh of a third, or 3, or 56 of l = 3
- 29 29 7 59 35
The author observes that it is some times convenient to
interchange the order of factors of a product. This leads to can-
cellation of today.
Example (3): by b
*
-
b
y
-:
1.
lº
J
l.
y
#
4 by 6 by 7.
5 7 3.
#
3. 7.
4. - 7
of l or 2 and l of l or l an
5 4. 5 4.
:
d
#.
5
#
by 3 by 3 = s.
4. 5 4.
To multiply a series of fractions by a fraction, multiply
each fraction of the multiplicand by the multiplier, and divide
these products by the respective denominators of the multiplicand,
and combine the results.
Example (4): (6 - 7 - 8) & = i.2 thirds 14 thirds, 16
#-F # tº # Tº ++;
- 3. Tº -
thirds = (1 a thirds): (1 - 3 thirds) (1 - 7 thirds) = (1 l
# + # 3 7. 3 9 # #"
1) - (5 3 || 7) thirds = 1 + (2 + 6l) thirds = 1 + 2 - 61.
3. –– # F#-F 9 - 52 - 3 H 756
This rule may be extended to the case where the multiplier,
also , contains several fractions . Then multiply the multiplicand by
6|ach term of the multiplier as above , and combine the products ob-
tained.
A second method is to reduce multiplicand and multiplier to

(129
one fraction each and of a common denominator, and then divide the
product of their numerators by the square of their common denomin-
ator a
Example (5): (12 + 3 − 4) by (21+ 2 + 3) = 3348 by 4035 =
5 9 3 4 180 130
9474,180 - 292 )
TCO2
o
#4 º'
(b) Multiplication of sexagesimal fractions.
Th the multiplication of sexagesimal fract, ions, the given
fractions are arranged from left to right, as shown in figure 33
where 7090 integer's, 40", 51* *, 3” 2 are multiplied by 83 inte-
geºs, or , 57*. observe that the product of a fraction by arl
integer gives a fraction of the same order, of a fraction by a
- - - T TI III TV IT III, VI
T The series of the multiplier 33 9 57
The series of the multiplicand 7090 40. 51 3
-- - - 112 || 13 30 3343||37 tº 2 | Bl
1063 30 6. 7|39 j27
5334.70 || 55+l 10 + 2.0 4 + 33 9 | | |
Result Series. 589646 42 7 23 || 36|| 06 || 51 i.
Figure 38. (Te Wi ben Gerºon, p. 100.)
fraction give an order 'whose distance from the order of the multi-
plier in front is equal to the distance of the multiplicand from the
class of unity.' Multiply the first number of the upper series and
the first number of the lower series and place the product , if less
than 60, in its proper column. If the product is more than 60,
divide it by 60 and carry the integer’s of the quotient to the next
column (to the left), retaining the remainder, if any, in its proper
column. If the integers of the quotient carried to the next column
are greater than 60 and are not of the units column, then repeat the
last, step . Proceed thus until all numbers of the lower series are
multiplied by the respective numbers of the upper series, there
being a 'result series* for each number of the upper series. Then


(130)
add all of the result, series, reducing any sum greater than 60 LO
numbers of a higher order, as in the example shown above. This
final result is the result desired.
XIX. PLANUDES. *
| pp. 34–35.
For the multiplication of degrees, minutes, or seconds by
degrees, minutes, seconds . . . . . . See p. 107 *
Example: To multiply 1.4 degrees and 23 minutes by 3 degrees
and 16 minutes. By the method explained under the multiplication of
integers, one has 3 degrees times 14 degrees, or 112 degrees; lº
degrees times 16 minutes, or 224 minutes, which added to 8 degrees
__
times 23 minutes, or 134 minutes, gives 408 minutes; º 23 EA 3
14 23
8 16
- Fig. 39.
368 seconds equal to 6 minutes and 3 seconds. Write (Planudes p.
and 23 minutes times 16 minutes, or 363 seconds. But
3.
5
)
down the 3 seconds as shown, and add the 6 minutes to the 408 min-
utes , giving 414 minutes, Which are equal to 6 degrees and 54 min-
utes. Write down the 54 minutes as shown, and add the 6 degrees to
the llº degrees previously found. One has lla degrees, or 3 signs
and 28 degrees, which Write as shown.
Similarly other examples may be solved.
XXI. T.TVERTUS. * - t
| | fr. 23-25. |
(a) Multiplication of common fractions.
Common fractions are multiplied, as one would multiply them
today.
Examples: (1) 2 by 3 = 2.3 = 6; (2) 3 by 4 = 3.4 - 12;
- 5 7 5, 7 35 5 5 T5
two integers and 2 by two integers and 2' = 10 fourths by 8 thirds
4. 3
- * 6 integers a
#9


(131)
(b) Multiplication of sexagesimal fractions.
Multiply the numerators of tº given sexagesimal fractions,
and denominate the product by the number Which is the sum or the die-
nominatºrs of the fractions multiplied; or give the product a de-
nominator such that its distance from one of the fractions is equal
to the distance of this fraction from integers; thus, fifths by
thirds give eighths . . Any fraction by integers, or conversely,
always gives a fraction of the same denomination.
Fractions of airferent denominations may be multiplied by
fractions of different denominations in two Ways: reduce fractions
of the multiplicand and of the multiplier respectively to one de-
nomination and then multiply as above, or multiply the fractions
of the multiplier by each fraction of the multiplicand, and add the
results a -
XXTT - TOHANNES DE MURTS.
Rule : Put, if you Wish to multiply a fraction by a fraction,
multiply those by themselves under the line, and next those above,
multiply above the line. Do nothing more but collect what has been
done.
L. 1."
Example (1): 'l and l |-e-e: ; that is multiply (1+ 1) by
- 3 2, 4, 5 - - ZT-3
l of l. The author obtains 5 by l or 5 or l.
7. 5 - 6, 20 120 24.
2
- 4 5
Example (2). Multiply 2 by € 9. In modern notation,
5 6
3.
- 4. - -
the author has 2 + 3)(#3) = 2.4 + 3:3: . 20 = 17 . 2 (written
2 - 3 4 & 6 3. 4. 3O 12 3
17 -º), or 34 or 17-
Tº 3 36 L3
Cºv. VON GMUNDEN. *
|| Fr. 50–55 ||
Sexagesimal fractions of different denominations one
(132)
multiplied as described in Liverius, p. 131 .
No examples are given. Neither does the author state how the
terms of the multiplicand and multiplier are arranged With reference
to each other.
XXVI. PEURBACH. -
| F. 46.
To multiply a fraction by a fraction proceed as one does
today.
To multiply several fractions by one or by Several fractions,
combine the fractions of the multiplicand and multiplier respective-
ly into one fraction, and then multiply as above a -
'secondary fractions are converted into 'primary' frac-
tions, and then multiplied as above.
Whole numbers and Whole numbers With fractions are converted
into fractions as one converts then today, and then treated aS any
'primary' fraction.
No examples are given.
XXVTTT. AT-KAL SADT. “
|| Atti dell'Accademia Pontificia de' Nuovi Lincei, 1859, XII,
pp. 270-272. ||
sº-ºne, pp. 12°–12.

CHAPTER vr
DIVISION OF FRACTIONS.
Almost invariably division of fractions immediately follows
their multiplication. Al-Hajjar and Teonard of Pisa have this order: i.
multiplication, addition, subtraction and division.
In common fractions, all four of the Hindu arithmetics
examined invert, the advisor and multiply: *.
Gerºnardus and ben Gerson multiply cross-Wise ; that is , they multi-
ply the numerator of each fraction by the denominator of the other,
and then divide the products. Jordanus, Gernardus, Liverius and
Peurºach treat , also , the special case where the numerator and de-
nominator of the dividend are exactly divisible by the numerator
and denominator respectively of the divisor; that is , a 3 c_ a + c =
- - - b d T b + d
h, Where a = hg, and b = kół. But the most common rule is to reduce
the given fractions to a common denominator, and then divide the
numerator of the dividend by that of the divisor. When the dividend,
or divisor consists of more than one fraction, or of a whole number
with one or more fractions, reduce it to one fraction and then pro-
", seed as usual a
( - -
In dividing sexagesimal fractions, the earlier authors re-
-
duce the fractions of the dividend and of the divisor to the low-
est denomination found in either of them and then divide, whereas
the later ones, before dividing , usually reduce them to the lowest
denomination respectively found in them. Ben Gerson stands alone
in his method of dividing them.*
|º. 149-150.
A
Two special cases of division, 'restoration' and 'abase-
º
- º
ment," are given by Al-Haggar, Al-Banna, and Al-Kalsadi. *
- / A -
| See p. igo ||, -


(134)
TT. BRAHMAGUPTA. -
- - º **** #.
|| pp. 278-279, pań. 4 and footneſſes; p. 292, pº and foot, -
notes. |
- f -
"Both terms being rendered homogenous, the denominator and
numerator of the divisor are transposed: and then the denominator of
the dividend is multiplied by the (new) denominator; and its numer-
ator, by the (new) numerator. Thus division ( is performed). *
"Example: In a rectangle, the area of which is given, a
hundred and twenty-two and a half; and the side, ten and a half;
tell the upright.
Statement, : 112 • Reduced to homogeneous form 245
2
#
+
º
£º
1.
p
& -
Here the side is divisor. Its denominator and numerator
are ******* The numerator of the dividend, multiplied by
this *:: becomes 490, and the denominator of the dividend,
taken into the denominator, makes 42. The one, divided by the
other, gives the quotient 11 1/2. rt is the upright."
The method of finding the third assimilation consists in º
division: the upper numerator is multiplied by the denominatore
The "upper numerator" is the dividend. The middle Quah-
tity together with its denominator is the divisor. Then the rules
given above for the interchange of the terms of the divisor applied,
"Example: In what time will (four) fountains, being let
loose together, fill a cistern, Which they would severally fill
-
in a day; in half a one; n a quarter; and in a fifth part :
1.
| | | 1. -
"Statement: l l l l - 'The rule being observed; 1 2 4 5.
2 4 5 - T I I T
The sum is 12. So many are the measures in a day. With all the
fountains. Then, by the rule of three, if so many fillings take
(135)
place in one day, in what time will one? Statement: 12 || 1 | l .
- 1. 1. 1.
Answer: +: In this portion of a day , all the fountains, loose
12 - -
together, fill the cistern.”-
| This example is given by Bhaskāra also , Colebrooke, p. 42. |
III. AL-KHOIART2Mr. *
| | Trattati I, pe. 20–21. ||
"When you wish to divide a number with a fraction by some
number with a fraction; or a number with a fraction by an integral
number ; or an integral number by a number with a fraction, make
each number of one class ; that is, convert both numbers into the
lower orders . Next divide what you wish over what you wish. . . .
and the result is degrees, that is an integral number. "
Example (1): ' If xv. thirds are divided by six thirds, two
and a half result from the equality of division: Since . KV. thirds
make .V. integers; Which When you divide by .V.I. thirds which are
two integers, two and a half result: * And similarly when halves
| This examples is found on p. 142 also . | |
are divided by halves, and fourths by fourths, minutes by minutes,
and seconds by seconds, and thirds by thirds.'
Example (2): Here it is required to divide x seconds by
V minutes. Expressed in modern symbols, the steps of the text are:
lo" – lo." - 600" - 2:
5? 3OO" 3OO"
observe that the text writes the quotient two seconds, duo secunda,
instead of bivo minutes, duo minuta. *
|| It is significant that both X and XI contain this example,
and both make this same error - Trattati, T , p. 20-21; Trattati
TI, p. 54: Abh' l zur Gesch - der Math. , VIII, p. 22; this
study pp. 141, 142 - ||


(136)
Example (3): "Likewise when you wish to divide .x. minutes
above .V. thirds, convert minutes into thirds, and there are
• XXXVI, thousand thirds; and divide over .V. thirds, and there are
.VII. thousand two hundred degrees. "
TV. MAHAVTRACARYA.
- Rule: "After making the denominator of the divisor its
numerator (and vice versa), the operation to be conducted then is
as in the multiplication (of fractions ), or, when (the fractions
constituting) the divisor and the dividend are multiplied by the de-
nominators of each other and (these two products) are (thus reduced
so as to be ) without denominators, (the operation to be conducted)
is as in the division of Whole numbers."
X dº (2) a . g = ad tº be , (foot-
C p d -
|| That is ; (1) a , 2 = a
b
|
Example: "When the cost of half a pola of asa foetida is
note, D - 39 °
3. of a pona, what does 3 per’SOh get if he sells l pola at that
ame ) rate?"
Three other examples are stated. None are solved sº
"Thus ends the division of fractions."
W. SRIDHARACARYA. *
| atºlognees Mathematica, VIII.g., pp. 209-210.
Exchange the numerator and denominator of the divisor and
multiply."
- No examples are given.
"When unity is divided by a fraction the quotient represents
the value"; that is, l. = b .
37ET a
"What is called a complex fraction is that in which several
forms of fractions appear together." They are solved by rules al-
ready given.

(137)
VI. AT-NASA'ſ I.
See p - 34 . The methods given are entirely modern.
WTT. ATI-KARKHT. *
|| Chapters XXXIV-XXXV; XXVII-XXXVIII. ||
(a) Division of common fractions •
"You determine a common general denominator for the frac-
tions of the dividend and of the divisor, and multiply the same
into both. The result, Which you obtain by the multiplication of
the dividend, you divide by that Which the multiplication of the
divisor gives. What results is the solution. "
- Example : Divide *H #4 + 4 + by 4 + # tº #: The general
denominator for the fractions of the dividend and divisor is lä2 a
If you multiply this * intº * , +++ # it gives 3401;
and, if into 4+ + ++. it gives 627. Whence dividing 3401 by 627
"one obtaing 5 alºnem and 266 parts of 627 parts of the unit..."
"Sometimes it happens that the products of the terms have
common factors a Tf this is the case , then you may divide each Of
them by the greatest common factor. After the division you proceed
as I have directed. '
Example: Divide 5 Dirhem # l, by 1 dirhem + 1 + l\l-
- t 7 || 2 | 7 || 4 7 7 || 4.
The general denominator is 28. Multiplying with it, you obtain for
the dividend 143, for the divisor 33. Dividing by 11, their great-
est common factor, the dividend is 13, the divisor 3. Divide l&
by 3, and you obtain 4 l- And this is the solution.
3
C.
Further examples are given in phapters xxxv.
(138)
(b) Division of sexagesimal fractions.
The division of the degree and its parts may be produced
if you divide the whole number of it by the Whole number accord-
ing to the rule given earlier. The remainder you can divide in
two ways, in Which you either divide the smaller number by the
larger, or the larger by the smaller’s
If you divide by a smaller number, then you resolve each
term A^it into the order of the smallest part found in them, and
observe how many times the dividend contains the divisors The re-
sult is of the order of degrees. And this is the solution.
&
But if the smaller number is divided by the larger thin
you subtract the order number of the divisor from that of the
dividend. The remainder is the order number of the quotient.
Example: Divide 20 ninths by 4 thirds. Subtract the 3
from the 9; then 6 remain. The result consists of sixths. Then
divide 20 by 4 and the quotient is 5. The result then is 5 sixths.
Chapter XXXVTTT contains several more examples,
vTTT. AT-HAssaſº. +
| | Bibliotheca Mathematica II.3, pp. 35–37. | |
Division is described in chapter VI of this work. It is
divided into two general parts: (a) "division of a smaller by a
* The former,
larger" and (b) "division of a larger by a smaller.
is furthor Subdivided into 23 subheads, and the latter in 25
subheads, each dealing with different combinations of the element-
ary forms, whole numbers, fractions, fractions of fractions, and
SC Oil a
(a) The translator gives two examples of the former:
Example (1), Divide 1 by 4. This is written # It rº
- - • It reduces
3. - 4.
a.

(139)
i.
to _l(= l of l), Wrriten -
13 2 6 6 2
The original text gives here a general rule for divid-
ing a fraction by a whole number.
- 3 7
Example (2): Divide 3 + 7. by 6. This is written 5 §
-- 5 8 6
and is equal to 3.8 + 7.5 = 59, written 2 & 2.
- 5 - 6 - 8 º, 10 3 3 LO
(b) Example (1): Divide 10 by 1. This is weriten 1 .
- - - 5 - * >
The result is 10.5 = 50.
A general rule for dividing a whole number by a fraction
follows a
- - - - 9 -
Example (2): Divide 9 by 3 , Written 3, and reduces to
- - 3 8
9-3 = 24-
-g *. 10
#xample (3): Divide 10 by l of l Written l. It becomes
- 3 5 5 3
10 - 5.3 = 150.
Example (4): Divide 3 of 5 by 2 of 3, written
- - 4. 5 -
- -
3. -
# 9. This reduces to 3.5- 3 l
2 3 2. 3.4 8
5 -
Two special cases of division are given in this chapters
They are called, (1) restoration of fractions and (2) abasement
of fractions. The latter is, in a sense, the inverse of the former.
This meaning may be readily understood from an illustrative example
of each: (1) "If it is asked with what must you restore 1 in order
- 3
to obtain l that is the same if it is asked with what number must
you multiply l in order to obtain lº The procedure is that you
}
divide 1 by l, which gives 3 &
3
(2) "If it is asked with what number must you abase l
so that you may obtain l, then divide l' by l, this gives 1," and
- 2 - 2
this is the result sought -
Tx. BHASKARA. *
|| Colebrook, sections 40-41 - p. 17. ||
*

(140)
Rule for division: *"After reversing the numerator and de-
| One of the eight rules of Arithmetic applied to fractions. ||
nominator of the divisor, the remaining process for division of
fractions is that of multiplication."
ºxample: "Tell me the result of dividing five by two and a
third; and a sixth by a third; if thy understanding, sharpened into
confidence, be competent to the division of fractions.”
"Statement . 207)5 l l.
l 3 1 3 6
"Answer: lö and l."
7 2
X. ANONYMOUS - T,IBER ALGORISMI DE PRATTCA ARISMETRICE. -
| Tratvati II, pp. 53-54. ||
)a) Division of sexages imal fractions.
When you wish to divide a whole number with a fraction by
a whole number. With a fraction, a Whole number only by a whole num-
ber with a fraction, or a Whole number with a fraction by a Whole
number only, or fractions only by fractions you should reduce both
the number dividing and the number to be divided to the lower class
of fractions found in either so that they shall both be of the same
class . . . . . . . . º since, in division, it is necessary to divide similar
fractions by similar fractions; as minutes by minutes , or seconds
by seconds, and go on, so that the result will be a whole number. . . .
But if you wish to divide minutes by minutes, which are too
few to be divided, reduce them to seconds by multiplication by 60;
the result will then be minutes. Similarly for other classes.
Example (1) : Given one degree and 30 minutes to be divided
by two degrees.
Reducing dividend and divisor to minutes We have 90 minutes

(141)
to be divided by 120 minutes, which can not be done. Then reduc-
ing 90 to seconds one has 5400 seconds to be divided by 120 min-
utes, Which gives 45 minutes = - -
Example (2): ' If it is given 10, seconds to be divided by
5. minutes, two seconds result by division.'”
| Si proponatur - 10: secunda dividenda per ...5 minuta,
exibunt de divisione duo secunda. Trattati, II, p. 54. ||
observe that the author does not give any solution, but
merely states that the quotient is 'two seconds', thus making the
same error found in IT and XI.
The fractions of the dividend and of the divisor are arrang-
ad in two columns, each of which is then reduced to a common de-
nominator, numerºus denominationis, and then further reduced to one
fraction each, the numerator of which is the numerºus collectionis.
These two fractions are now reduced to a common denominator, rium-
erºus communis. Whence division is performed by dividing the numer-
at or of the dividend by that of the divisor. It is readily seen
that, except for the last step, division of common fractions is
similar to their multiplication.
--- TSTT3 |
Example (1): The author uses the same 1. 1.
: 2 3
example that was used in multiplication. Tn I TT
4. 9
modern symbols, it is (C + 1 + 1 + 1) ** 1 +
- - 2 4 5 5. ---
(3 + 1 + 1) = 333 + 33 - ???? - #722 = 4. 0 27.
3 9 40 57 TOGO TOGO | ió. 8.6.
9666 - 2 22:26. The text gives the solution 9666. 3720.
3.720 37.50 - 222 2
in full. See Fig. 40. - - 6
372 O
Fig. 40. (Tºtati
TI, p. 69. )
Example (2): 'Given to be divided .2.0. integers, and two

(142)
thirteenths by 3, and a third.'
S - 20- 3.
Tr, modern Aymbols, one has 2. 1.
- 13 3
(20 + 2) * (3–1) - 780 + 6 + 117 -H 13 * 736 13. 3
13 3 39 39 39 -
* 130 – 6 6_. The text gives the complete 7. 3.. 6. l. 30.
130 - - 6
solution. See figure 41. This example does 6
- 130
not occur in ITT, but it does in XI. * Fig. 41. (Trattati
- II, p. 71. )
| See below. ||
XT. ANONYMOUS – A 12th CENTURY AT GORTSM. *
|| Abh' l zur gesch. der ºath. , vol., VIII, pp. 22, 24–25.
Both in sexagesimal fractions and in common fractions, the
author's exposition is too biºief a
(a) In sexagesimal fractions two examples are proposed and solved:
example (1), Divide 15 thirds by 6 thirds; and example (2), divide
10 seconds by 5 minutes. Example (1) has been given in TIT,” and
| |
| p. 135 || -
4.
example (2) in IIT and X. In all three works the answer to example
| 141
| pp. 135, 7 |
(2) is incorrectly given as 2 seconds.
(b) Division of common fractions is explained in less than
two lines, 'But We divide these fractions instruction having been
stated above and having reduced them to the last order. ‘ One ex-
- - lº
ample is givene Divide; 20 and 2 thirteenths by 39 fºſys6
o
- º
3 and a third. See above for solution, and Fig. 6. 6.130 1
39 || 3 || |
42 for the form in which this text gives the 1.
result. - Fig. 42 . (Abh' l. p. 25. )
- - 5
XITT. ABRAHAM BEN EZR.A.
ſº, º ºs. |
W
Only the division of common fractions is considered, and

(143)
* @ex-
less than half a page is devoted, them. Nºra says, 'Proceed as T
have shown you in which you reduce the fractions into one, and if
whole numbers are With fractions, proceed according to the rule. "
Example (1): ' Divide 3 2 by 2 4 - The denominator is 35.
- 5 - - -
The 3 integers are 105 and the 2 la , together l19. The 2 whole
- 5 -
units We make into 70 and the 4 into 20, gives 90. We divide llo
- 7
With it, then the quotient is 1 whole number and 29 remaining over,
Which a lºe ; + 1. "
* T TO -
Example (2) for pure fractions: divide 7 by 2 . Here again
- 9 7
the author reduces the fractions to a common denominator, and then
divides their numerator’s a The result is at 3 + +"
KTV. LEONARD OF PTSA,
- Division of fractions is treated in Chapter VII, pp. 63-
83 in subdivisions l, 2, 3, 4, and 5, immediately following the
respective cases” of addition and subtraction. In general each
| see pp.40-44
illustrative example is used to illustrate addition, subtraction
and division,
(a) Division of one line by another. *
| Liber Abaci, pp. 63-65.
Example: ' If you wish to divide 4 by 2 you divide lº by
5 3
10, 1 1 results. ' -
E *-
Here 4 and 2 have previously been reduced to a common de-
§ 3 -
nominator for addition 4 divided by 2 is equal to 12 fifteenths
5 3
divided by 10 fifteenths. Hence the quotient of 12 by 10-
No general rule is given.
(b) Division of two lines by two lines. *
| risia, pp. 65-68.

(144)
Example : Divide l l by l 1 by
7 5 4.
l; that is, divide
3 7
l
5
*
+
under addition,” these expressions have already been reduced
|| This study pp. 79-80 ||
to a common denominator. Whence one has 14.4 divided by 245, equal
to 144 = 144 - 41-4 of 1 of * Written 4 O 4.
7 5
345 E. 7.7 - - 7) 5-7 7
No general aule i iven.
S
.
(c) Division of whole numbers by whole numbers and fractions, and
conversely. * -
|| * Abaci, pp. 69-71. ||
This subhead of chapter VIT is devoted entirely to division.
A general rule is given: , -º
"when you wish to divide any whole number with one fraction,
} -
or many, r, conversely, a whole number with a fraction by another
Z-S
whole number, make fractions of each number What ever it may be , or
*-* -
those which are proposed with one number. Next divide the sum of
the fractions of that number by the sum of the fractions of the
other, and you have such division as you wish "
Example (1): ' If you wish to divide 93 by 2 5, make thirds
3 -
of each number thus: multiply 5 by 3, which are .
- r 1 7 2 4: 9
under the line, and add 2, there are 17 thirds: 2 5 9 3
- 3
and multiply 83 by 3, so that you make thirds l l 14
l 7 ---
3.
of them, there are 249 thirds: then divide 249 Fig. 43. (T.A. p. 69. )
by 17, H 14 result for the required division.'
from this then it is clear, that the same is the division
of 83 into 2 5, as of 249 into 17; and this is what Euclid, the most
skilled in Geometry, declared in his book.'


(145)
Example (2): Divide 217 by 1 2 12, that is, divide 217
- : 3 -
by 13 - 2 1. Ti a 7-3 a 5-4. T
+ 3 + + l 2, 13 2 1 7
2 is used as the common denomina- 4 3
99 Tº
tor. The quotient is 15 99, written 99 is, Tº
- 6 167 -
See Fig. 44. - Eig. 44. (L.A. 2 p. 69. )
similarly, other examples in this case are solved.
(d) Division of whole numbers with fractions. -
| roid, pp. 71-75. -
Rule: The rule stated briefly is this: write the smaller
number on the left, reduce dividend and divisor to improper frac -
tions and proceed as in (a) above-
Example : To * , 12 * } 126, since these reduce: to
149 twelfths and 1521 twelfths respectively. * -
- L G 2 T. 1 4 3
| | In addition p. 30 - 3 1. 2 6 l, l 2
4. 3
One divides 148 by 1521. The quotient, is divisio minoris
per maiorem
lá8 = 148 - l 3 of º of l of 4 3 1
Tº gºis T3 Tº Tºt 3 tº TIET:
i. Written 4 3 l. Fig. 45: ) Fig. 45. (L.A., p. 73.)
13 JTET3 -
Other examples are solved by the author.
(e) Division of parts of numbers with fractions. *
| Liber Abaci, pp. 75–77.
Example : Divide 2 is a by 2, 29 33 that is , divide 5 of
5 *** 7 5 4. 7
128 2 by 3 of 29 2.
9 4. - 5 - -- -
Under addition*these numbers have already been reduced to
| | p. 81 of this study. |
a common denominator, with numerators 115400 and 27783 respectively.
Therefore one has llc.400 divided 27783 – 4 4268 - 4 4268 =
27783 7. 7.7. 9.9
# 5 0 3 3 L 4
9
7 7 7 g ºf

º
5
.
T.1.
i
- :
9
Fig. 46. (T.A., p. 76 . )
29 3 divided by
2 . See Fig. 46.
4. 9
#
;
128 & is l 8
7 2 7
xv. Jordanus NEMORATIUS.
See sections 26-30 of the synopsis of this algorism, p. 48 °
wr, gºnnarous.*
| sections 18, is, 22, 27.
(a) Division of common fractions.
(1) Special case: make one fraction of those fractions that
31}^3 to be divided and one fraction of those by which division is to
be made • Then divide the numerator and denominator of the dividend
by the numerator and denominator respectively of the divisor, if such
divisions can be performed without remainders. The results will be
the numerator and denominator respectively of the quotient.
Example: If the fraction c is to be divided by the fraction
d
, and c can be exactly divided by a , and f obtained, and d by b and
- - - • f. • * *
obtained, then the fraction f is the - C - - d.
& * 9. e. • b =
quotient sought (See figure 47. ) Fig. 47 (Gerºnardus, Fig. 14. )
:
(2) General 6ase: When the Special case given can not apply,
a more general rule is given. tated in symbols it is (after
= be - k, where c and a are the two
Eneström), -
ad Il º lº
#
C.
d
given fractions

(147)
-
- c. x b = bo., a fact observed by Gernardus, for he says,
d 3. ad - -
'Whenever one fraction is to be divided by another take the numer-
ator of the dividend into the denominator of the other and make the
number which is produced the numferator. Then take the denomina-
t
or of the dividend into the numerator of the other and What re-
sults make the denominator, and you have the fraction resulting.' *
|| Gernardus, p. 19. |
An example is given in Which letter's and lines are used
instead of numbers -
To divide any integer - b. by any other integer - a . , b
being larger, one has ordinary division of integers with the remain-
der expressed in the form of a fraction.
(b) Division of sexagesimal fractions.
Reduce dividend and divisor to their lowest respective de-
nominations, and then divide the result of the first by that of the
second. The quotient will be as far from integers as is given by
the difference of the distances of the dividend and divisor from
integers.
Ti' there be a remainder it may be multiplied by 60, reduc-
ing it to the next lower place toward the left, and division per-
formed again. This may be repeated if there be a remainder stiii.
If the dividend is nearer to the integers than is the
divisor, it should be multiplied successively by 60 until it is
removed 3. S far from integers as is the divisor, and then division
performed.
Example: . c. fourths divided by . a. • 3 e a C+.
minutes, if .. c. contains - a . exactly - b.
times, is , b, thirds. If .c. contains - a . . b. times With a remain-
(148) -
der of , d. multiply - d. by 60 and divide .600. by . a. obtaining
.g., which is fourths. (Fig. 48). - - - -
XVTT. TEN AT-BANNA. - -
|| Atti dell'Accadenie Pontificia denuovi Lince i, xv.11,
p. 310. | -
Rule: 'The operation consists in this that you multiply
the numerator of each fraction by the denominator of the other, and
that you divide the result of the dividend by the result of the di-
visor, or that you denominate the first by the second. When the
denominators of the two fractions are equal, you divide the numer-
ator, or else denominate them the one by the other. (When the num-
erators are equal, divide the denominator of the divisor by the de-
nominator of the dividend, or else denominate it by this last. ). "
No examples are given. *
t
- º - -
| º (ibid, Vol. XII, p. 272, gives this example:
5 3 -
7 4; that is divide (3 + 3 of l) by (2 - 6 of l). Redic-
6 2 4. 7 4. 5 7 5
7 5 -
ing dividend and divisor to one fraction each, and apply-
ing the rule given above by Al-Banna, which is, in effect,
the same as Al-Kalsadi's, one obtains 91.0 divided by 560
(= 7.8.10), or successively by 7, 8, and 10. This gives
1 - 6 2 of 1, written o 2 g . . . .
T Tö - # "Tâ’ #–E–rs l
XVIII. LEVIT BEN GERSON. 5
| pp. 98, loo-104. | |
(a) Division of common fractions.
To divide a whole number by a whole number, 'divide as far
º - cº- - * +
as you can and represent the remainder as so many parts of the
divisor. "

(149)
To divide a Whole number With a fraction by a Whole number
With a fraction, reduce both dividend and divisor to a common de-
nominator , and then divide the numerator of the dividend by that,
of the divisor. -
(b) Division of Sexagesimal fractions.
Stated briefly the rule is: Write the dividend above the
divisor, leaving a space between them for the quotient, as shown in
figure 49, where 700 integers,
- - - 4. 28
40' 50" are divided by 9 integei 2 52 29
40 12 3C
2O" O'' 30" " . Consider as units 47 20 15 - Series of
700 40 50 dividend
the last column (on the left). 75 4. 18 Series of result
9 2O O 30 Series of
Let the number of each column divisor
- 653 2O 35
be considered increased by l., 46 40 2 39
- 36 1 20 2
Here one would have 21 in the 2 42 6 ()
- Fig. 49. (Lewi ben Gerschon, p. 100)
column of primes, and 10 in the -
column of integers. Divide the last column ºf the dividend by this
number and place the quotient in its proper Column of the result,
series. Here the result is first 70 and then later 5. Multiplying
the divisor by 70, the first partial quotient, the product series is
653, 20 35. Place this series below as shown, and then subtract it
from the dividend above. The remainder series is 47. 20' 15". This
constitutes a new dividend. Dividying it further, one obtains 5
units. Repeating the steps just taken above , the next remainder
series is 40' 12" so". Proceeding as before the next quotient is
4 primes, since primes divided by units gives primes. In general, if
the division can not be performed, reduce the number of the last col-
umn of the dividend to a number of the column nearest to the right
and then divide • But if division can not yet be performed, repeat

(150)
the reduction just taken until a number larger than the divisor
is obtained. Place the quotient in that column of the result
series such that its distance from the units place is equal to the
distance of its dividend from the divisor.
XTX. PIANUDES e “
| pp. 35-38. I
The method of dividing sexagesimals is the method used
| See also Gernardus, p. 147||
Example : Divide 3 degrees, 23 minutes, and 54 seconds, by
2 degrees, 34 minutes and 24 seconds. Reducing dividend and divisor
to seconds, one has lºg 34 seconds divided by 92.64 T3 23 54.
2 34 24
- -
seconds, or 1 and 2970 seconds remaining. The l 12234 it.
9264 -
is called degree, for seconds by degrees give 2970
- 173200 L9
seconds. Reducing 2970 seconds to thirds, and 9.264
- 2134
dividing as before , one has 19 minutes and 2134 13104-0 14.
92.64.
thirds remaining. Treating this remainder as Fig. 50. (*::::
p. 35.
before, one obtains next 14 seconds and 13444 fourths remaining,
which may be treated as before, and so on indefinitely.
XXIT. JOHANNES DE ºunts.
Rule: Two fractions having been given for division, the one
by the other, take the denominator of the divisor into the numerator
of the dividend; What results make the numerator. Next take the
denominator of the dividend into the numérator of the divisor; what
results make the denominator, and the division is made •
Examples: Applying the rule given, de Muris divides 1 by l
2 3
and obtains 3, 4 by 2 and obtains 12 divided by 2 or 6, 2 by 4 and
2 . 3 - - 3
obtains divided by lº or l-
2
3
3 6

(151)
XXI. LIVERTUS. *
| r. 25. I
(a) Division of common fractions.
To divide common ractions, divide the numerator and de-
nominator of the dividend by the numerator and denominator respect-
ively of the divisor; for example, 6 divided by 3 - 6 - 3 - 2.
- - 35 7 35 + 7 5
"But since We can not always thus precisely divide, there-
fore another general rule is given,' which, stated briefly, is:
invert the terms of the divisor and proceed as in multiplication.
Fractions are divided by integers, and integers are divided
by integer's just as one divides them today 3 for example, if 23
loaves are divided by 5 men each man will receive 4 loaves and 3 of
- 5
one loaf's
(b) Division of sexagesimal fractions.
To divide sexagesimal fractions, reduce the fractions of the
dividend and divisor respectively to the same denomination and divide
as in integers, denominating the result by the order which is as far
from the place of integers as the divisor is from the dividend; for
bw -
example, sixths divided, fo rths gives seconds, and fourths by fourths
give integers.
The author assumes the order of the divisor to be equal
or x lower than that of the dividends
XXV. VON GMUNDEN*
| | f f. 51-52. -
Sexagesimal fractions of different denominations are divided
.*.*.*… *
XXVI. PEURBACH. *
|| ºf. 46-47. ||

(152)
When the dividend is exactly divisible by the divisor,
divide the numerator and denominator of the dividend by the num-
erator and denominator respectively of the divisors
otherwise invert the terms of the divisor and multiply
the fractions. -
Integers are regarded as fractions, whose denominators
are unity. Whole numbers With fractions are reduced to improper
fractions, and treated as above.
No examples are given.
XXVTIT. AL-KALSADT. *
|| Atti dell'Accademia Pontificia de'Nuovº Lincel, 1859, XII,
p. 272. ||
*
Seº-Banna, p - 148 °


CHAPTER VII.
conclusion
Ancient civilizations developed three different systems of
fractions: unit, fractions, sexagesimal fractions and Roman frac -
tions. The first system was developed by the Egyptians and was
already in general use among them. When the scribe Ahmes produced
his famous arithmetic preserved in the Ahmes papyrus, written
-
about 1700 B.C. This system involves the use of fractions with a
constant numerºat or of l. The second system was developed by the
Babylonians, and was probably in use 2000 s.c. It deals with
fractions with denominators of 60 and powers of 60 * The third
system, of which more will be said later” was developed by the
pp. 156-157.
Romans.
---
Unit fractions passed from the Egyptians to the Greeks,
Whose notation enabled them to express any common reaction, unit,
or non-unit The sexagesimal system spread from Babylon to various
centers of India; and 3:..., being adopted by the great astronomer
Hipparchus of Rhodes (between 200 and loo B.C.), came into general
use in Greece. The treatment of fractions by Ahmes includes prob-
lems involving the four fundamental operations, but no systematic
discussion of these operations is given; nor are any definitions
given. The same may be said of the Hindu works. However, all of
the Hindu authors studied show considerable ingenuity and in some
instances, some entirely modern methods. * Unit fractions pre-
º
|| For example p. 133 ||
a
dominate • A mixed number a b is Written b : a fraction b, b; and an
C. : C 3 &
integer a , a . The Greeks defined their terms, developed the ir sub-
- l - -
ject more methodically and, in general, were much more scientific •

(154)
Along with the rise of Bagdad to its high intellectual
position in the 8th century A. D. came the Arabic acquisition of the
mathematical knowledge of almost the Whole civilized World. From
all centers of Learning scholars were called to the courts of the
Caliphs, where the great Works of Euclid, Archimedes, Ptolemy, Bra-
hmagupta and other mathematicians and astronomers of Greece and
Thdia were studied zealously. In particular, there came , to the
court of Al-Mamun, in the first, half of the 9th century, Mohammed ibn
Musa Al-Khowarizmi, who wrote ‘treatises on arithmetic, the sundial,
i
the astrolabe, chronology, geometry and algebra." In his arithme-
| Smith-Karpinski p. 97.
tic, he treats both sexagesimal and common fractions, defines the
divisions and subdivisions of the sexagesimal unit, gives rules of
operation, and follows a definite order: multiplication, division,
arrangement, addition, subtraction, duplation and mediation of sexa-
gesimals and multiplication of common fractions. He probably treatº
next in order division of common fractions and extraction of square
root * * He gives prefereº to sexagesimals, for he treats them &
- ºgº.º.º.ºrºſiº’s
| She-Woº-ººººººººººeºevºdºntºy-missing: pp. |
first and more fully and employs them in multiplying common frºac-
tions. Al-Khowarizmi specifically refers to the Hindus, * but as
| pp. 32,102 . . .
#
Brahmagupta does not treat sexagesimals, it is clear that Al-Khow-
| p. 31 , ||
arizmi obtained his knowledge of them through other sources than
tº a ſº ºn. º - ºf
through Brahmaguptº - º ** tºº. alº (a / & º ſu. º º
The works of Al-Nasawi (c. 1030), Al-Karkhi (c. 1010-1016),
Jº -
Al-Hassar ( 12th century), Al-Banna (13th century), *




(155)
(d. 1486), prominent Arabic mathematicians of succeeding centuries,
are very similar. Their forms are Egyptian and Arabic, but their
methods are Greek as well, and not far from modern. Only Al-NasaWi
and Al-Karkhi treat sexagesimals, but they , too, give preference to
common fractions. Al-Nasawi, in his treatise , entitled the Satis-
factory. One, makes a conscious effort at simplification, and, accord-
ingly, produces a very modern Work. The order of his operations a
Which is in itself modern, is addition, subtraction, multiplication,
division, and extraction of square and of cube roots. Al-Karkhi
really treats only multiplication and division of fractions – common
and sexagesimal - for his only exception is a very brief treatment ºf
-
of addition of common fractions incidental to and in the midst of
their multiplication. His numerous examples in common fractions
show a great preponderance of unit fractions. The treatments of
* - - -
Al-Banna and Al-Kalsadi are essentially the same. The
A
separate treatments of 'denomination, ' ' restoration', 'abasement."
Al-Hassar,
and 'transformation' as found in them were found in no other works.
The Hebrew author’s, ºan ben Ezra and Lewi ben Gerson,
are well-informed and original writers. Ben Ezra gives careful
introductions both to his treatment of common fractions and to that,
of sexagesimals; he uses unit fractions frequently, he solves first
degree equations of the type and after the manner of Annes, he spec-
ifically mentions Archimedes and Ptolemy and their works, he follows
the Arabic custom of reducing a fraction by dividing the numerator
successively by the prime º of the denominator, he multiplies
sexagesimals precisely as Al-Karkhi does - in short, he reflects the
influence of the Hindu, of the Egyptian, of the Greek, and of the
Arabic - Yet he uses many methods that are quite modern. The Work


(156)
of Ben Gerson, by the omission of introductory remarks and by the iaci.
of a treatment of certain operations, * leaves an impression of incom-
| see p. 53. | -
ple teness. His methods and his solutions of examples resemble in
part those of Al-Karkhi" and of Ben Ezra. * But he is the first of
- | Compare example (5), p. 129 with example (1) p. 105; also ,
rule bottom p. 128 with rule p. 106 (d); the methods of
multiplying sexagesimals are the same. |
|| Compare Example (1) p. 128 with example (1) p. 117 (d); Com-
pare , also, the steps in the reduction of the result in (4)
p. 128 With that of (2), p. 78; compare , also, their methods
of multiplying sexegesimals. | | -
the authors studied who divides Sexagesimals Without first reducing
the dividend and divisor to the same order, or te the lowest order
in them respectively.” The arrangement of the Work in multiplying
|| Fig. 49 and solution. p. 149. |
a sºlº and his manner of applying cancellation seem to be
|ºs, Figure 38, p. 129. | | | See example (3), p. 128 ||
original, also.
The three chief sources through Which Europe obtained her
fractions are the Latin Works on the computation with the abacus,
the arithmetics of Al-Khowarizmi and other Arabs, and the Liber
Abaci Öf Leonard of Pisa. The Roman fractions Were probable devel- Y
oped before the 2nd century B. C. The unit of the system is the
- - . . ºcº ſº.
uncia, or 1. While th? appeared in the Works on the abacus reck-
-- Tº
oning until the 12th century, still, of the works examined, only
the arithmetic of Raoul de Laon (first half of the 12th century) em-
ploys them. Laon employs the abacus, but uses With it the Hindu- .
(157)
ow!
º, translations; With the exception of certain excerpts pp. 93-136,
Arabic numerals • The conclusion seems to be justified that, from
- a ºn ſº sº. º Tºº Cºe º ſº
the twelfth century º the Roman fractionis rapidly disappeared
from the instruction in arithmetic in Europe .
The arithmetic of Al-Khowarizmi found its way to Europe
following the conquest of Spain by the Moors and, in the l2th
century, was translated into Latin. The name of the translator
is unknown, but the translation itself is now accepted as being
--- *- -º-º-
the Latin work entitled the Algoritimi de numero Indorum. The
r − i = − - - -
treatment of fractions contained in it - and perhaps in other
Arabic arithmetics now lost - furnished the basis for many sub-
sequent discussions of fractions,
Of these those ºf the Liber algorismi de pratica arisme-
ºrice ascribed to Joannis Hispalensis (or Gerard of Cremona), and
of the anonymous 12th century algorism described by Maximilian
Curtze are important illustrations. The Algoritmi de numero In-
dorum employs Roman numerals and the language of the abacus, con-
tains many errors and, in general, appears to be one of first
-
º - N.
ſº
the Liber algorismi is a more complete and finished work; while the
third work, intended to prepare students for the study of astronomy,
º -
ſis only a summarized treatment. Although the author of the second
work had before him some other Arabic Work, or Works, he certainly
numero Indorum, or its original
- N
the third work is based directly or indirectly on the first and
seems to have used the Algoritimi º
second, probably only upon the first. These conclusions are justi-
fied by the following reasons: (1) The same order of treatment is
common to all three works; (2) the language of the first two is
similar in many places; (3) many numerical illustrations are

(158)
common to all three Works ; (4) a certain error made in the first Work
second Work specifically mentions t,
|| see p. iia . ||
perpetuated in the se cond and third, and (5) the author of the
alcorismus. *
(1) The general order* of t, reat, ment, of the three Works is
multiplication, division, aprangement,
addition , subtraction , du-
plation, and mediation of sexagesimais, and multiplication and di-
vision of common fractions and extraction of square poot. In the
first Work the last two operations are not given, but it is evident,
that they Were once present .
In dupla
tion and mediation , all three
Works begin doubling at, the top and halving at, thc bottom, the given
integer and fractions being arranged in a column.
-t-
-
(2) As an illustration of
following passages are Worth' citing:
Scito quare cum volueris dividore
--_
-* -
numerum cum fractione aliquem num-
erum cum fractione ; uel numerum cum
-
- -
fractione per numerum integrum; aut.
_n
numerum integrum per numerum cum frac-
tione , facies utrumque numerum unius
the similarity in language the
cum numerum integrum cum
{ frac Lione per numerum integ-
rum cum fractione , aut numer-
i -
{um integrum tantum per numerum
integrum cum fractione , aut
numerum integrum cum fractione
-
generis, idest vertes utrosqug numer- | Per numerum integrum tantum,
s in inferiorem different, iam. aut fractiones tantum pcr frac-
- - - ® _s - - -
lGoritmi de numero indArum • p • 20 | tiones uolueris dividere, debes
** **$* * * * ***** …. Tt -
utrumque numerum dividentem,
t.
-
scilicet, dividendum ad infer-
ius genus fractionis, que fuerit.
in quolibet eorum reducere, ut.
[sit. uterque unius generis.
l Libe- algorismi de pratica
aEiETEtEIGENTE. 55. Γ

(159)
The rule on the right may be a revision of the one on the left.
(3) The following is a tabulation in order, in modern no-
tation, of the examples in fractions of the three works :
A. - B. C.
(Algoritimi de numero (Liber algorismi de (12th century algorism)
àndorum) pratica arismetrice) -
multiply - Multiply - Multiply -
(a) l l by l l; (a) l l by l l; (a) l l by l is
2 26 - 2 2 - 2 2
(b) 294.5' by 3°10' 30"; (b) 2°10' by tº 2' 30" (b) 2°45' by 3910' 30";
Divide – - Divide - Divide -
(c) 15 thirds by 6 (c) 15 thirds by 6 thirds;
thirds; O -
( (d) lºgo' by 29;
(3) 10" by 5' ; |(e) lo” by 5': (e) 10” by 5' ;
(f) 10' by C" ' -
Arange - Arºange - Arange -
(g) 13930'45" 50+V; |(g) 12°20' 45" 50”; (g) 1224030' 45°50'’;
Multiply –
(h( 84. 2
(
1.
)
º
lº
J
Divide -
(m) & Ll,l by 3+l 1
** if: # * -
Multiply - Multiply- Multiply-
(n) 3 by 4; (n) & 3 by 3 l; (n) g by 4 ;
7 g 11. 2 7 9
(o) 3 l by C 3: (n) 3 by 4: (o) & 1 by 8 3:
2 11. 7 9 2 Ll
Divide - Divide -
(g) 20 2 by 3 l l (q) 20 2 by 3 l
3 3 3 3
Multiply -
1.
(r) & by
3
#


(160)
Note that, with the exception of an obvious error in (g) of C and of
example (f), which is similar to (3), every example of A occurs in C
and , with the further exception of (b), in B; that example (b) of .
A and C differs but partially from (b) of B; that examples (n) and
(o) occur in A, B, and C, but that their order, and also the order
of the multiplicana and multiplier of (o) is interchanged in B; that
(p) of C and (q) of B and C is missing from A, but occupies a position
corresponding to the assumed last portion of A; that, down to the
point corresponding to the point where A suddenly breaks off, every
example of C occurs in A and in the same order; and that, examples
(h) to (m) inclusive and (r) of B, which do not occur in A and C,
are of Arabic and Egyptian form.
(4) Example (e) above is sived in detail in A but the re-
sult is given as 'two seconds', duo secunda, instead of two minutes.
In B and C, the solution is om Eted but the result is again given
as 'two seconds'. The perpetuation of this error is a particularly
ºtrong argument in support of the theory that neither B nor C were
º ------------- --
ºved from an original source, but from A.
- considering these facts together With [ 5) mentioned above ;
i.e. , that the author of B specifically mentions "alcorismus," the
conclusicn is Warranted that B is derived from A and from some other,
Arabic work or works, and that C is derived entirely from A as A.
stood originally *
Further arguments may be found in the fact that the example ſº
º'
in Square root in B and C support ºne theory advanced for the exam-
ples in fractions; and, also , that C ends precisely where A is assumed
to have ended. Were C based on B, Wholly or in part, it probably
Would have gone further than A. It may be that C is even older than B. -

(161)
Granting the above arguments, this question may now be
asked, did the author of the Liber algorismi make use of the
arithmetic of Al-Karkhi: The examples in fractions in the Liber
Anºmero Indorum are examples
in multiplication and division, in fractions, the arithmetic of
algorismi not found in the Algoritmi
Al-Karkhi treats only multiplication and division. The methods
of solving them are similar in some respects but they are dissi-
milar in other respects. Also, Al-Karkhi uses no equivalent for
the numerous technical terms found only in the Liber Algorismi. *
| pp. 112–11z ||
The form of some of the examples of the Liber, Algorismi suggests
those of the arithmetic of Al-Karkhi, yet none of them are dupli-
cated in it, as would have been true had the work of Al-Karkhi been
used since the author of the Liber algorismi obviously aimed only
to translate and Weave together material already in existence.
Neither are the examples of the Liber algorismi as decidedly made
up of unit fractions as are those of Al-Karkhi's work. Therefore
the treatment of fractions of the arithmetic of Al-Karkhi was not
directly, and probably not indirectly used in constructing the
treatment of the Liber Algorismiſ
Tncidentally then an argument is here advanced in support
- - - *2.
of the belief that other important, but unpublished, Arabic treat-
ſº *. º ---- *2.
- - º - - - --- - w -
ºises on fractions were in existence in Europe during their period.
A third source through Which Europe received its fractions
is the Liber abaci of Leonard of Pisa, a work of importance in it-
self, but of little permanent effect. Leonard of Pisa was a
capable, practical, and Well-informed Writer, and produced a work
embodying the best Arabic arithmetic of his time. Leonard used
(162.)
º *
º
Euclid's methods-, and he reflected the Egyptian influence of his
| p. 144 | -
time * Moreover, he wrote at a time when the West was responding to
| pp. º . . . -
the touch of, East. But, his excellent Work exerted no very great.
influence, and, of the Works examined, only that of Joannes de Muris
shows any trace of the Liber, abaci. "This may have been because the
Western World Was not ready for so a dvanced a treatise, or because -
the monks, copying from monastery to monastery spread the other type
of algorism." "
|| Benedict, p. 120.
Many algorisms of the 13th century, among them the very
popular and important algorisms of Alexander de Villa Dei and of
John of sackobosco, the Carmen de Algorismo and the Algorismus vul-
garis, do not treat fractions at all, probably because only the
commercial classes and the astronomers employed fractions and the
scholars who studied arithmetic as a liberal art found operations
with integral numbers sufficient.
º -
Conspicuous except ions are the algorisms of Jordanus
Nemorarius (d. 1237) and of Gernardus, the latter of which is based
ºn the former. These algorisms are very abstract and theoretical,
hence are tedious and difficult to read. But they present very
*
and
thorough and systematic treatments of operations, both with integers
and with fractions, and contain a very great variety of forms, and
methods, both medieval and modern. Jordanus uses the phrases
fractiones eiusdem generis and fractiones diversorum generum; sim-
ilarly, the Liber algorismi has fractiones alterius denominationis
and an anonymous 12th century wºrk (XT) * minutiae diversorum generum
46.37, 33 º ****
Compare pp. 46 , ºf , of º -


(163)
- the
Gernardus frequently Writes a for a 3 in the same manner/Hindus -
- - b º -
|| Compare pp. 50, 36 - || 2
write their fractions. Jordanus writes 10 for 2° 10' 32". 52"'; in
32 -
- 52 12 - -
the same manner, Al-Khowarismi” writes 30 for 12° 30' 45" 50+V.
- 45 -
OO
| compare pp. 45,67-68,76,77. || 50
gernardus, in dividing sexagesimals, multiplies the successive re-
mainders by 60, and divides the corresponding products by the given
divisor as before; in the same manner theons divides sexagesimals.
| Compare pp. 147,23-24 º || -
Gernardus explicitly defines his terms, and, using letters and lines
to represent numbers, gives rigid proofs of his propositions after
the manner of Euclid, to whom he refers frequently. Jordanus and
Gernardus use many technical terms 4 that are common to much older.
| pp. 46, 50 - ||
Writers and many that are new. They are the first of the authors
examined to call sexagesimal fractions fractiones phisice ; their
-re
definitions of terms are used by Liverius, de Muris, Von Gmunden,
5
Peurbach and Exxxxxxix others; and Liverius (and after him Peur bach)
| compare pp. 49,54, 55,60-61,61-62.
follo Ws. ºrnardus in his method of dividing a by c where a = fo
and b = ga, proposes ſo tº division e by 3 and *.*.* 6 * 3 * 6
35 7 - 37 ± 7
|| Compare pp. 146, 191 . |
7
Even today many of their methods are still in use .
| See synopäis, p. 48. ||
other treatises studied offer nothing really news in general,
they are modern in arrangement of material and in methods employed.
Some of them show individual peculiarities: Planudes (c. 1337) treats

(164)
only sexagesimals and gives a careful introduction suggestive of
that of Ben Ezra; Liverius (c. 1300-1350) treats both common and
sexagesimal fractions, calling the latter fractiones phisice, and
in addition to the special case just mentioned above for dividing
a fraction by a fraction, uses the general case ºf inverting the
divisor and multiplying; Joannes de Muris (c. 1310-1360) treats
systematically only common fractions, employs 'multiplication in a
cruss' common to later writers, uses an unusual number of technical
terms many of Which are his own, possibly º the fact that part
of his work is in verse; John Killingworth (d. 1445), an English
astronomer and mathematician of ºº: only Sexagesimals,
includes a number of tables and employs the check by 59; Von Gmunden
(c. 1380–1442) treats only sexagesimals and makes a futile attempt
to introduce greater uniformity by dividing the circle into 6 signs,
"physical signs' , of 60 degrees each in place of 12 signs, common
signs, of 30 degrees each and a day into 60 hours instead of 24
hours; Peurbach (1423-1461), probably influenced by Joannes de
Muris, divides fractions into 'primary' and 'secondary'; and
~ - - -
Sibt El-Miradini checks sexagesimals by 7 and 8, and gives illus-
trations in their division in Which the quotients are periodic,
RULES coverING USE OF MANUSCRIPT THESEs.
IN THE UNIVERSITY OF MICHIGAN LIBRARY -
AND THE GRADUATE SCHOOL OFFICE 3.
Unpublished theses submitted for the doctor's degrees and deposited in the
University of Michigan Library and in the Office of the Graduate School are open
for inspection, but are to be used only with due regard to the rights of the authors.
For this reason it is necessary to require that a manuscript thesis be read within
the Library or the Office of the Graduate School. If the thesis is borrowed by
another Library, the same rules should be observed by it. Bibliographical refer-
ences may be noted, but passages may be copied only with the permission of the ,
authors, and proper credit must be given in subsequent written or published work.
Extensive copying or publication of the thesis in whole or in part must have the
consent of the author as well as of the Dean of the Graduate School.
This thesis by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
has been used by the following persons, whose signatures attest their acceptance
of the above restrictions.
A Library which borrows this thesis for use by its readers is expected to secure
the signature of each user.
NAME AND ADDRESS , DATE
UNIVERSITYQ.Māº -
§§§§76403384
º
º
sº
º
ºº
º º
º
º -
- º
º
º
º
-
- **
- - -
º -
º
- º
º
* º
- …
-
º º
- - -