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HUMANISTIC SERIES 

VOLUME XI 



CONTRIBUTIONS TO THE HISTORY 

OF SCIENCE 



PART I. ROBERT OF CHESTER'S LATIN 
TRANSLATION OF THE ALGEBRA OF 
AL-KHOWARIZMI 



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Plate I. 






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Codex Vindoboneksis 4770, Fol. l a . 



ROBERT OF CHESTER'S 

LATIN TRANSLATION 



OF THE 



ALGEBRA OF AL-KHOWARIZMI 



WITH AN INTRODUCTION, CRITICAL NOTES 
AND AN ENGLISH VERSION 

BY 

LOUIS CHARLES KARPINSKI 

UNIVERSITY OF MICHIGAN 



Neto gork 
THE MACMILLAN COMPANY 

LONDON : MACMILLAN AND COMPANY LIMITED 

I915 

jill rights reserved 




OCT i 4 KISO 

Sf7 J* 



Copyright, 1915, 
By FRANCIS W. KELSEY, EDITOR. 



Set up and electrotyped. Published December, 1915. 



J. S. Gushing Co. — Berwick & Smith Co. 
Norwood, Mass., U.S.A. 



PREFACE 

From the point of view of the history of science, no justification 
is needed for the publication of a mathematical text of the twelfth 
century, for the available material representing this period is 
meagre. A wider acquaintance with Robert of Chester's Latin 
translation of Al-Khowarizmi's Arabic treatise on algebra will per- 
haps contribute to a more just estimate of the services rendered to 
science by the Arabs. In the English version I have not at- 
tempted to give a literal translation of the Latin, but rather to 
express the thought in a phraseology which the modern student of 
mathematics will find easy of comprehension ; by consulting the 
Latin text and footnotes the reader will be able to examine Robert 
of Chester's own words. For the convenience of readers interested 
in the text I have added a Latin Glossary in which are noted 
many variations from the usage of classical writers. In the 
Introduction I have presented a study of the significance of the 
treatise in the history of mathematics, and a description of the 
manuscripts upon which the text is based. 

It is a pleasure to express indebtedness to Professor David 
Eugene Smith for having suggested the work ; to Mr. George A. 
Plimpton for the generous use of his unique mathematical library; 
to the librarians of the libraries of Vienna and Dresden for photo- 
graphic reproductions of manuscripts containing this text ; to the 
librarian of Columbia University for the loan of the Scheybl man- 
uscript, and to the librarian of the Cleveland Public Library 
for the use of works from the John G. White collection. I am 
much indebted also to colleagues of the University of Michigan, 
particularly in the Department of Latin and the Department of 
Mathematics. 

I am under special obligation to Mr. William H. Murphy for 
making possible this publication. 

LOUIS C. KARPINSKI. 

Ann Arbor, Michigan, 
November I, 1915. 



CONTENTS 



PAGE 

INTRODUCTION: 

I. Algebraic Analysis before Al-Khowarizmi i 

II. Al-Khowarizmi and his Treatise on Algebra .... 13 

III. Robert of Chester and Other Translators of Arabic into Latin 23 

IV. The Influence of Al-Khowarizmi's Algebra upon the Develop- 

ment of Mathematics 33 

V. The Arabic Text and the Translations of Al-Khowarizmi's 

Algebra 42 

VI. Preface and Additions found in the Arabic Text of Al-Khowa 

rizmi's Algebra 45 

VII. Manuscripts of Robert of Chester's Translation of Al- 
Khowarizmi's Algebra 49 

ABBREVIATIONS 65 

LATIN TEXT WITH CRITICAL NOTES: 

Liber Algebrae et Almucabola (Even Pages) . . . . 66 ff. 

Regule 6 Capitulis Algabre Correspondentes 126 

Addita Quaedam pro Declaratione Algebrae (Even Pages) . 128 ff. 

ENGLISH VERSION WITH EXPLANATORY NOTES: 

The Book of Algebra and Almucabola (Odd Pages) . . . 67 ff. 

Rules Corresponding to the Six Chapters of Algebra . . .127 

Some Additions in Explanation of the Algebra (Odd Pages) . 129 ff. 

LATIN GLOSSARY 159 



PLATES 

I. Codex Vindobonensis, 4770, Folio i° Frontispiece 

(Text, p. 66, I. 8 In . . . p. 68, 1. 19 substantiarum) 

FOLLOWING PAGE 

II. Codex Dresdensis, C. 80, Folio 340^ 40 

(Text, p. 68, 1. 28 Eodem . . . p. 72, 1. 12 radicum unius) 

III. Codex Dresdensis, C. 80, Folio 342° 48 

(Text, p. 78, 1. 7 applicantur ... p. 80, 1. 21 mul\tiplicatione~\') 

IV. Columbia University Manuscript, X 512, Sch. 2, Q, Page 93 . .64 

(Text, p. 90, 1. 9 [propd\sitae ... 1. 23 diminuendo) 

vii 



./if 



INTRODUCTION 

CHAPTER I 

Algebraic Analysis before Al-Khowarizmi 

Arabic contributions to science have, in the past, been some- 
what neglected by historians. More recent studies are recogniz- 
ing our indebtedness to Mohammedan scholars, who kept the 
embers of learning aglow while Europe was in the darkness of 
the Middle Ages. Much of our knowledge of Greek mathematics 
comes to us from Arabic sources ; the early Latin versions were 
frequently based upon Arabic texts rather than the Greek origi- 
nals. Similarly, Hindu arithmetic and astronomy were trans- 
mitted to Europe by Islam. The services of the Arabs to science 
were not limited to the preservation and transmission of the 
learning of other nations. They made independent contributions 
in many fields. 

Among these achievements is the Arabic algebra of Al-Khow- 
arizmi, which for centuries enjoyed wide popularity in the origi- 
nal, and for further centuries extended its popularity through 
translations and adaptations. A study of the content of this 
work is an excursion into mediaeval thought. By a study of the 
text in a form as nearly like the original as possible, we discover 
the reason for its long-continued appeal to the Occidental as well 
as the Oriental mind, its interest for the Englishman, the Ger- 
man, and the Italian, as well as for the Arab. Even to-day 
teachers of elementary mathematics may find this book fruitful 
in suggestion : the geometric solutions of quadratic equations 
presented by the Arabic writer more than a thousand years ago 
may be used with profit in our classrooms. 

Simple equations of the first degree in one unknown, of the 
type ax=b, are found in the oldest mathematical text-book which 
we possess, the Ahmes papyrus of about 1700 B.C., which was 
published with a German translation by Eisenlohr. 1 This Egyp- 

1 A. Eisenlohr, Ein mathematisches Handbuch der alien Aegypter, Leipzig, 1877; 
Facsimile of the Rhind Mathematical Papyrus in the British Museum, with preface by 
E. A. Wallis Budge, London, 1898. 

1 



2 INTRODUCTION 

tian document presents not only first degree equations together 
with symbols for the unknown quantity and for the operations of 
addition and subtraction, but also shows traces of a study of 
simultaneous linear equations some two thousand years before the 
Christian Era. Later, but still before the golden age of Greek 
mathematics, the quadratic equation appears in Egypt. The 
problems found involve simultaneous quadratic equations, thus: l 
" Another example of the distribution of a given area into 
squares. If you are told to distribute ioo square ells (units of 
area) over two squares so that the side of one shall be f of the 
side of the other: please give me each of the unknowns." The 
solution follows by assuming the side of one square to be unity, 
and the other Jk The sum of these areas is ff, of which the root 
is f. The root of ioo is 10 ; 10, then, is to the required side as f 
is to i, whence one side is 8 and the other 6. The algebraical 
equivalent of this geometrical problem is, evidently, 

x 2 +y 2 = ioo, 
y=\x. 

Noteworthy also is the fact that a symbol for square root occurs 
in the discussion of these problems. 

The solution above leads to the number relation, 6 2 + 8 2 =io 2 , 
which connects directly with the simpler form, 3 2 + 4 2 =5 2 , and to 
the same relation other problems of this kind reduce. 2 This 
makes connection, of course, with the so-called Pythagorean theo- 
rem that the sum of the squares on the sides of a right triangle 
equals the square on the hypotenuse. Even though the Egyp- 
tians had no logical proof for this proposition, their familiarity 
with it is well established. In the time of Plato, and for some 
centuries afterwards, the Egyptians were famed as surveyors, and 
the principle stated seems to have been applied by them in laying 
out right angles by means of a long rope knotted at equal inter- 
vals. Two pegs situated three units apart are set out along the 
line to which it is desired to draw a perpendicular. From one peg 

1 M. Cantor, Vorlesungen iiber Geschichte der Mathematik, Vol. I, third edition (Leip- 
zig, 1907), pp. 95-96. To this work we shall refer as Cantor, I (3), and to the other vol- 
umes similarly. 

See also Max Simon, Geschichte der Mathematik im Altertum (Berlin, 1909), pp. 41- 
42; Schack-Schackenburg, Zeitschrift fur Aegyptische Sprache, Vol. XXXVIII (1900), 
pp. 135-140, and Vol. XL, pp. 65-66. 

2 Cantor, I (3), p. 96. 



ALGEBRAIC ANALYSIS BEFORE AL-KHOWARIZMI 3 

an arc is swung with a radius of four units, while from the other 
end an arc is swung with a radius of five units. The intersection 
is connected with the peg from which the shorter arc is swung, 
forming thus a right angle with the desired line, for in any tri- 
angle with sides in the ratio three to four to five, a right angle 
lies opposite the longest side. 

The Pythagorean theorem was applied also in India, before the 
time of Pythagoras, in the construction of altars. With this 
theorem as developed in the Apastamba Sulba Sutras, 1 the rules 
for altar construction, are associated careful approximations of 
square root, pure quadratic equations, and even, as Milhaud has 
shown, 2 the possible solution of the complete quadratic equation, 

ax 2 + bx=c. 

The ancient Babylonians, furthermore, constructed tables of 
squares and cubes. Such tables are found upon the famous 
tablets of Senkereh, 3 which are contemporary with the Ahmes 
papyrus. Application of these quadratic numbers to problems 
similar to those of Egypt already mentioned has not been dis- 
covered, but the fact is evident that such tables were a step toward 
the study of quadratic equations. Cantor 4 shows that the ancient 
Hebrews were probably familiar with the 3, 4, 5 right triangle. In 
China, too, students mathematically inclined had come upon this 
number relation, 5 and evidently were studying quadratic numbers. 

Familiarity of Greek mathematicians with the geometrical 
solution of quadratic equations in the time of Pythagoras is now 
well established. 6 Hippocrates (fifth century b.c.) writing on the 
quadrature of the lunes, in an attempt to square the circle, 
assumes a construction which is equivalent to the solution of the 
equation, 7 ,_ 

x 2 + \ 2 ax = a 2 . 
— > 

1 Biirk, Das Apastamba-Sulba-Sutra, Zeitschrift der deutscheti Morgenl'dndischen 
Gesellschaft, Vol. LV (1901), pp. 543-591, and Vol. LVI (1902), pp. 327-391. 

2 G. Milhaud, La Geometrie d'' Apastamba, Revue getierale des Sciences, Vol. XXI (1910)5 
pp. 512-520; see also T. L. Heath, The thirteen books of Euclid \r Elements (3 vols., Cam- 
bridge, 1908). Vol. I, pp. 352-364. We shall refer to this latter work as Heath's Euclid. 

3 Cantor, I (3), pp. 25-31. 4 Cantor, I (3), p. 49. 

5 Cantor, I (3), pp. 181 and 679-680. 6 Heath's Euclid, Vol. I, pp. 386-387, 403. 

7 T. L. Heath, Diophantus of Alexandria, A study in the history of Greek algebra, 
second edition (Cambridge, 1910), p. 63 ; more detailed in Rudio, Der Bericht des Sim- 
plicius uber die Quadraturen des Antiphon und des Hippokrates (1907), p. 58, and same 
author in the Bibliotheca Mathetnatica, Vol. Ill, third series (1902), pp. 7-42. 



4 INTRODUCTION 

Several propositions of Euclid present geometrical equivalents 
of the solution of various types of quadratic equations, not involv- 
ing negative coefficients, and further study of similar problems 
appears in Euclid's Data. Of this nature are the fifth, sixth, and 
eleventh propositions of the second book of the Elements and the 
twenty-seventh, twenty-eighth, and twenty-ninth of the sixth book, 
and problems 84, 85, 86, and others of the Data} Problem 84, 
for example, reads : 

" If two straight lines include a given area in a given angle, 
and the excess of the greater over the less is given, then each of 
them is given." 

This corresponds to the equations : 

xy = k 2 

x—y = a. 

The two following problems (85 and 86) correspond to the 
simultaneous quadratic equations : 

xy = k 2 , 

x +y = a, 

and xy = k 2 , 

x 2 —y 2 = a 2 . 

The eleventh proposition of the second book of the Elements 
furnishes the solution of the equation 

x 2 + ax = a 2 
or even more general, 

x 2 + ax = b 2 . 

As this so well illustrates the geometrical solution, it is given in 
full, following Heath's Euclid. 

Book II of the Elements of Euclid, Proposition ii 

" To cut a given straight line so that the rectangle contained by the whole and 
one of the segments is equal to the square on the remaining segment. 

" Let AB be the given straight line ; thus it is required to cut AB so that the 
rectangle contained by the whole and one of the segments is equal to the square on 
the remaining segment. 

1 References and citations from the Elements are to Heath's Euclid and the Data (Greek 
and Latin) edited by H. Menge, Leipzig, 1896, being Vol. VI of Euclidis opera omnia, ed. 
Heiberg et Menge. An English translation of the Data is found in the numerous editions 
of The Elements of Euclid 'by Simson. The numbering of the problems is slightly differ- 
ent in the two versions. 



ALGEBRAIC ANALYSIS BEFORE AL-KHOWARIZMI 



" For let the square ABDC be described on AB (I. 46) ; let AC be bisected at 
the point E, and let BE be joined ; let CA be drawn through to F, and let EF be 
made equal to BE ; let the square FH be described on AF, and let GH be drawn 
through to K. 

" I say that AB has been cut at H so as to make the rectangle contained by AB, 
BH equal to the square on AH. 

" For, since the straight line AC has been bisected at E, and FA is added to it, 
the rectangle contained by CF, FA together with the square on AE is equal to the 
square on EF. (II. 6.) 

" But EF is equal to EB ; therefore the rectangle CF, FA together with the 
square on AE is equal to the square on EB. 

" But the squares on BA, AE are equal to the 
square on EB, for the angle at A is right (I. 47) ; 
therefore the rectangle CF, FA together with the 
square on AE is equal to the squares on BA, AE. 

" Let the square on AE be subtracted from 
each ; therefore the rectangle CF, FA which re- 
mains is equal to the square on AB. 

" Now the rectangle CF, FA is FK, for AF is 
equal to FG ; and the square on AB is AD ; 
therefore FK is equal to AD. 

" Let A K be subtracted from each ; therefore 
FH which remains is equal to HD. 

" And HD is the rectangle AB, BH, for AB is 
equal to BD ; and FH is the square on AH ; 
therefore the rectangle contained by AB, BH is 
equal to the square on HA. 

" Therefore the given straight line AB has been cut at H so as to make the 
rectangle contained by AB, BH equal to the square on HA. q. e. f." 




The ordinary algebraical solution of the corresponding equation 
x 2 + ax = a 2 , from a{a —x) = x 2 , 
parallels this geometrical demonstration. 



a' 



To complete the square in the left-hand member, — is added to 

4 
both members. This corresponds to marking the point E on the 



a- 



figure, for then the square on BE equals a 2 H — or AB 2 + AE 2 . 

4 
Extracting the square root of both members, we have, alge- 



x + 



a 



-w 



5* s 



braically, 

2*4 

the negative sign being disregarded. The right-hand member 
corresponds to the line BE and the left-hand member to EF, 
which is equal to BE. 



INTRODUCTION 
Algebraically we proceed by subtracting -- from both members, 

2 

5 a 2 a 



-^ 



giving x 

This corresponds to the line AF in the figure which is 
BE-AE. 

Analytical solution of the quadratic equation appears quite 
definitely in the works of Heron of Alexandria, who flourished 
about the beginning of the Christian Era. Heron states in effect 
that given the sum of two line segments and their product then 
each of the segments is known. 1 However, he goes farther than 
any work of Euclid in applying this to a numerical example, 

144 #(14 — X) = 6'J20. 

Without putting this into the form of an equation, Heron states 
that the approximate value of x is 8|, and this evidently indicates 
an analytical solution. The geometrical garb is absolutely dis- 
carded in a problem in the Geometry doubtfully attributed to 
Heron. 2 The problem is to compute the diameter of a circle 
given the sum of the area, the circumference, and the diameter, 
summing an area and lengths, entirely contrary to Greek usage. 
The form of the result, practically 

x= V(i54 x 212 + 840-29 ^ 
1 1 
indicates that the equation 

j^x 2 + 2 ^-x = 212 
was put in the form 

1 2 1 x 2 + 638 # = (212X1 54). 

Somewhat similar problems 3 in which lines and areas are 
summed appear in Greece in the period between Heron and 
Diophantus (about 250 a.d.) as well as in the works of the 
latter. One of these problems is to find a square whose area and 
perimeter together equal 896 (x 2 + 4x= 896). The solution pro- 

1 Heron, Metrica, ed. Schbne (Leipzig, 1903), pp. 148-151. 

2 Cantor, I (3), p. 405 ; Heron, Geometria, ed. Hultsch (Berlin, 1864), p. 133 ; Heronis 
Opera, ed. Heiberg, Vol. IV, Geometria. p. 381 ; Heath, Diophantus, pp. 63-64. 

3 Heiberg and Zeuthen, Ueber einige Aufgaben der unbestimmten Analytik, Bibliotheca 
Mathematica, Vol. VIII, third series, pp. 1 18-134. The date is conjectural. See Heath, 
Diophantus, pp. 118-121. 



ALGEBRAIC ANALYSIS BEFORE AL-KHOWARIZMI 7 

ceeds in the ordinary manner by adding to 896 the square of half 
the coefficient of x and then taking the square root of this sum. 
From this is subtracted one-half the coefficient of x, giving the 
side 28. Four other problems of this series deal with right tri- 
angles having rational sides and hypotenuse, in which the sum of 
the area and perimeter is to equal a given number. If a, b are 
the sides, c the hypotenuse, 6" the area, r radius of the inscribed 
circle, and s = \{a + b + c), then the solution depends upon the 
following formulas: 

S=rs = jj ab, r + s = a + b, c = s — r. 
a I _ r + s± V(r+i-) 2 -8 rs. 



In the sixth book of the Arithmetica Diophantus treats rational 
right triangles in which the area plus or minus one side is a 
given number, or the area plus or minus the sum of two sides or 
one side and the hypotenuse, is a given number. Again, such a 
problem appears in an algebraic work by Shoja ben Aslam, Abu 
Kamil, an Arabic writer of the tenth century. 1 

In respect to analysis Diophantus is the greatest name among 
the Greeks. Recently it has been established that he flourished 
in the third century of the Christian Era, when Greek supremacy 
in mathematics was waning. No doubt whatever exists that this 
Alexandrian was familiar with the analytical solution of the vari- 
ous forms of quadratic equations, neglecting negative roots and, of 
course, imaginary roots, which did not receive serious treatment 
for more than a millennium after Diophantus. The three types of 
complete quadratic equations, involving only positive coefficients, 
are the following : 

ax 2 + bx= c, 
ax 2 + c= bx, 
ax 2 = bx + c. 

All three types appear in the Arithmetica of Diophantus, not 
systematically treated but solved as incidental to the solution of 
other problems. In fact, after dealing with the solution of equa- 
tions of the form 

ax" 1 = bx n , 

1 Suter, Die Abhandlung des Abu Kamil ShogcC b. Aslam " liber das Funfeck und 
Zehneck" Bibliotheca Mathematical Vol. X, third series (1910-11), pp. 15-42. 



8 INTRODUCTION 

Diophantus makes the following explicit statement regarding his 
intention of writing a systematic treatise on the quadratic 
equation : 1 

" This should be the object aimed at in framing the hypotheses of propositions, 
that is to say, to reduce the equations, if possible, until one term is left equal to one 
term ; but I will show you later how, in the case also where two terms are left equal 
to one term, such a problem is solved." 

So far as we know this promise was never fulfilled. 
An equation of the first type is presented by the sixth problem 
of the sixth book, and this we reproduce from Heath : 2 

" 6. To find a right-angled triangle such that the area added to one of the per- 
pendiculars makes a given number. 

" Given number 7, triangle (3 x, 4 x, 5 x). 

"Therefore 6 x 2 + 3-x - = 7." 

" In order that this might be solved it would be necessary that (half coefficient 
of x) 2 + product of coefficient of x 2 and absolute term should be a square : but 
(ii) 2 + 6 • 7 is not a square. Hence we must find, to replace (3, 4, 5), a right-angled 
triangle such that 

" (^ one perpendicular) 2 + 7 times area = a square ; " 

and the subsequent work leads to the equation, 84.x 2 + 7 x= 7, x = \; 
and the solution (6, %, %£). 

In the following problem (VI. 7) the value of x is given as ^ for 

the equation 

84 x 2 — 7 x = 7, 

which equation is of the third type when the negative term is 
transposed after the manner of Diophantus. The equations 

6t ) ox 2 + 'J3x=6, 

630 x 2 — 73: x= 6, 

630^ + 81 x = 4, 

and 630 ^- 2 — 81 x = 4, 

occur in the next four problems (VI. 8-1 1). Another problem of 
the third kind (IV. 39) is of especial interest because the rule is 
given for solving this type : 3 

" To find three numbers such that the difference of the greatest and the middle 
has to the difference of the middle and the least a given ratio, and also the sum of 
any two is a square." 

1 Heath, Diophantus, p. 131. 

2 Heath, Diophantus, pp. 228-229. 

3 Heath, Diophantus, pp. 197-198. 



x = 


1 

18' 


X — 


6 
35' 


x= 


4 
10 5' 


x = 


1 
6' 



ALGEBRAIC ANALYSIS BEFORE AL-KHOWARIZMI 9 

The discussion leads to the inequality, 2 m 2 > 6 m+ 18, which, 
since only integral solutions are desired, explains the use of 7 as 
the approximate square root of 45, in the following paragraph : 

" When we solve such an equation, we multiply half the coefficient of x into itself, — 
this gives 9, — then multiply the coefficient of x 2 into the units, — 2 • 18 = 36, — add 
this last number to the 9, making 45, and take the side [square root] of 45, which 
is not less than 7 ; add half the coefficient of x, — making a number not less than 
10, — and divide the result by the coefficient of x 2 ; the result is not less than 5." 

Of the second type, ax 2 + c = bx, Diophantus gives several illus- 
trations, requiring frequently only the approximate value of the 
root. Problems of this kind are the following: 

72 m > 17 m 2 +17, m not greater than |^-, (V. 10) 

19//2 2 + 19 > 72 w, m not less than ^-|, (V. 10) 

w 2 + 6o > 22 m, m not less than 19, (V. 30) 

m 2 + 6o< 2\m, m not greater than 21, (V. 30) 

and 172.3:= 336. r 2 + 24, (VI. 22) 

of which the statement is made that the root is not rational, and in 
the same problem 

78848^ — 8432^+ 225 = 0, 

which has the rational root $*g. 

Commentaries on the Arithmetica began to appear very early. 
Probably the most interesting commentary from the modern point 
of view was the one written in the late fourth or early fifth cen- 
tury by Hypatia, the daughter of Theon of Alexandria. Unfortu- 
nately her writings are all lost, although there is ground for the 
belief x that some remarks made by Michael Psellus (eleventh 
century) concerning Egyptian arithmetic and algebra were based 
on her commentary. She came naturally by her mathematical 
ability ; her father Theon wrote a commentary on Ptolemy's 
Almagest and makes in this the earliest known reference to 
Diophantus. 

Cossali, 2 writing in 1797 on the history of algebra, conjectures 
that the step from the geometrical to the analytical form of solu- 
tion took place in the period between Euclid and Diophantus. 

1 Diophantus, ed. P. Tannery (Leipzig, 1893), Vol. II, pp. 37-38; Heath, Diophantus, 
pp. 18, 41. 

2 Origine, trasporto in Italia, primi progressi in essa delP algebra (Parma, 1797), Vol. 
I, pp. 87-91. 



io INTRODUCTION 

Now the Arabic Book of Chronicles l (987 a.d.) states that the 
astronomer Hipparchus (second century B.C.) wrote a treatise on 
algebra, and Cantor 2 inclines to the belief that there actually was 
such a work. No trace, however, has been found of it, and the 
probability is that Hipparchus did not write any systematic trea- 
tise on algebra or on quadratic equations. The word " algebra " 
indeed is Arabic in its origin and the use of it as a title goes back 
only to the time of our author, Mohammed ibn Musa. Neverthe- 
less it is possible that Greek mathematicians of the time of 
Hipparchus did occupy themselves with problems of the kind in 
question, because this was a natural development out of the con- 
sideration of rational right triangles, as given by Pythagoras and 
Plato, in connection with the geometrical treatment of quadratic 
equations as given by Euclid. Quadratic equations connect even 
more directly with the application of areas, of Pythagorean origin, 
which is extensively treated by Euclid. 

Some two centuries after the period of Diophantus, Aryabhata, 
one of the earliest Hindu mathematicians of prominence, was born 
(476 a.d.). In the work of Aryabhata as presented by Rodet 3 
we find the solution of a quadratic equation assumed in the rule 
for finding the number of terms of an arithmetic series when the 
sum, difference, and first term are given. Nor does Aryabhata 
in India stand alone in the study of analysis, as Diophantus 
does in Greece. Brahmagupta of Ujjain, the centre of Indian 
learning, wrote on algebra in the early part of the seventh century 
and gave a rule 4 for the solution of quadratic equations: 

" To the absolute number multiplied by the [coefficient of the] square, add the 
square of half the [coefficient of the] unknown, the square root of the sum, less half 
the [coefficient of the] unknown, being divided by the [coefficient of the] square, is 
the unknown." 

In formula this corresponds to the solution 



_ V {b/if + ac-b/i 
a 

1 Das Mathematiker- Verzeichniss im Fihrist des Ibn Abi Jdkub an-Nadim, translated 
by H. Suter, Abhandl. z. Geschichte der Mathematik, Vol. 6 (Leipzig, 1892), p. 22, and 
note, pp. 54-55- Suter holds that there is some error in the text, and this seems probable. 

2 Geschichte, Vol. I (3), pp. 362-363. 

3 Lecons de Calcul d } Aryabhata, Journal Asiatique, seventh series, Vol. XIII (1879), 

PP- 393-434- 

4 Colebrooke, Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brah- 

megupta and Bhdscara (London, 1817), p. 347; Cantor, I (3), p. 625. 



ALGEBRAIC ANALYSIS BEFORE AL-KHOWARIZMI n 

of the equation, 

ax 2 + bx = c. 

Contemporary with Al-Khowarizmi is the Hindu writer Mahavi- 
racarya, whose arithmetical and algebraical work has been trans- 
lated into English by M. Rahgacarya. 1 Rules are given in the 
sections devoted to algebra for the three types of complete quadratic 
equations. A peculiarity of the treatment is that the unknown 
quantity and the square root of the unknown appear, rather than 
the unknown and its square. The significance of the work is 
that it shows a persistence of interest in algebra in India from the 
time of Aryabhata. Three centuries later Bhaskara (b. 1114 a.d.), 
another Hindu mathematician, made important contributions to 
the advance of the science. 

The brief survey which we have given of the study of algebra 
before the time of Mohammed ibn Musa does not at all purpose 
to present the sources from which the great Arab drew his inspi- 
ration. Greece undoubtedly took mathematical ideas from Egypt, 
as Rodet 2 some years ago pointed out with reference to algebra. 
Even more definite evidence is presented by the Greek use of 
unit fractions as well as by the references to Egyptian mathe- 
matics which were made by Plato and Herodotus, and much later 
by Michael Psellus. Babylon and Greece were constantly exchang- 
ing ideas ; 3 a striking proof of this is the Greek use of sexagesimal 
fractions. India, too, was not out of touch with these neighbors 
to her west. Especially in the fields of religion and the closely 
associated astrology we have abundant evidence not only of inter- 
change of ideas between the East and the West but also of the 
recurrence in mediaeval times of ideas advanced by more ancient 
civilizations. Yet we need to notice that we are dealing with 
the independent appearances of algebraic ideas, and that the 
mathematics of Egypt, Babylon, China, Greece, and India was 
developing from within. Algebra is not, as often assumed, an 

1 M. Rangacarya, The Ganita-Sara-Sangraha of Mahavlracarya (Madras, government 
press, 1912). Professor D. E. Smith gave a brief preliminary report of the work in the 
Bibliotheca Mathematical Vol. IX, third series, pp. 106-110. 

2 L. Rodet, Sur les notations numeriqnes et algebriques anterieurement an XVI' siecle 
(Paris, 1881), pp. 43-51. 

3 F. Cumont, Babylon und die griechische Astronomie, Neite Jahrbilcher f. das klassische 
Altertum . . ., Vol. 27 (191 1), pp. 1-10; The Oriental Religions in Roman Paganism 
(Chicago, 191 1) ; and Astrology and Religion among the Greeks and Romans (New 
York, 1912). 



12 INTRODUCTION 

artificial effort of human ingenuity, but rather the natural expres- 
sion of man's interest in the numerical side of the universe of 
thought. Tables of square and cubic numbers in Babylon; geo- 
metric progressions, involving the idea of powers, together with 
linear and quadratic equations in Egypt ; the so-called Pythago- 
rean theorem in India, and possibly in China, before the time of 
Pythagoras; and the geometrical solution of quadratic equations 
even before Euclid in Greece, are not isolated facts of the history of 
mathematics. While they do indeed mark stages in the develop- 
ment of pure mathematics, this is only a small part of their signifi- 
cance. More vital is the implication that the algebraical side of 
mathematics has an intrinsic interest for the human mind not 
conditioned upon time or place, but dependent simply upon the 
development of the reasoning faculty. We may say that the 
study of powers of numbers, and the related study of quadratic 
equations, were an evolution out of a natural interest in numbers; 
the facts which we have presented are traces of the process of this 
evolution. 



CHAPTER II 

Al-Khowarizmi and his Treatise on Algebra 

The activity of the great Arabic mathematician Abu 'Abdallah 
Mohammed ibn Musa al-Khowarizmi marks the beginning of that 
period of mathematical history in which analysis assumed a place 
on a level with geometry ; and his algebra gave a definite form to 
the ideas which we have been setting forth. The arithmetic of 
Al-Khowarizmi made known to the Arabs and, through an early 
twelfth-century translation, to Europeans also, the Hindu art of 
reckoning. Any consideration of the difficulties attending arith- 
metical operations with the Greek letter numerals, 1 or even with 
the Roman numerals, shows how essential the adoption of numer- 
als with place value was for the development of the analytical 
side of mathematics ; the way was being prepared also for the 
appearance of decimal fractions, many centuries later, and for 
logarithms, both indispensable tools of modern science. Quite as 
important as the arithmetic for the development of mathematics 
was the systematic treatise on algebra 2 which Mohammed ibn 
Musa gave to the world. This is the work of which we present 
the Latin translation made by Robert of Chester while living in 
Segovia in 1183 of the Spanish Era (1 145 a.d.). 3 

Our chief source of information in regard to the life and the 
writings of our Arabic author is the Book of Chronicles 4 {Kitab 

1 Nine letters represent the units from i to 9, nine further letters the tens from 10 to 90, 
and nine others the hundreds. The thousands up to 900,000 are represented by the same 
letters with a kind of accent mark. 

- The Arabic text as given in the MS. Hunt, 214, of the Bodleian library, a unique copy, 
was published with English translation by Frederic Rosen, The Algebra of Mohatnmed 
ben Musa (London, 1831 ), being one of the volumes published for the Oriental Translation 
Fund ; a French translation of the chapter on Mensuration was published by Marre, based 
on Rosen's Arabic text, Nouvellcs Annates de Mathhnatiqnes, Vol. 5 (1846), pp. 557-581, 
and also later revised in Annali di Maternal., Vol. VII (1866), pp. 268-280. 

3 A preliminary note by the author concerning this translation appeared in the Bib- 
liotheca Mathematical third series, Vol. XI (191 1), pp. 125-131, with the title, Robert of 
Chester's Translation of the Algebra of Al-Khowarizmi. 

4 The Kitab al-Fihrist was edited with notes by G. FlUgel and published after Professor 
Fliigel's death by J. Roediger and A. Mueller (Leipzig, 1871-1872). I quote from the 
German translation by H. Suter, Das Mathematiker-Verzeichniss im Fihrist, Abhandlungen 
zur Geschichte der Mathematik, Vol. VI, Leipzig, 1892. 

13 



i 4 INTRODUCTION 

al-Fihrist) by Ibn Abi Ya'qub al-Nadim. This work, which was 
completed about 987 a.d., gives biographies of learned men of all 
nations, together with lists of such of their works as were known 
to Al-Nadim. We quote the passage relating to our author, re- 
garding whose life and activity we have only meagre information. 

" Al-Khowarizmi " 

" Mohammed ibn Musa, born in Khowarizm (modern Khiva), worked in the 
library of the caliphs under Al-Mamun. During his lifetime, and afterward, where 
observations were made, people were accustomed to rely upon his tables, which 
were known by the name Sind-Hind (Hindu Siddhanta). He wrote : The book of 
astronomical tables in two editions, the first and the second ; On the sun-dial ; 
On the use of the astrolabe ; On the construction of the astrolabe ; The Book of 
Chronology.'''' 

Neither the date of the birth nor the date of the death of 
Al-Khowarizmi has been definitely established. However, the 
fact, mentioned by Al-Nadim in the Fihrist, that he worked in 
the library of the caliph Al-Mamun, who reigned from 813 to 833 
a.d., indicates the period of his literary activity. The introduction 
to the algebra, which is not found in the extant Latin translations, 
is here given in English (p. 45), following Rosen. This brings 
further evidence of the acquaintance with Al-Mamun, for Al- 
Khowarizmi states that the interest of the caliph in science en- 
couraged him to write the treatise. The probability is that early 
in the reign of Al-Mamun our author began work upon the Hindu 
astronomical tables which, as the Fihrist account implies, brought 
him almost immediate fame. This stimulated him to undertake 
the work upon algebra, and the success of the second work 
induced him to write the treatise on arithmetic in which reference 
is made to the algebra. The height of his literary activity may 
reasonably be placed about 825 a.d. 

The bibliography of Al-Nadim does not include four works 
from the hand of Al-Khowarizmi which have come to us. These 
are his arithmetic, his algebra, a work on the quadrivium, and an 
adaptation of Ptolemy's geography. To Sened ibn 'Ali, the Jew, 
whose biography immediately follows that of Mohammed ibn 
Musa, are ascribed works entitled, On Increasing and Decreasing 
(algebraical), The Book of Algebra, and On the Hindu Art of 
Reckoning. The probability is, as Suter points out, 1 that an inter- 

1 Fihrist, loc. cit., pp. 62-63. 



AL-KHOWARIZMI AND HIS TREATISE ON ALGEBRA 15 

change has taken place here, although this must have been rela- 
tively early since Ibn al-Qifti, 1 who died in 1248 a.d., in his 
Chronicles of the Learned, gives the same account of Al-Khowa- 
rizmi as Al-Nadim. Furthermore, the author of the Fihrist knew 
of the algebra, for he mentions no less than three men as com- 
mentators on the algebra of Mohammed ibn Musa ; Sinan ibn 
al-Fath of Harran, 'Abdallah ibn al-Hasan al-Saidanani, and 
Abu'1-Wefa al-Buzjani are credited with such commentaries. 2 

The arithmetic of Al-Khowarizmi has come down to us only in 
a Latin translation, and this survives in a unique copy belonging 
to the library of the University of Cambridge, published in 1857 
by Prince Baldassare Boncompagni. 3 Several references 4 in this 
work to the writer's other book on arithmetic make it evident 
that the Al-Khowarizmi in question is the author of the algebra. 
Our word algorism, as well as the obsolete form augrim? used by 

1 Suter, loc. tit., pp. 62-63 i Casiri, Bibliotheca Arabico-Hispana Escurialensis (Madrid, 
1760-1770), I, pp. 427-428. Ibn al-Qifti mentions the arithmetic. 

2 Suter, loc. tit., pp. yj, 36, and 39 respectively. 

3 Trattati cf aritmetica (Rome, 1857), I. Algoritmi de numero indorwn, pp. 1-23, which 
is the arithmetic in question, and bears internal evidence that it is a translation from the 
Arabic; II. Joannis Nispalensis liber algorismi de pratica arismetrice, evidently an 
adaptation and enlargement of the preceding. Some other manuscripts of the second 
treatise are anonymous, while others ascribe it to Gerard of Cremona. 

4 One passage is in Trattati, I, pp. 2-3 ; Etiam patefeci in libro algebre et almucabalah, 
idest restaurationis et oppositionis, quod uniuersus numerus sit compositus, et quod 
uniuersus numerus componatur super unum. Vnum ergo inuenitur in uniuerso numero; 
et hoc est quod in alio libro arithmetice dicitur. Quia unum est radix uniuersi numeri, et 
est extra numerum. Radix numeri est, quare per eum inuenitur omnis numerus. . . . 
Reliquus autem numerus sine uno inueniri non potest. . . . Reliquus autem numerus 
indiget . . . uno : quare non potes dicere duo uel tria, nisi precedat unum. Nichil 
aliud est ergo numerus, nisi unitatum collectio. . . . 

Inueni, inquit algorizmi, omne quod potest dici ex numero, et esse quicquid excedit 
unum usque in ix., id est quod est inter, ix. et unum, id est duplicatur unum et hunt duo ; 
et triplicatur idem unum, fiuntque tria, et sic in ceteris usque in ix. De inde ponuntur, 
x. in loco unius, et duplicantur, x. ac triplicantur, quemadmodum factum est de uno, 
fiuntque ex eorum duplicatione, xx, et triplicatione, xxx, et ita usque ad xc. Post hec 
redeunt c in loco unius, et duplicantur ibi atque triplicantur, quemadmodum factum est de 
uno et x ; efficiunturque ex eis cc, et ccc, et cetera usque in dcccc. Rursum ponuntur 
mille in loco unius ; et duplicando et triplicando, ut diximus, fiunt ex eis 11 mila, et in et 
cetera usque in infinitum numerum, secundum hunc modum. 

Compare with this our text, p. 66, lines 10-21. 

Another passage is Trattati, I, p. 10. Etiam patefeci in libro, quod necesse est omni 
numero qui multiplicatur in aliquo quolibet numero, ut duplicetur unus ex eis secundum 
unitates alterius. 

Compare with this our text, p. 90, lines 6-8. 

5 See my paper on Augrim-stones, Modern Language Notes, Vol. XXVII (1912), 
pp. 206-209 5 compare also the use of the term Algaurizin in our text, p. 102. 



1 6 INTRODUCTION 

Chaucer, is derived from the use of the name Al-Khowarizmi in 
the opening sentence of the arithmetic, which reads, Dixit 
algoritmi or ' Algorithm says ' ; the word ' algebra ' is derived 
from its use as a title by Al-Khowarizmi in the work which we 
are presenting. 1 Up to the eighteenth century the common name 
for the new arithmetic with the ten figures of India, i 234567 
890, was algorism, or in Latin algorismus. Interesting also is 
the fact that a Spanish transliteration of Khowarizm, guarismo, is 
used for ' numerals,' corresponding to our ordinary use of ' ciphers.' 
Aside from Al-Khowarizmi's arithmetic in Latin translation, which 
was never widely used, the works which served to introduce the 
numerals into Europe were the Carmen de Algorismo, 2 in verse, 
written by Alexander de Villa Dei (about 1 220), and the Algorismus 
vulgaris* by John of Halifax, commonly known as Sacrobosco 
(about 1 2 50). Both of these works were somewhat dependent upon 
Mohammed ibn Musa's arithmetic, and both continued in wide use 
for centuries. Many manuscript copies of the Carmen are found 
in European libraries, and rather more of the Algorismus vulgaris. 
Even after the invention of printing Sacrobosco's Algorism was 
widely used for university instruction in arithmetic, many editions 
appearing in the fifteenth and sixteenth centuries. 4 Detailed and 
extended commentary upon the work was given by Petrus de 
Dacia in 1291, in his lectures, evidently, and probably in a similar 
manner by many other university lecturers. 

Another arithmetical treatise ascribed to Al-Khowarizmi is 
found in the Liber ysagogarum Alchorismi in artem astronomi- 
cam a magistro A. compositus. h The principles of arithmetic, 
geometry, music, and astronomy are explained in five books, or 
chapters, and in two manuscripts there follow three books on 

1 See my note on Algebra, Modern Language Notes, Vol. XXVIII (1913), p. 93 ; com- 
pare also the use of the term in our text, p. 2. 

2 Published by J. O. Halliwell, Rara Mathematica (London, 1839). 

3 There were many early editions ; see Curtze, Petri Philomeni de Dacia in Algorisnntm 
vulgarem Johannis de Sacrobosco commentarius, una cum Algorismo ipso (Ed. M. Curtze, 
Copenhagen, 1897.) 

4 See Smith, Rara Arithmetica (Boston, 1908), pp. 31-33 ; see also my article, Jordanus 
Nemorarius and John of Halifax, American Mathematical Monthly, Vol. XVII (1910), 
pp. 108-113. 

5 Nagl, in Zeitschrift f. Math. u. Physik, Hist.-litt. Abth., XXXIV, pp. 129-146, 161- 
170; the first three books complete, and summary of the other two, by Curtze, Ueber eitie 
Algorismus-schrift des XII. Jahrhunderts in Abhandl. z. Gesch. d. math. lVissen.,Vlll, 
pp. 1-27 ; Haskins, in The English Historical Review, XXVI, p. 494. 



AL-KHOWARIZMI AND HIS TREATISE ON ALGEBRA 17 

astronomy. Three of the books which deal particularly with 
arithmetic have been published, 1 but no study has yet been made 
of the three books on astronomy. The writer A. is supposed to 
be Adelard of Bath, who was active at the time that this work 
was written ; he translated the tables of Al-Khowarizmi. So far 
as these five books of the introduction are concerned, the work 
may well be a summary of the elementary teachings of Al-Khow- 
arizmi according to the unknown writer's conception of them. 

Al-Mas'udi (885-956 a.d.) in his Meadows of Gold 2 mentions 
Mohammed ibn Musa among the historians and chroniclers, bas- 
ing his reference doubtless on the Book of Chronology above men- 
tioned. Al-Biruni (973-1048 a.d.), whose work on India 3 has 
recently (1910) appeared in a second English edition, refers to 
the tables and the astronomical work of our author. No less 
than three works by Al-Biruni 4 are explanatory of works written 
by the distinguished mathematician and astronomer who was his 
fellow-countryman, both being from Khowarizm (or Khiva). 
Not only in the fields of astronomy, chronology, and mathematics 
did Mohammed ibn Musa achieve fame, but also as a geographer. 
His contribution in this field has been set forth by Nallino, who 
states, by way of conclusion to his article on Al-Khowarizmi and 
his reconstruction of the geography of Ptolemy? that this geog- 
raphy is not a servile imitation of the Greek model, but an elabo- 
ration of Ptolemaic material made with more independence and 
ability than is displayed by any European writer of that period. 
The trigonometric tables, in Latin translation by Adelard of Bath, 
appear to be the earliest of Al-Khowarizmi's works to be known 
in Europe, and indeed one of the earliest mathematical treatises 
taken from the Arabic; it was translated in 11 26 a.d. A study 

1 Curtze, loc. cit. ; Haskins, loc. cit., gives the incipit and explicit of the astronomical 
work : Quoniam cuiusque actionis quantitatem temporis spacium metitur, celestium mo- 
tuum doctrinam querentibus eius primum ratio occurrit investiganda. . . . Divide quoque 
arcum diei per 12 et quod fuerit erunt partes horarum eius, si deus inveniri consenserit. 

2 Les prairies d'or, translated by Barbier de Meynard and Pavet de Courteille, I-IX 
(Paris, 1861-1877), I, 11. 

3 Alberunfs India, translation by C. E. Sachau (London, 1888, and second edition, 
London, 1910). 

4 Suter, Der Verfasser des Bitches Gr'unde der Tafeln des Chowarezini* Bibliotheca 
Mathemalica, third series, Vol. IV, 1903, pp. 127-129. 

5 Al-Hnwarizmi e il sno rifacimento delta Geografia di Tolomeo, Atti della R. Accade- 
mia dei Lincei, fifth series, Memorie, Classe di scienze morali, storiche e filologiche, Vol. II 
(1896), pp. 11-53. 



18 INTRODUCTION 

of these tables, with extracts from the Latin text, was made by 
A. A. Bjornbo J and the completed work, edited by Suter, has 
recently been published. Suter has found 2 that this translation is 
not of the original work by Al-Khowarizmi, but is based on a revi- 
sion of that work made by Maslama al-Majriti (about iooo a.d.). 
The work is of considerable importance for the history of trigonom- 
etry, for even though the tangent function which appears therein 
may be the addition of Maslama, yet the introduction of the sine 
function is certainly carried back as far as Al-Khowarizmi. 

The combination of mathematical, geographical, and astronomi- 
cal interests exhibited by Al-Khowarizmi renders plausible the 
hypothesis advanced by Suter, 3 on chronological grounds, that 
this Mohammed ibn Musa took part in the measurement of a 
degree of the earth's circumference which was made at the request 
of the caliph Al-Mamun. Some early Arabic chroniclers join the 
three sons of Moses, the Beni Musa, in this task. The oldest of 
these brothers was Mohammed and so he was called after the 
Arabic custom, Mohammed ibn Musa, meaning Mohammed, the 
son of Moses. Abu Ja'far is the prefix to his name, and this has 
been incorrectly given by numerous modern dictionaries 4 as the 
name of our author. 

Concerning Mohammed ibn Musa's fame among the Arabs as 
an algebraist abundant evidence exists. Not only the commen- 
taries cited bear witness to this fame but also the recurrent 
appearance for centuries of the numerical examples, x 2 + io.r= 39, 
x 2 +2i = 10 .r, 3x + /\. = x 2 and many others, which Al-Khowarizmi 
used. Some of the later authors, like Abu Kamil Shoja' ibn 
Aslam 5 (about 925 a.d.), explicitly acknowledge their indebtedness 

1 Bjornbo, Al-ChwarizmPs trigono7netriske Tavler, in Festskrift til H. G. Zeuthen, 
Copenhagen (1909), pp. 1-26; the sudden death of this brilliant student of mediaeval 
mathematics is a great loss, for his systematic studies in this field were beginning to bear 
fruit in numerous and important publications. 

2 Suter, Memoires de VAcad. Rayale des Sciences et des Lettres de Danemark, Series 7 
(Lettres), Vol. 3, pt. 1 (Copenhagen, 1914). The dependence upon Maslama's version was 
noted by C. H. Haskins, Adelard of Bath, The English Historical Review, Vol. XXVI 
(191 1 ), pp. 49 r -49 8 - 

3 Suter, Die Mathematiker und Astronomen der Araber und Hire Werke, in Abhandl. 
z. Gesch. d. tnath. Wissenschaften, Vol. X (Leipzig, 1900), p. 20. 

4 Century, Webster'' s, The New English Dictionary, Encyclopaedia Britannica. 

5 L. C. Karpinski, The algebra of Abu Kamil Shojd ben Aslant, Bibliotheca Mathematica, 
Vol. XII, third series (191 2), pp. 40-55; The Algebra of Abu Kamil, American Mathe- 
matical Monthly, Vol. XXI (1914), pp. 37-48. 



AL-KHOWARIZMI AND HIS TREATISE ON ALGEBRA 19 

to our author. Others, like the poet Omar ibn Ibrahim al-Khay- 

yami (about 1045-1123 a.d.), familiar to English readers as Omar 

Khayyam, and Mohammed ibn al-Hasan al-Karkhi (died about 

1029), do not consider it necessary to state the source of these 

problems and the proofs, which had become classic. Thus the 

equation 

x 1 + \ox = 39 

runs like a thread of gold through the algebras for several cen- 
turies, appearing in the algebras of the three writers mentioned, 
Abu Kamil, Al-Karkhi 1 and 'Omar al-Khayyami, 2 and frequently 
in the works of Christian writers, centuries later, as we shall have 
occasion to note below. 

Recognition of the fame of Al-Khowarizmi is to be found in the 
explicit statement by Ibn Khaldun (1 332-1406) in his encylopasdic 
work 3 : " The first who wrote upon this branch (algebra) was Abu 
'Abdallah al-Khowarizmi, after whom came Abu Kamil Shoja' ibn 
Aslam." Haji Khalfa, some two hundred years later, makes a 
similar statement. The Chronicles of the Learned 'by Ibn al-Qifti 
(d. 1282 a.d.) speak very highly of his ability as an arithmetician, 4 
and a contemporary of Al-Qifti, Zakariya ibn Moh. ibn Mahmud 
al-Qazwini, refers to him as translating the art of algebra for the 
Mohammedans. 5 

I have recently made a study 6 of the algebra of Abu Kamil, whose 
name we have seen associated by Arabic historians with that of 
Al-Khowarizmi. I have shown that he drew extensively upon the 
work of his predecessor, and further that Leonard of Pisa (1202 
a.d.) drew, in turn, even more extensively from Abu Kamil. Thus 
the influence of the Khowarizmian was carried over to Italy as 
well as to the Arabic commentators and the European translators 

1 F. Woepcke, Extrait du Fakhri (Paris, 1853) ; A. Hochheim, Die Arithmetik des 
Abu Bekr Muhammed ben Alhusein Alkarkhi (Programm, Magdeburg, 1878), and Kafifil 
Hisab {Genilgendes iiber Arithmetik) (Halle, 1878-1880). The Kafi fil Hisdb is an 
earlier work by Al-Karkhi which includes a treatment of algebra. 

2 F. Woepcke, Ealgebre d^Omar Alkhayyami (Paris, 185 1). 

3 MacGuckin de Slane, Brolegomines historiqnes dVbn Khaldoun, Notices et Extraits 
des Manuscrits de la Bibliotheque Imperiale et autres Bibliothiques (Text, Vol. XIX and 
XX, Translation, Vol. XXI, 1868), Vol. XXI, p. 136. 

4 Casiri, Bibliotheca Arabico-Hispana Escurialensis, p. 427. 

5 Casiri, loc. cit., p. 427; Cossali, Origine, trasporto in Italia, primi progressi in essa 
delP algebra (Parma, 1797), Vol. I, p. 178. 

6 The Algebra of Abu Kamil, loc. cit. 



20 INTRODUCTION 

of Abu Kamil's algebra. Two of these commentaries on Abu 
Kamil's work appeared in the tenth century and are listed in the 
Fihrist. The Persian Al-Istakhri, 1 known as The Reckoner, was 
the author of one, and 'Ali ibn Ahmed al-Tmrani 2 wrote the other. 
Neither commentary has come down to us. The author of the 
Latin translation upon which my study is based is not known, but 
the probability is that the translation was made about the time of 
Gerard of Cremona. Another commentary of uncertain date in 
Arabic by a Mohammedan Spaniard, Al-Khoreshi, is probably of 
later origin. In the fifteenth century a Hebrew translation was 
prepared by Mordechai Finzi of Mantua (about 1475) ; this is of 
especial interest since it bears internal evidence, in the termi- 
nology employed, that it was based upon a Spanish original. The 
author of the Spanish treatise, doubtless a Christian, is unknown. 
Ibn Khaldun mentions the commentary written by Al-Khoreshi, 
but no trace has yet been found of either Spanish translation or 
commentary. A unique copy of the Latin translation of the 
algebra is in Paris (Mss. Lat. 7377 A) and copies of the Hebrew 
translation are found in Paris and Munich. 3 

The widespread interest among Arabic scientists in the study 
of algebra is attested by the number of works upon the subject. 
Even as early as the tenth century besides Al-Khowarizmi there 
were three other writers 4 of sufficient prominence to warrant 
their appearance in the Fihrist. Abu 'Otman Sahl ibn Bishr. ibn 
Habib ibn Hani, a Jew, wrote an algebra which Al-Nadim states 
was praised by the Romans, and Ahmed ibn Da'ud, Abu Hanifa, 
al-Dlnawarl (d. 895 a.d.) and Abu Yusuf al-Missisi also wrote 
treatises on the subject. These works are preserved only in 
title. In the thirteenth century Ibn al-Banna, 5 including in his 
arithmetic a brief exposition of algebra, employs the title, 
"algebra and almucabala," as given by Al-Khowarizmi, and 
follows the same peculiar order of the six types of quadratic 
equations. Al-Banna adds no numerical illustrations but states 

1 Suter, Die Mathematiker und Astronomen der Araber, p. 51. 

2 Suter, loc. tit., pp. 56-57. 

3 Steinschneider, Die Hebraeischen Uebersetzungen des Mittelalters (Berlin, 1893), pp. 
584-588; Sitzungsber. d. Akad. d. Wissen. (Vienna), Phil.-hist. Klasse, Vol. 149 (1905), 

P- 3i- 

4 Suter, Die Mathematiker, pp. 15, 31, and 66 respectively. 

6 Suter, Die Mathematiker, pp. 162-163 ! A. Marre, Le Talkhys d' 1 Ibn Albanna, Atti 
deW accad. ftontif. dei nuovi Lincei, Vol. XVII (1864), pp. 289-319. 



AL-KHOWARIZMI AND HIS TREATISE ON ALGEBRA 21 

simple general rules for the solution of equations. A separate 
work on the same subject by Ibn al- Banna is included among the 
Arabic manuscripts in Cairo. In modern times Arabs have used 
lithograph copies of an arithmetic and algebra by Mohammed 
ibn al-Hosein, Beha al-din al- e Amili (Beha ed-dm), who died in 
1622. Two editions * with a translation into the Persian language 
and commentary were published in the nineteenth century. 
Beha ed-din continues the use of some of Al-Khowarizmi's 
problems as well as the six types of quadratics. 

Greek influence on Arabic geometry is revealed by the order 
of the letters employed on the geometrical figures. These letters 
follow the natural Greek order and not the Arabic order. The 
same is true of the order of the letters in the geometrical figures 
used by Al-Khowarizmi for verification of his solutions of 
quadratic equations. However, the somewhat ingenious hypothe- 
sis put forth by Cantor 2 that this fact shows that these demon- 
strations are from Greek sources is hardly tenable. The Arabs 
were much more familiar with and grounded in Euclid than are 
mathematicians to-day, and it was entirely natural in constructing 
new figures that they should follow the order of lettering to which 
they had grown accustomed in their study of Euclid. Until 
further and more definite historical evidence to the contrary is 
brought to light we must regard Al-Khowarizmi as the first to 
bring out sharply the parallelism between the analytical and 
geometrical solutions of quadratic equations. 

The Arabic students of algebra included poets, philosophers 
and perhaps even kings. Omar Khayyam, whom we have men- 
tioned among the writers on the subject, was too excellent 
a mathematician and too true a poet to woo the muse of algebra 
in verse. But in the library of the Escurial at Madrid there is 
preserved a poem treating of algebra, written by a native of 
Granada, Mohammed al-Qasim. Needless to say, the content, 
adapted to the exigencies of verse, does not compare with the 

1 Calcutta, 1812, and Constantinople, 1851-1852 ; Arabic with German translation by 
Nesselmann, Berlin, 1843 ; French translation, A. Marre, Paris, 1864. 

2 Cantor, I (3), pp. 724-725 ; Simon has noted the weakness of Cantor's argument in 
the article, Zu Hwariz»i?s hisab al gabr wal muqabala, ArcJiiv der Mathematik und 
Pliysik, third series, XVIII (191 1), pp. 202-203 ; see also Bjornbo and Vogl, Alkindi, Tideas 
und Psejido-Euklid, Abhandl. z. Geschichte d. math. Wissen., XXXVI (Leipzig, 1912), p. 
156, and the review of the same by A. Birkenmajer, Bibliotheca Mathematica, Vol. XIII, 
third series (1913), pp. 273-280. 



22 INTRODUCTION 

prose of Omar Khayyam. The King of Saragossa, Jusuf al- 
Mutamin (reigned 1081-1085), was a devoted student of the 
mathematical sciences. The title of one of his works suggests 
the possibility that it was algebraical. For the study of law a 
knowledge of algebra seems to have been necessary, as various 
questions of inheritance were treated by this science, and even 
to-day in the great Mohammedan schools at Cairo and Mecca the 
study aljebr w al-muqabala is considered of peculiar value to the 
prospective lawyer. 



CHAPTER III 

Robert of Chester and Other Translators of Arabic 

into Latin 

When towards the beginning of the twelfth century European 

scholars turned to Islam for light, the works of Mohammed ibn 

Musa came to occupy a prominent place in their studies. One 

of the first of these students was styled John of Seville, 1 or of 

Luna, or of Spain, whose name is attached to some manuscript 

copies of an adaptation of Al-Khowarizmi's arithmetic. Thus, 

the version, mentioned above, published by Prince Boncompagni, 

bears the title, Joannis Hispaleusis liber A Igorismi de pratica 

arismetrice, and in a sub-head the editing of the arithmetic is 

ascribed to John of Spain. This work contains also a very brief 

treatment of algebra, entitled Exceptiones de libro, qui dicihir 

gleba mutabitia? Res is employed for the square of the unknown, 

and radix for the unknown, the usage being, in this respect, 

unique. The problems, 

x 2 + iox= 39, 

occur as in Al-Khowarizmi, with a variation in the second type, 

x 2 + 9 = 6 x. 

The authorship is in question, since some manuscripts ascribe the 
work to Gerard of Cremona, some to John of Spain, and others 
are anonymous. However, no doubt now exists that John of 
Spain was familiar with the arithmetic of Al-Khowarizmi, for 
Dominicus Gundisallinus, co-laborer with John in translating from 
the Arabic, mentions the Liber algorismi in the chapter on arith- 
metic of the De divisione philosophiae? About 1 133 a.d. Bishop 
Raimund of Toledo commissioned John of Spain to work with 
Gundisallinus on translations from the Arabic. John made the 
translation into Spanish, and this was put into Latin by Gun- 

1 Steinschneider, Die Hebr. Uebers., p. 981, and note 82, p. 380. 

2 Unintelligible Latin forms for "algebra w' almucabala." 

3 Edited by L. Baur, Beitrage 2. Gesch. d. Philos. d. Mittelalters (Vol. IV, Munster, 
1903), p. 91. 

23 



24 INTRODUCTION 

disallinus. The probability is that the De divisione philosophiae 
is their joint production. 1 A somewhat similar method was pur- 
sued at first by Gerard of Cremona, collaborating with an Arab 
named Galippus or Galib. 2 

Adelard of Bath translated the astronomical tables which we 
have mentioned and possibly another astronomical work by 
Al-Khowarizmi. 3 His life is typical of the life of learned men 
of that period. Although born in England, he evidently went to 
France at an early age. There he studied at Tours and delivered 
lectures in Laon. At least seven years of his life were spent in 
study and travel in the East. Tarsus, Antioch, and Salerno are 
mentioned by him as cities which he visited. While no direct 
evidence is known that he studied in Spain, yet many of his 
works are based on Arabic documents transmitted to Europe 
through the Spanish schools at Toledo and Segovia. Learning 
was quite as international in that time as to-day. 

Gerard of Cremona, too, desiring to find the works of Ptolemy, 
journeyed to Spain and there took up the study of the Arabic 
language in order to understand the Arabic version of Ptolemy, 
with the result that he devoted his life to translations from the 
Arabic. Included in an early list 4 of his translations is the alge- 
bra of Al-Khowarizmi, and it seems probable that the Latin 
version published by Libri 5 is from his hand. However, Boncom- 
pagni in his discussion of the life and works of Gerard of Cre- 
mona has published another mediaeval adaptation of the algebra 
which is ascribed to Gerard. The words res and census for the 
unknown and its square, and also the title aliabre et almuchabala, 
are used by Gerard in his translation of Ababucri's Book of the 
measurement of the earth and of solids, as yet in manuscript. 6 
Plato of Tivoli was also doubtless familiar with Mohammed ibn 

1 Steinschneider, Die Hebr. Uebers., p. 981 and note 82, p. 380. 

2 Valentin Rose, Ptole»iaeus und die Schule von Toledo, Hermes, Vol. VIII, pp. 332 ff. 

3 C. H. Haskins, Adelard of Bath, The English Historical Review, Vol. XXXVI (191 1), 
pp. 491-498, and Adelard of Bath and Henry Planlagenet, ibid., Vol. XXXVIII (1913), 
pp. 515-516. 

* Boncompagni, Delia vita e delle opere di Gherardo Cremonese, etc., Atti delP Acca- 
de/nia pontificia de'' nuovi Lincei, Vol. IV (185 1), pp. 378-493. 

5 Libri, Histoire des sciences mathematiques en Italic, Vol. I (Paris, 1838), pp. 253-297; 
Bjbrnbo, Gerhard von Cremona^s Uebersetzung von Alkwarizmfs Algebra und von Eu- 
klid^s Elementen, Bibliotheca Mathematica, Vol. VI (1905), third series, pp. 239-248. 

6 My statements are based upon the Paris MS. Latin 9335 and the Cambridge Uni- 
versity Library MS. Mm. 2, 18, both of which contain the work in question. 



ROBERT OF CHESTER AND OTHER TRANSLATORS OF ARABIC 25 

Musa's works, for he mentions him as one of the Arabic math- 
ematicians. 1 Contemporary with these men was Robert of Chester. 

The assertion was made by Curtze 2 that the Liber ernbadorum, 
Book of Measures, in Hebrew, by Abraham bar Chiyya Ha Nasi, 
known as Savasorda, was the first work to appear in Latin show- 
ing to the Western world how the solution of quadratic equations 
is accomplished. This statement is made on the basis of the date 
DX of the Hegira for the translation of the work made by Plato 
of Tivoli. It has recently been shown by Haskins, 3 on the basis 
of astronomical data in the work, that this date is undoubtedly 
a scribe's error for DXL, corresponding to 1 145 a.d. Savasorda 
was approximately contemporary with his translator. Of the early 
Jewish writers many were familiar with the works of Al-Khowa- 
rizmi. Thus Abu Masar 4 cites the tables of our author while 
Abraham ben Esra 5 refers frequently to the same tables. 

Three other Englishmen besides Adelard of Bath are known to 
have been students of Arabic mathematical science as taught in 
the schools of Spain in the twelfth century. The names and 
dates, as given by Wallis, 6 are : Adelard of Bath in 1 130, Robertus 
Retinensis in 1140, William Shelley (de Conchis) in 1 145, and 
Daniel Morley (Merlac) in 1180. This Robertus Retinensis 7 
was also known as Robertus Ketenensis, de Ketene, Ostiensis, 

1 Favaro, Intorno alia vita ed alle opere di Prosdocimo de* Beldomandi, Bullettino di 
bibliografia e di storia delle scienze matematiche e fisiche, Vol. XII (Rome, 1879), pp. 
1-74, 1 15-251. See pp. 122-123. 

2 M. Curtze, Der Liber Ernbadorum des Saitasorda in der Uebersetzung des Plato von 
Tivoli, Abhandl. z. Geschichte d. math. Wissen., Vol. XII (1902), p. 7. 

3 C. H. Haskins, The Romanic Review, Vol. II (191 1), p. 2, note 5 ; The English His- 
torical Review, Vol. XXVI (191 1), p. 491, note 1. 

4 Steinschneider, Zum Speculum des Albertus Magnus, Zeitschrift f. Mathematik und 
Physik, Vol. XVI (1871), p. 376. . 

5 Steinschneider, Zur Geschichte der Uebersetzungen aus dem Indischen in^s Arabische 
und ihres Einflusses auf die arabische Literatur, Zeitschrift der deutschen morgenldnd- 
ischen Gesellschaft, Vol. XXIV (1870), pp. 339, 355 et al. 

6 Wallis, A Treatise on Algebra, both historical and practical (London, 1685), pp. 10-12. 

7 The Dictionary of National Biography includes two accounts of the life of Robert of 
Chester; Vol. X (London, 1887), p. 203, Chester, Robert, by A. M. Clerke ; Vol. XLVIII 
(New York, 1896), pp. 362-364, Robert the Englishman, Robert de Ketene, or Robert de 
Reiines, by T. A. Archer. Wustenfeld, Die Uebersetzungen arabischer Werke in das 
Lateinische, Abhandl. d. Koniglicheti Gesellschaft der Wissenschaften zu G'ottingen, Vol. 
XXII (1877), pp. 44-47; L. Leclerc, Histoire de la medecine arabe (Paris, 1876), Vol. II, 
pp. 380-387 ; Jourdain, Recherches sur les anciennes traductions Latines d^Aristote (Paris, 
1843, revised edition), pp. 100-104; Thomas Wright, Biographia Britannica Literaria, 
Anglo-Norman Period (London, 1846), pp. 116-119. 



26 INTRODUCTION 

Astensis, or Cestrensis, the final form being the most common. 
Retinensis (Retenensis) has been somewhat doubtfully referred to 
Reading, England, while Cestrensis certainly refers to Chester. 
Similar peculiarities in the dual or multiple designation of writers 
have been noted in connection with the name of John of Spain; 
such variations seem to have been common in this period. Robert 
of Chester, known to fame chiefly as the first translator of the 
Qoran, 1 was doubtless educated in the well-known school located 
at Chester. Of the more personal side of Robert's life we have 
but scattered facts. His nationality is established not only by 
his name and his return to England in 1 150, but also by the direct 
statement made by Peter the Venerable in a letter 2 of 1143 con- 
cerning the Qoran, addressed to Bernard of Clairvaux. Peter 
states in this letter that Robert was then archdeacon of Pampe- 
luna, in northern Spain. Hermann the Dalmatian, commonly 
known as Hermannus secundus, but also spoken of as Scho- 
lasticus, Sclavus, or Chaldasus, refers to Robert as his " special 
and inseparable comrade, his peerless partner in every deed and 
art." In the year 1141 Robert and Hermann were living in 
Spain near the Ebro, studying the arts of astrology. There in 
that year Peter the Venerable found them and "by entreaty and 
a good price " induced them to take up studies in Mohammedan 
religion and law, and also to translate the Qoran. 3 

1 Machumetis Saracenorum principis, eiusquc successorum vitae, doctn'na, ac ipse 
Alcoran . . . cum doctiss. uiri Philippi Melancthonis praemonitione. . . . Haec omnia 
in unum uolumen redacta sunt, opera et studio Theodori Bibliandri (1550. place of publi- 
cation, Basle, not given in book itself), Vol. I, pp. 213-223 ; a different edition was edited 
by Wallis (Basle, 1643) i possibly the first edition was printed at Basle, 1543. I have 
used a copy of the edition of 1550, loaned to me from the John G. White Collection, 
Cleveland Public Library, by the courtesy of the librarian. 

2 Qoran, 1550 edition, pp. 1-2; Migne, Patrologia Latina, Vol. 189 (Paris. 1890), 
col. 649-652 ; see also col. 1073-1076, and col. 339. 

3 As there is some question as to the real translator of the Qoran it seems desirable to 
add from the 1550 edition of the Qoran the evidence that the translation is due to Robert, 
pp. 1-2 : Epistola Domini Petri Abbatis, ad Dominum Bernhardum Claraeuallis Abbatem, 
de translatione sua, qua fecit transferri ex Arabico in Latinum, sectam, siue haeresim, 
Saracenorum. . . . Mitto uobis, charissime, nouam translationem nostram, contra pes- 
simum nequam Machumet haeresim disputantem. Quae nuper dum in Hispaniis morarer 
meo studio de Arabico uersa est in Latinam. Feci autem earn transferri a perito utriusque 
linguae uiro magistro Petro Toletano. Sed quia lingua Latina non ei adeo familiaris uel 
nota erat, ut Arabica, dedi ei coadiutorem doctum uirum dilectum filium et fratrem Petrum 
notarium nostrum, reuerentiae uestrae, ut extimo, bene cognitum. Qui uerba Latina 
impolite uel confuse plerumque ab eo prolata poliens et ordinans, epistolam, imo libellum 
multis, ut credo, propter ignotarum rerum notitiam perutilem futurum perfecit. Sed et 



ROBERT OF CHESTER AND OTHER TRANSLATORS OF ARABIC 27 

In the prologue to his treatise 1 against the "sect" of the Saracens, 
Peter says : " Contuli ergo me ad peritos linguae Arabicae, . . . 
eis ad transferendum de lingua Arabica in Latinam perditi hominis 
originem, vitam, doctrinam, legemque ipsam quae Alchoran voca- 
tur, tam prece quam pretio, persuasi. Et ut translationi fides 
plenissima non deesset, nee quidquam fraude aliqua nostrorum 
notitiae subtrahi posset, Christianis interpretibus etiam Sarracenum 
adjunxi. Christianorum interpretum nomina: Robertus Kecenen- 
sis, Armannus Dalmata, Petrus Toletanus; Saraceni Mahumeth 
nomen erat. Qui intima ipsa barbarae gentis armaria perscru- 
tantes, volumen non parvum ex praedicta materia Latinis lectoribus 
ediderunt. Hoc anno illo factum est quo Hispanias adii, . . . 
qui annus fuit ab Incarnatione Domini 1141." 

The letter to Bernard seems to be of date 1143, and that is 
the date, evidently, of the completion of Robert's translation, and 
so it is given at the end of the Qoran : " Illustri gloriosoque viro 
Petro cluniacensi Abbate praecipiente, suus Angligena, Robertus 
Retenensis librum istum transtulit Anno domini mcxliii, anno 
Alexandri mcccciii, anno Alhigere dxxxvii, anno Persarum quin- 
gentesimo undecimo." 

Robert and Hermann appear to have been associated in trans- 
lating scientific works particularly along the lines of astrology 
from Arabic into Latin, as well as the Qoran. 2 In connection 
with these astrological treatises a mediaeval reference, 3 probably 
contemporary, mentions Robert as "a man most learned in astrol- 
ogy." Peter the Venerable refers to Robert and Hermann as most 
acute and well-trained scholars, while Peter of Poitiers, in a letter 4 
of unknown date addressed to Peter the Venerable, cites Robert 

totam impiam sectam, uitamque nefarii hominis, ac legem, quam Alcoran, id est, col- 
lectaneum praeceptorum appellauit, sibique ab angelo Gabriele de coelo callatam miserrimis 
hominibus persuasit, nihilominus ex Arabico ad Latinitatem perduxi, interpretantibus 
scilicet uiris utriusque linguae peritis, Roberto Retenensi de Anglia, qui nunc Papilonensis 
ecclesiae archidiaconus est, Hermanno quoque Dalmata acutissimi et literati ingenii 
scholastico. Qubs in Hispania circa Hiberum Astrologicae arti studentes inueni, eosque 
ad haec faciendum multo precio conduxi. 

1 Patrologia Latina, Vol. 189, col. 671. 

2 See the article by Bjbrnbo, Hermannus Dalmata als Uebersetzer astronomischer 
Arbeiten, Bibliotheca Mathematica, Vol. VI, third series (1905), pp. 130-133 and p. 28 below. 

8 Steinschneider, Die Europaischeti Uebersetzungen aus dem Arabischen, Sitzungsbe- 
richte der Philosophisch-historischen Klasse der kaiserlichen Akademie der IVissensc/ia/teu, 
Vol. 149 (Vienna, 1905), p. 72. 

4 Patrologia Latina, Vol. 189, col. 661. 



28 INTRODUCTION 

as an authority on Mohammedan customs. A further manuscript 
reference 1 seems to indicate that in the year 1136 Robert was 
studying in Barcelona with Plato of Tivoli, while Fabricius 2 states, 
but upon what authority does not appear, that Robert travelled in 
Italy, Greece, and Spain. The time and the place of Robert's 
death are equally uncertain. 

To Peter the Venerable, Abbot of Cluny, who induced Robert 
to undertake the translation of the Qoran, the latter addressed a 
Saracenic chronicle, Chronica mendosa et ridicnlosa Saracenorum? 
which was published in 1550 with the Qoran. Other names have 
been connected with this translation of the Qoran, but a study of 
Peter's letters, and the introduction to Peter's treatise, " Against 
the Sect of the Saracens," 4 shows that Robert and Hermann were 
definitely requested to undertake the translation, which it appears 
from Robert's preface was finally completed by Robert alone. 
Possibly Peter of Toledo made an earlier, 5 unsatisfactory transla- 
tion of the same work. A Mohammedan by the name of 
Mohammed was engaged by Peter the Venerable to scrutinize 
the various treatises with a view to correcting errors due to 
mistranslation. 

Of Robert's works the version of the Qoran was completed in 
1 143. In a prefatory letter he states that this task was regarded 
by him only as a digression from his principal studies of as- 
tronomy and geometry, 6 but posterity knew of him through the 

1 Archer, Dictionary of National Biography, Vol. XLVIII (New York, 1896). p. 362 ; 
Professor Haskins examined the MS. but found no evidence for this statement. 

2 Bibliotheca med. et infim. Lat. (Florence, 1858-1859), Vol. VI, p. 107. 

3 Qoran, 1550 edition, pp. 213-223. 

4 Migne, Patrologia Latiua, Vol. 189, col. 659-720, Petri Venerabilis, Abbatis Clunia- 
censis noni, adversus nefandam sectam Saracenorum libri duo. 

5 Since Peter the Venerable expressly refers to a new translation by the labors of 
Robert and Hermann. In the letter to Bernhard of Clairvaux Peter says : " Mitto vobis, 
charissime, novam translationem nostram," which may mean, possibly, another document 
on Arabic customs. 

6 Machnmetis Saracenorum principis, eiusque successorum vitae, doctrina, ac ipse 
Alcoran (Basle, 1550), pp. 7-8 : Praefatio Roberti translatoris ad Dominum Petrum 
Abbatem Cluniacensem, in libro legis Saracenorum, quern Alcoran uocant, id est, Collec- 
tionem praeceptorum quae Machumet pseudopropheta per angelum Gabrielem quasi de 
coelo sibi missa confinxerit. Begins, Domino suo Petro diuino instinctu Cluniacensi 
abbati, Robertus Retenensis suorum minimus in Deo perfecte gaudere ; and ends, Istud 
quidem tuam minime latuit sapientiam, quae me compulit interim astronomiae geometriae- 
que studium meum principale praetermittere. Sed ne proemium fastidium generet, ipsi 
finem impono, tibique coelesti, coelum omne penetranti, coeleste munus uoueo : quod 
integritatem in se scientiae complectitur. Quae secundum numerum, & proportionem 



ROBERT OF CHESTER AND OTHER TRANSLATORS OF ARABIC 29 

earlier work rather than through the sciences to which he dedi- 
cated himself in this letter. Even the mathematician John Wallis, 
editor of one edition of this translation, did not connect the trans- 
lator with the Robertus Retinensis who was mentioned, as we 
have seen, by Wallis in his Algebra. An unpublished letter 
written by Robert, beginning Cum jubendi religio, is preserved 
in the Selden manuscript, sup. 31, in the Bodleian library. 

Several sets of astronomical tables are included among the 
products of Robert's literary activity. The Canones in motibus 
coelestium corporum ad meridiem urbis Londoniarum in duos 
partes, prior 11 49 ad Jidem tabularum toletanarum Arzachelis, 
altera pro anno 11 50 iuxta Alba tern Haracensis 1 evidently in- 
cludes two sets of tables. This reference to the methods of 
Al-Battani may account for the fact that Robert is sometimes 
credited with a translation of Al-Battani's tables. 2 Wallis asserts 
that the tables accommodated to the meridian of London were 
adjusted to the year 1150 and in fact the text in MS. Savile 21, 
fol. 86 r , states the date as March first, 11 50. Reference is made 
also to a preceding work as accommodated to the meridian of 
Toledo for the year 1148 or 1149 and based on tables by Rabbi 
Abraham ibn Esra. 3 Extracts, apparently from these tables of 
Robert, are found under the title, De diuersitate annorum ex 
Roberto Cestrensi super tabulas toletanas* and this contains a 
reference to a sexagesimal multiplication table, evidently con- 
structed by Robert, in which appear all products from 1 x 1 up to 
60 x 60. 

atque mensuram coelestes, circulos omnes, & eorum quantitates & ordines & habitudines, 
demum stellarum motus omnimodos, & earum eft'ectus atque naturas, & huiusmodi 
caetera diligentissime diligentibus aperit, nunc probabilibus, nonnunquam necessariis argu- 
ments innitens. 

1 Steinschneider. Sitzungsber., loc. cit.; MS. Savile 21, 63 ~95 v . Fol. 63 r , lucipiunt 
canones in motibus coelestium corporum, which begins, Quoniam cuiusque accionis quanti- 
tatem metitur celestium spacium, and proceeds with a discussion of various chronological 
systems. Fol. 86 r , Incipit pars altera huius operis que videlicet ad meridiem urbis Lon- 
doniarum iuxta Albatem Haracensis sententiam per Robertum Cestrensem contexitur : 
Begins, Premissa uero electe facultatis . . . ; ends, Ergo iuxta hanc scienciam planetarum 
loca figura ponenda sunt. See above, p. 17, note 1. 

2 See Nallino, Al-Battani opus astronomicum, Pubbl. del reale Osservatorio di Brera 
(Milan), Vol. XL (1903), Introduction. 

3 In MS. Savile : ebenza (?) 

4 MS. Digby 17. Beg. fol. 156, Diuersi astronomi secundum diuersos annos tabulas 
faciunt et quidam secundum annos Alexandri seu grecorum, alii secundum ierdaguth 
(Yezdegerd) seu persarum. 



30 INTRODUCTION 

In 1 144 Robert completed the translation, entitled De composi- 
tione alchemiae, or De re rnetallica, by one Morienus Romanus. 
This was published in Paris in 1546, and later by Manget. 1 
Another of the products of his literary activity, dated 1185 of the 
Spanish Era {circa 1 150 a.d.), was the work, De compositione astro- 
labii 2 which is ascribed to Ptolemy. The place of composition is 
given in one manuscript copy as London. 3 Like Adelard of Bath, 
Robert seems also to have written a chemical work, dealing with 
pigments and other associated topics, entitled Liber metricus qui 
dicitur Mappa claviculae. This version, Greek in its origin, is in 
verse. 4 

An astrological work, entitled Judicia Alkindi astrologi or 
De judiciis astrorum, which was written by the great Arabic phi- 

1 J. J. Manget, Bibliotheca Chemica Curiosa (Geneva, 1702), Vol. I, pp. 509-519, 
Liber De compositione Alchemiae, quern edidit Morienus Romanus, Calid Regi Aegyp- 
torum ; quern Robertus Castrensis de Arabico in Latinum transtulit. 

The prologue begins : Praefatio Castrensis. Legimus in Historiis veterum divinorum, 
tres fuisse Philosophos, quorum unusquisque Hermes vocabatur. . . . Sed nos, licet in 
nobis juvene sit ingenium, et latinitas permodica : hoc tamen tantum ac tarn magnum opus 
ad transferendum de Arabico in Latinum suscepimus. Vnde et de singulari gratia nobis 
a Deo inter modernos collata, summas illi Deo vivo, qui trinus extat et unus, grates referimus. 
Nomen autem meum in principio Prologi taceri non placuit, ne aliquis hunc nostrum 
laborem sibi assumeret, et etiam ejus laudem et meritum sibi quasi proprium vendicaret. 
Quid amplius ? humiliter omnes rogo et obsecro, ne quis nostrorum erga meum nomen 
mentis livore (quod saepe a multis fieri consuevit) tabescat. Scit namque Deus omnium, 
cui suam conferat gratiam : et spiritus ex gratia procedit, qui quos vult inspirat. Merito 
igitur gaudere debemus quum omnium Creator et conditor omnibus quasi particularem 
suam monstret divinitatem. 

The treatise itself begins : Morieni Romani, Eremitae Hierosolymitani, Sermo. Omnes 
Philosophiae partes, mens Hermetis divina plenarie attigit . . . and ends, Quod si quando 
Alchymia confecta fuerit, ejus una pars inter novem partes argenti ponatur, quoniam totum 
in aurum purissimum convertetur. Sic ergo Deus benedictus Amen, per cuncta seculorum 
secula. 

Explicit Liber Alchymiae de Arabico in Latinum translatus, anno millesimo centesimo 
octuagesimo secundo, in mense Februarii et in ejus die undecimo. 

2 Steinschneider, Zeitschrift f. Mathematik und Physik, Vol. XVI, p. 393, with refer- 
ence to the Vienna MS. 531 1 8 , De compositione astrolabii universalis liber a Roberto Cas- 
trensi translatus, which begins : Quoniam in mundi spera, and ends : ad altitudines 
accipiendas ; also to the Oxford MS. Cod. Canonic. Misc. 61 6 , Liber de officio astrolabii 
secundum mag. Rob. Cestrensem, in thirty-five chapters, which begins: 1. De gradu solis 
per diem et diei, and ends : et cetere ceteris per diametrum ut jam dictum est opponuntur. 
Revised after 11 50, as it cites the tables of that year (Bodleian MS. Canon Misc. 61, f. 22 2 ; 
examined by Professor Haskins). 

3 Steinschneider, ibid., p. 393, refers to the Gonville and Caius College, Cambridge, 
MS. 35 14 , Ptholomaei de compositione astrolabii, translatus a Rob. Cestiensi in civitate 
London; also MS. Digby 40, dates this " Aera 1185 in civitate London:" 

4 Steinschneider, loc. cit., p. 72. 



ROBERT OF CHESTER AND OTHER TRANSLATORS OF ARABIC 31 

losopher Ya'qub ibn Ishaq ibn Sabbah al-Kindi (died about 874 a.d.), 
was translated by Robert of Chester and not by Robertus Angli- 
cus in 1272 as usually 1 stated. This treatise in forty-five chapters 
is preceded by an introduction 2 which definitely establishes the 
authorship of the translation. In these prefatory remarks the 
writer addresses himself to " my friend Hermann, second to no 
astronomer of our time, of those who speak Latin," and further 
makes reference to the translation as being made at the request 
or wish of Hermann. As some manuscripts ascribe this work to 
Robert of Chester who was associated with Hermann the Dal- 
matian, the reference to Hermann would be almost conclusive, 
and the terminology of the introduction by its similarity to that of 
the Qoran introductory letter establishes the fact beyond question. 
A rearrangement of the Al-Khowarizmian tables as translated 
by Adelard of Bath was made by Robert of Chester. 3 The basis 

1 Suter, Die Mathemaliker und Astronomen der Araber und ihre IVerke, loc. cit., p. 26; 
Steinschneider, Die europdischen Uebersetzutigen aus dem Arabischen, loc. cit., p. 66, et al. 

2 Introduction, for which I am indebted to the courtesy of Professor C. H. Haskins, 
following MS. Ashmole, 369, fol. 85*. The heading and explicit are from the Cotton MS. 
App. vi. Numerous other manuscripts of the same are found in European libraries. 

Incipiunt indicia Alkindi astrologi, Rodberti de Ketene translatio. 

Quamquam post Euclidem Theodosii cosmometrie libroque proportionum libentius in- 
sudarem, unde commodior ad almaiesti quo praecipuum nostrum aspirat studium pateret ac- 
cessus, tamen ne per meam segnitiem nostra surdesceret amicitia, vestris nutibus nil preter 
equum postulantibus, mi Hermanne, nulli latinorum huius nostri temporis astronomico 
sedere (sedem?) penitus parare paratus, eum quern commodissimum et veracissimum inter 
astrologos indicem vestra quam sepe notauit diligencia, voto vestro seruiens transtuli non 
minus amicitie quam pericie facultatibus innisus. In quo turn nobis turn ceteris huius 
scientie studiosis placere plurimum studens, enodato verborum vultu rerum seriem et 
effectum atque summam stellarium effectuum pronosticationisque quorumlibet eventuum 
latine brevitati diligenter inclusi. Cuius examen vestram manum postremo postulans non 
indigne vobis laudis meritum si quod adsit communiter autem fructus pariat, mihique non 
segne res arduas aggrediendi calcar adhibeat, si nostri laboris munus amplexu fauoris elu- 
cescat. Sed ne proemium lectori tedium lectionique moram faciat vel afferat, illius prolixi- 
tate supersedendo, rem propositam secundum nature tramitem a toto generalique natis 
exordiis texamus, (P)rius tamen libri totius capitulis enunciatis ad rerum evidentiam 
suorumque locorum repertum facilem. Explicit prohemium. Incipiunt libri capitula. 

Primum igitur capitulum, zodiaci diuisiones, earumque proprietates, tarn naturales quam 
accidentales generaliter complectitur. . . . 

The text proper begins (Ashmole MS.) : Circulus itaque spericus cuius atque terre cen- 
trum est . . . and ends : sequitur in proximo. 

3 Haskins, The English Historical Review, XXVI, footnote on p. 498: Madrid MS. (no. 
10016) of the translation of the Al-Khowarizmian tables, which has this heading: Incipit 
liber ezeig id est chanonum Alghoariznii per Adelardum bathoniensem ex arabico sumptus 
et per Rodbertum cestrensem or dine digestus ! Suter, Die astron. Tafeln des al-Khwarizmi 
in der Bearbeitnng des Maslama ibn Ahmed al-Madjriti und der lot. Uebers. des Athelard 
von Bath auf Grund der Vorarbeiten von A. Bjornbo \ und R. Besthorn, loc. cit. 



32 INTRODUCTION 

of this revision was doubtless a translation of the same tables by 
Hermann, who refers to his own translation in his astrological 
work, Introductorium in astronomiam Albumasaris abalaclii, octo 
continens libros partiales} This work is dedicated to Robert ; 
the similarity of the phraseology with that of the prefaces by 
Robert bears witness to the intimacy of their literary labors. Also, 
in the prologue to Hermann's translation of Ptolemy's Planisphere? 
Robert is mentioned as an associate of Hermann and again in the 
text. This work on the Planisphere has been incorrectly ascribed 
in the printed edition of 1537 and elsewhere to Hermann's pupil, 
Rudolph of Bruges. 3 

1 H. Suter, in a letter (Aug. 13, 1914) to the author. 

The Introductorium was printed at Augsburg in 1489, and other editions followed. I 
am indebted to Professor Haskins for a transcription of the preface, from which the follow- 
ing passage is taken : Que cum ego prolixitatis exosus et quasi minus contentia : cum et 
hunc morem latinis cognoscerem preterire volens animo ipso potius tractatum exordiri 
pararem. Tu mihi studjorum olim specialis atque inseperabilis comes, rerumque et actuum 
per omnia consors unice, mi Rodberte, si memores obviasti dicens : Quanquam equidem 
nee tibi pro amore tuo, mi Hermanne, nee ulli consulto aliene lingue interpreti in rerum 
translationibus. . . . 

Steinschneider, Ueber die Mondstationen (JVaxatra) und das Buck Arcandum, Zeit- 
schrift d. deutschen morgenlandischen Gesellschaft, Vol. XVIII (1864), pp. 170-172, and 
Die Hebraischen Ueber set zungen, pp. 568-570, and in Die europdischen Ueber setzungen, 
loc. cit., p. 34, demonstrated from the introduction that Hermann was the author. This 
view was accepted by Bjornbo, Hermanmts Dalmata als Uebersetzer astronomischer 
Arbeiten, Bibliotheca Mathematica, third series, Vol. IV (1903), pp. 130-133. 

2 J. L. Heiberg, Claudii Ptolemaei Opera quae exstant omnia, Vol. II, Opera astrono- 
mica minora, pp. clxxxiii-clxxxvi : "• tuam itaque uirtutem quasi propositum intuentes 
speculum ego et unicus atque illustris socius Rebertus Ketenensis nequitie dispicere, licet 
plurimum possit, perpetuum habemus propositum, cum, ut Tullius meminit, misera sit for- 
tuna, cui nemo inuideat." 

In the text proper Hermann inserts the following note: "quern locum a Ptolomeo 
minus diligenter perspectum cum Albateni miratur et Alchoarismus, quorum hunc quidem 
ope nostra Latium habet, illius uero comodissima translatione studiosissimi Roberti mei 
industria Latine orationis thesaurum accumulat, nos discutiendi ueri in libro nostro de cir- 
culis rationem damus." 

8 After this chapter was in type a noteworthy article, The reception of Arabic science in 
England, by Professor C H. Haskins, appeared in The English Historical Review, Vol. 
XXX (191 5), pp. 56-69. The works of Robert of Chester are carefully considered. For- 
tunately, through the courtesy of Professor Haskins, I was enabled to incorporate in advance 
the results which bear directly upon our discussion. 



CHAPTER IV 

The Influence of Al-Khowarizmi's Algebra upon the 
Development of Mathematics 

By the translators of Arabic lore we are brought from Islam to 
Christendom. Mathematical science in Europe was more vitally 
influenced by Mohammed ibn Musa than by any other writer from 
the time of the Greeks to Regiomontanus (1436-1476). Through 
his arithmetic, presenting the Hindu art of reckoning, he revolu- 
tionized the common processes of calculation and through his 
algebra he laid the foundation for modern analysis. Evidence of 
the influence of the great Arab is presented by the relatively large 
number of translations and adaptations of his various mathematical 
works which appeared before the invention of printing. Un- 
doubtedly the earliest translation of the Arabic algebra, although 
not the most widely used, was that made by Robert of Chester. 
Probably the version published by Libri appeared shortly afterwards 
for, as we have mentioned, Gerard of Cremona employs the ter- 
minology of that version in algebraic work. Roger Bacon (1214- 
1294), too, has occasion 1 to mention the algebra, as well as the 
arithmetic, and uses terms not found in Robert of Chester's version. 
Bacon shows that he had but superficial familiarity with the subject, 
for he made incorrect statements about the fundamental elements of 
the algebra. Similarly, Vincent de Beauvais (about 1275) in his 
encyclopaedic work, Speculum Principale? refers under arithmetic 
to the book, qui apud Arabes mahalehe dicitur. Albertus Magnus 
(1 193-1280) mentions the tables of Al-Khowarizmi. 3 

Even earlier than Roger Bacon is Leonard of Pisa, whose monu- 
mental Liber abaci contains a chapter involving the title, Aljebra 

1 In his unfinished Scriptum principale. See J. H. Bridges, The '■'■Opus Ma/us' 1 ' 1 of 
Roger Bacon (London, 1900), Vol. I, p. lvii ; D. E. Smith, The place of Robert Bacon in 
the history of mathematics, in Roger Bacon, Essays, collected and edited by A. G. Little 
(Oxford, 1914), p, 177. 

2 Liber xviii. Cap. V, De Arithmetica ; Liber xviii. Cap. ix, is entitled, De Computo et 
algorismo, and takes up representation of numbers by the Hindu numerals. 

3 Steinschneider, Zum Speculum astronomicum des Albertus Magnus, Zeitschrift f. 
Mathematik unci Physik, Vol. XVI, pp. 375-376, and in Zeitschrift d. deutschen morgenl. 
Gesellschaft, Vol. XXV, pp. 404-405. 

33 



34 INTRODUCTION 

et almuchabala} The first draft of this work was written in 1202, 
and in 1228 a revised and enlarged version appeared, dedicated to 
Michael Scotus. Woepcke has shown that Leonard drew many 
of his problems from Al-Khowarizmi, 2 but some of these may have 
come indirectly through Abu Kamil, from whom, as I have 
shown, 3 Leonard took many of his algebraic problems. In the 
manuscripts of the Italian's treatise the only mention of Al- 
Khowarizmi is in the margin, simply Maumet, at the beginning 
of the section dealing with algebra; but the term algorismus 
occurs for arithmetic. 4 

In the century following Leonard of Pisa, another Italian 
mathematician, William of Luna, is reputed to have put Al- 
Khowarizmi's algebra into the Italian language. Raffaello di 
Giovanni Canacci, a Florentine citizen of the fifteenth century, 
states in an Italian work on algebra, 5 as yet in manuscript, that 
William had translated the rules of algebra out of Arabic into " our 
language." Reference to his work is also made by at least three 
writers of the sixteenth century, the Florentine Francesco Ghaligai, 
the Spaniard Marco Aurel, and another Spaniard Antich Rocha 
of Gerona. 6 An Italian manuscript of 1464 in the library of 
George A. Plimpton, Esq., of New York, does contain an Italian 
version of the Algebra of Al-Khowarizmi in which reference is 
made to William of Luna as a translator of algebra. The possi- 
bility is that we have here the version of William. The writer of 
the manuscript is not known, but he explicitly states that he bases 
his treatise on the labours of numerous predecessors in this field. 
One chapter, as I have shown in a recent study of this manu- 
script, 7 deals with the algebra of an unknown Maestro Biagio 
(died 1340) and to Leonard of Pisa another section is devoted. 
The writer purposed also to deal with the works of a " subtle 
Maestro Antonio," doubtless Antonio Mazzinghi da Peretola, who 
wrote a treatise on algebra called il fioretto, and a " Maestro 

1 Scritti di Leonardo Pisano (Vol. I, Rome, 1857 ; Vol. II, Rome, 1862), Vol. I, p. 406. 

2 Extrait du Fakhri, p. 29. 

8 The Algebra of Abu Kamil, loc. cit. 
* Scritti, Vol. I, p. 1. 

5 Codex Palat. 567, Biblioteca Nazionale, Florence. 

6 Ghaligai, Pratica d^arithmetica (Florence, 1552) ; Aurel, Libro primero, de arith- 
metica algebratica (Valencia, 1552) ; Rocha, Arithmetica (Barcelona, 1565). 

7 An Italian Algebra of the fifteenth century, Bibliotheca Mathematica, third series, 
Vol. XI, pp. 209-219. 



THE INFLUENCE OF AL-KHOWARIZMI'S ALGEBRA 35 

Giouanni," but either this manuscript is incomplete or the plan 
was not carried out. Prominent also in the discussions is an 
Augustinian monk, Gratia de Castellani (about 1340), famed as a 
theologian. The relatively large number of names of men who 
had evidently attained something more than local repute in the 
study of algebra shows the place which it had reached in in- 
struction. 

Another prominent writer of the fourteenth century, Johannes 
de Muris, included a discussion of algebra in the third book of 
his popular Quadripartitum numerorum} Of this section of the 
work of John of Meurs I have recently made a study, 2 showing 
that he drew extensively from Leonard of Pisa and from Al- 
Khowarizmi, thus continuing the Arabic influence. Regiomon- 
tanus included the work in a list of important early works on 
mathematics, and further Regiomontanus refers to algebra as the ars 
rei et census. This corresponds to a line of the Quadripartitum, : 

" Que tamen ars minor est quam sit de censibus et rei." 

Later the expression, Arte magiore? or Ars mayor? or Ars magna? 
was used for algebra, tracing back to this passage here given, in 
which ars minor refers to arithmetic as opposed to algebra. 
Adam Riese presents the problem, 6 x 2 + 21 = \ox, as being found 
in the eleventh chapter of the third book of the Quadripartitum ; 
we have mentioned that this problem is one of the type problems 
found in Al-Khowarizmi's algebra. Another French author who 
gives an adaptation in Latin of the Arabic algebra is Rollandus, 
Canon of St. Chapelle. At the command of John, Duke of Lan- 
caster, Rollandus wrote in the year 1424 a compendium of mathe- 
matics ; the labor of composition was considerably lightened by 
making large extracts from the Qtiadripartitum, including most of 
the arithmetic and algebra. A summary of the contents of the 

1 Two chapters of the second book, dealing with arithmetic, were published by A. Nagl, 
Das Quadripartitum etc., Abhaudl. 2. Geschichte der mathem. Wissenschaften, Vol. V 
(Leipzig, 1890), pp. 137-146. 

2 The " Quadripartitum numerorum " of John of Meurs, Bibliotheca Mathematica, 
third series, Vol. XIII, pp. 99-114. 

3 Paciuolo, Summa d'arithmetica (Venice, 1494). 

4 In Aurel's Libro primero, de arithmetica algebratica (Valencia, 1552). 

5 Cardan, Ars Magna (Nuremberg, 1545). 

c B., Berlet, Adam Riese, sein Leben, seine Rechenbucher und seine Art zu rechnen, Die 
Coss von Adam Riese (Leipzig, 1892), p. 37. 



36 INTRODUCTION 

manuscript is given in the Rara Arithmetical but the somewhat 
extensive treatment of algebra is not mentioned. 

The first work in the German language on algebra was an 
excerpt from Al-Khowarizmi which begins : 2 " Mohammed in the 
book of algebra and almucabala has spoken these words 'census, 
radix (root), and number." This is followed by two problems 
from the text. The manuscript which contains this brief dis- 
cussion, of date 1 46 1, is now in Munich, having been moved 
from the Benedictine Abbey of St. Emmeran. The first treat- 
ment in the English language appears to be that by Robert 
Recorde, The Whetstone of Witte, which was published in 
1557. This work which, as I have elsewhere shown, 3 does 
not display any marked originality on the part of Recorde, intro- 
duced our present symbol of equality, = , and contributed to the 
study of algebra in England by presenting the material in the 
mother tongue. 

Regiomontanus seems to have been familiar with Al-Kho- 
warizmi's work, for he not only refers to the art of thing and 
square {ars rei et census), but also uses certain technical expres- 
sions, restaurare defectus, for example, similar to those in the 
algebra. A manuscript copy 4 of Mohammed ibn Musa's algebra 
in Mr. Plimpton's collection shows astonishing similarity to the 
handwriting and abbreviations of Regiomontanus as well as to 
the form of equation used by the great German. Furthermore, 
some of the problems given in this manuscript, which are not 
part of Al-Khowarizmi's text, are discussed by Regiomontanus in 
his correspondence with Cardinal Blanchinus. 5 We must sup- 
pose him to have been familiar with this text if not actually, as 
we suspect, the transcriber of this copy. Regiomontanus was 
twenty years of age when this manuscript was written (1456), and 
we know that he did transcribe numerous mathematical and 
astronomical works of historical importance. Algebra was em- 

1 Rara, 446-447. 

2 Gerhardt, Znr Geschichte der Algebra in Dentschland, Monatsbericht der Konigl. 
Akad. d. Wissensch. zu Berlin, 1870, pp. 141—1 53. 

3 Karpinski, The whetstone of witte, Bibliotheca Mathematical third series, Vol. XIII, 
pp. 223-228. 

4 Rara Arithmetica, 454-456. 

5 M. Curtze, Der Briefwechsel Regiomontan''s mil Giovanni Bianchini, Jacob von 
Speier und Christian Roder, Abhandl. z. Gesch. d. math. Wissen., Vol. XII (Leipzig, 
1902), problem 6, p. 219. 



THE INFLUENCE OF AL-KHOWARIZMI'S ALGEBRA 37 

ployed by Regiomontanus in his trigonometry 1 in the solution of 
problems. 

Regiomontanus has been cited by Nesselmann 2 as an illustra- 
tion of one who employed rhetorical algebra as opposed to synco- 
pated or symbolical. Later writers have followed Nesselmann in 
the assertion that Regiomontanus used rhetorical algebra, but, 
whereas the statement in Nesselmann is correct in so far as the 
illustration which he gives is concerned, the assumption that this 
was the general practice of the great Teuton is an error. In 
fact, his correspondence with Cardinal Blanchinus shows that he 
had a form of equation little inferior to ours. The + sign which 
he uses is a ligature for et, the minus sign a ligature for minus, 
and for an equality sign he uses a single straight line. Further 
he has separate symbols for the various powers of the unknown 
up to the cube, so that Regiomontanus approached modern forms 
more closely than most mathematicians even of the sixteenth 
century. This attempted division of the history of algebra into 
rhetorical, syncopated, and symbolic periods is an excellent illus- 
tration of a plausible and taking theory, in historical matters, 
which lacks only the first essential for such a theory ; namely, 
historical evidence. Development in mathematics, as in art and 
literature, does not proceed in a logical manner, but rather in 
waves advancing and receding, and yet withal constantly 
advancing. 

We have mentioned the Hebrew translation of the algebra 
of Abu Kamil, which was made by Mordechai Finzi (about 
1475 a.d.) of Mantua. Another treatise on algebra, in Hebrew, 
dedicated to Finzi, was written by Simon Motot. 3 As the words 
cosa and censo are mentioned by Motot as being found in the 
works of Christian authors with which he was familiar the 
Italian source of his information is established, although the par- 
ticular writers in question are not known. As we have above 
indicated, the Italians were sufficiently active in this science, so 
that many Italian works on algebra were available, in manuscript, 
at this time. 

In the summer of i486 Johann Widmann of Eger is known to 

1 Regiomontanus, De triangulis libri quinque (Nuremberg, 1533), problem 12, p. 51. 

2 Nesselmann, Die Algebra der Griechen (Berlin, 1842), p. 303. 

3 G. Sacerdote, Le livre de I'algibre et le probleme des asymptotes de Simon Motot, 
Revue des etudes juives, 1 893-1 894. 



38 INTRODUCTION 

have lectured on algebra in the university at Leipzig, and the fee 
for the course was set extraordinarily high, being two florins. 1 
Widmann was in possession of the Dresden manuscript, 2 which 
contains Robert of Chester's version of the algebra of Al-Kho- 
warizmi, and himself added certain algebraic problems to another 
part of the same manuscript dealing with algebra. The Arab 
Al-Kalasadi, 3 contemporary with Widmann, wrote also on similar 
topics, and although he does not cite Al-Khowarizmi, yet he 
continues the old order of the six types of quadratic equations. 

Adam Riese wrote in 1524 a work on algebra entitled, Die 
Coss, which contains, as we have noted, the problem 

x 2 + 21 = 10 x. 

Riese refers to "that most celebrated Arabic master Algebra, 
learned in number, whose like in computation there never was, 
and hardly will any one exceed him." He refers also to the 
book, "named gebra and almucabola," by this mythical Algebra. 
A reference to Algum is also doubtless to Al-Khowarizmi. Sev- 
eral contemporaries, students of the Coss, are mentioned 4 by Adam 
Riese, and included among these is Grammateus, 5 also known as 
Schreiber or Scriptor, to whom is credited the first algebra in 
print in the German language. Interesting is Riese's note that 
Hans Conrad, to whom he frequently refers, paid the mathema- 
tician Andreas Alexander one florin in gold, to be taught how to 
solve certain types of problems by the Coss. This title is from 
the Italian cosa (Latin res, Arabic shcii) and connects with the 
use by Al-Khowarizmi, and subsequent Arabs, of the word shdi, 
meaning thing, for the first power of the unknown. For centuries 
the title continued in circulation in Germany, and even in English 
appeared in the form, "the arte of cosslike nombers." 6 

Luca Paciuolo, otherwise Luca de Burgo San Sepulchro, to 
whom is credited the first printed work on algebra, was evidently 

1 Wappler, Zur Geschichte der deutschen Algebra in 15. Jahrhnndert, in Programm, 
Zwickau, 1887, pp. 9-10; also in Zeitschrift f. Math, und Physik, Hist.-lit. Abtheil., 
Vol. 45, 1900. 

2 Codex Dresden C. 80. 

3 Woepcke, Atti deW accad. pont. de" 1 nuovi Lincei, Vol. XII (Rome, 1859), 230-275, 

399-438- 

4 Berlet, loc. at., pp. 33, 34, 36, and 62 for references. 

6 . . . Rechenbuchlin, including Etlichen Regeln Cosse, written 15 18 and published 1521. 
6 Recorde, loc. cit. 



THE INFLUENCE OF AL-KHOWARIZMI'S ALGEBRA 39 

influenced by Al-Khowarizmi. Paciuolo gives * the equation 

x 2 + \ox= 39 

and presents the geometrical explanation as given by the Arab. 
In certain others of the early printed algebras the fundamental or 
type equations as given by Al-Khowarizmi do not appear. Of 
such are the works by Grammateus appearing in 15 18, by Chris- 
tian Rudolph in 1525, 2 and by Estienne de la Roche in 1520, 3 
whose work is known to have been a plagiarism of the Triparty 
by Nicolas Chuquet (1484). 4 However, many other writers did 
continue the type problems of the first systematic treatise. Thus 
Elia Misrachi 5 in an arithmetic which appeared in Constantinople 
in 1534, eight years after the author's death, devotes a section to 
algebra, and this is to a large extent an adaptation of the Algebra 
and Almucabala. In the work by Perez de Moya (1562) G and in 
the arithmetic of 1539 by Cardan we come upon the type equations. 
Mennher de Kempten, a Dutch mathematician, states that Algo- 
rithmo was the first writer on algebra. Ghaligai, the Italian, and 
the Spaniard Pedro Nunez follow the peculiar order of equations 
found in Al-Khowarizmi. In these and other ways we might 
trace through the centuries the persistent influence, direct and 
indirect, of our Arabic author, but that is beside our present 
purpose. 

Among the writers who made a serious study of Robert of 
Chester's translation we must place Johann Scheybl (1494-15 70), 
who was professor of mathematics at Tubingen from about 1550 
to the time of his death. He was the author of an algebra which 
appeared in two editions in Paris, 1551 and 1552. This treatment 
of algebra was first published in 1550 by Scheybl, prefixed to his 
Greek and Latin edition of the first six books of Euclid. Scheybl 

1 Summa de arithmetica (Venice, 1494), fol. 146 rec. 

2 I have not seen a copy of this edition. My remark is based upon Die Coss Christoffs 
Rudolffs (Konigsberg, 1553) by Stifel. From certain notes about the history of the ter- 
minology and the words dragma, res and substantia, and the like, it appears that Stifel 
had seen a copy of Robert of Chester's version. 

3 Smith, Rara Arithmetica, p. 128. 

4 A. Marre, Notice sur Nicolas Chuquet et son Triparty en la science des nombres, 
Boncompagni's Bulletino di bibliografia e di storia delle scienze matematiche e fisiche, Vol. 
XIII (1880), pp. 555-592; the text of the Triparty, same volume, pp. 593-659 and pp. 
693-814; the third section from 736-814 deals with quadratic equations and problems. 

5 G. Wertheim, Die Arithmetic des Elia Misrachi (Braunschweig, 1896), pp. 54-59. 

6 Arithmetica practica, y speculativa (Salamanca, 1562). 



40 INTRODUCTION 

prepared for publication the Latin version of Al-Khowarizmi's 
algebra as translated by Robert of Chester. His manuscript copy 
is now in the Columbia University library. The title page * reads, 
in translation : ' A brief and clear exposition of the rules of algebra 
by Johann Scheybl, Professor of Euclid in the famous University 
of Tubingen. To this is added the work, On given numbers, by 
that most excellent mathematician Jordanus. Furthermore there 
is presented the book containing the demonstrations of the rules 
of algebraic equations, written some time ago in Arabic. All of 
these are now published for the first time by the above-mentioned 
Scheybl. These are corrected as far as possible and illustrated 
by appropriate and useful examples.' 

The algebra contained in this manuscript is not the same as 
the published work mentioned above. However, the method of 
treatment is not materially different. The work by Jordanus 
Nemorarius, entitled De numeris datis, dates from the early part 
of the thirteenth century. The importance of the work, chiefly 
with respect to the development of algebra, is well attested by the 
fact that Regiomontanus and Maurolycus 2 both planned to publish 
the work, although neither carried the plan to completion. The 
work was published in 1879 by Treutlein. 3 The Scheybl version 
contains the complete list of 113 propositions to which Chasles 
made reference in 1841. These are divided into four books con- 
taining respectively 30, 26, 22 and 35 propositions. Scheybl adds 
solutions by the rules of algebra 4 in which he employs the same 
algebraic notation as in his published algebra, but he does not 
give the complete text of the work of Jordanus. The other two 
works in the manuscript are presented in this monograph. The 
manuscript was carefully prepared, but, for some reason which we 
do not know, the publication was not accomplished. 



1 Breuis ac dilucida regularum Algebrae descriptio, autore Joanne Scheubelio, in inclyta 
Tubingensi academia Euclidis professore ordinario. 

Huic accedit liber consumatissimi mathematici Jordani, de datis. 

Liber praeterea, continens demonstrationes aequationum regularum Algebrae, Arabice 
olim conscriptus. 

Quae (corr. Qui) ambo ab eodem Scheubelio nunc primum, quantum fieri potuit, emen- 
dato (corr. emendati) in lucem aedita, et aptissimis atque utilibus exemplis illustrata sunt. 

2 Treutlein, Abhandlungen zur Gesch. d. Math., Vol. II (Leipzig, 1879), pp. 127-128. 

8 Treutlein, loc. cit., pp. 127-166; corrected by Curtze, Zeitschrift f. Mathematik und 
Physik, Vol. XXXVI (1891), pp. 1-23, 41-63, 81-95, an( * 121-138. 
4 Frequently adding, Sequitur solutio ex regula Algebrae. 



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THE INFLUENCE OF AL-KHOWARIZMI'S ALGEBRA 41 

A few words about the life of Scheybl * may be of interest. His 
student days include an early stay at the University of Vienna, 
made famous in mathematical studies by Peurbach and Regiomon- 
tanus. In 1532 Scheybl matriculated at Tubingen, which was a 
stronghold of Protestantism, and in 1535 he was a student there. 
In 1540 he was Magister in Tubingen, and four years later Docent 
in mathematics. After another period of about five years we find 
him Professor (ordinarius) of Euclid, and in 1555 Professor of 
Euclid and Arithmetic. How little some aspects of university 
life have changed during four centuries is shown by the fact that 
Scheybl twice, in 1551 and 1562, requested of the university 
authorities an increase of salary in order that he might pay his 
debts and obtain the necessaries of life. In addition to the 
treatises on geometry and algebra which have been mentioned, 
Scheybl published other works on geometry and arithmetic. 

As late as the end of the sixteenth century an able mathema- 
tician, Adrien Romain, deemed the algebra of Al-Khowarizmi 
sufficiently worthy of serious study to justify him in publishing a 
commentary on the work. The version upon which Romain 
based his study is that of Robert of Chester. In the course of 
his commentary he gives small portions of this translation. At 
the time of writing this, according to Bosnians 2 in 1598 or 1599, 
Romain was teaching in Wiirzburg. His student and teaching 
life covered periods of residence in Germany, Italy, and Poland, 
besides his native Louvain where he studied and taught. 

1 H. Staigmiiller, Johannes Scheubel, ein deutscher Algebraiker des XVI Jahrhunderts 
Abhand. zur Gesch. d. math. Wissenschaften, Vol. IX, pp. 431-469. 

2 H. Bosmans, S.J., Le fragment du commentaire d' "Adrien Romain stir Valgebre de 
Mahnmed ben Musa el-Chowarezmi, Annates de la Societe scientifique de Bruxelles y 
Vol. XXX (1906). 



CHAPTER V 

The Arabic Text and the Translations of Al-Khowarizmi's 

Algebra 

The Arabic text of Al-Khowarizmi's algebra, together with an 
English translation, was published by Frederick Rosen in 1831. 
This excellent * work is unfortunately out of print, and so is not 
available to most students of mathematics. The translation is 
made with care and intelligence, but not literally. Thus the 
Arabic invocation to the Deity is frequently omitted, just as it 
often is by modern translators. 

The translation of the Algebra into Latin was made not only 
by Robert of Chester, but also, as we have indicated, by some 
other student of Arabic science who lived about the same time as 
Robert. This Latin version, as found in two Paris manuscripts, 
was published by Libri ; 2 Bjornbo believed that he had established 3 
this to be the work of Gerard of Cremona, which indeed is probable. 
A list of the numerous translations due to Gerard was made soon 
after Gerard's death by some friend and admirer, and the list was 
published by Boncompagni. 4 Included among the titles is the 
algebra, Liber alchoarismi de iebra et almucabtila tractatus I. 
However, the question is somewhat complicated by the fact that a 
mediaeval adaptation of the algebra which was published by Bon- 
compagni 5 bears the name of Gerard of Cremona. The text of 
this version does not follow the Arabic at all closely, and there is 
little reason for considering it as a direct translation. Probably 
the meaning of the title 6 is that the text of this version is based 
upon Gerard's translation. 

1 Portions of the Arabic text and translation have been examined by Professor 
W. H. Worrell, to whose courtesy I am indebted for the information about the character 
of the translation. 

2 Histoire des sciences mathanatiques en Italie (Paris, 1838), Vol. I, pp. 253-297. 

3 Gey-hard von Cremona's Ubersetzung von AlkwarizmPs Algebra und von Euklid's 
Elementen, Bibliotheca Mathematica, third series, Vol. VI (1905), pp. 239-248. 

4 Delia vita e delle opere di Gherardo Cre/nonese etc., Atti dell" 1 Accademia de' nuovi 
Lincei, Vol. IV (185 1), pp. 4-7. 

5 Loc. tit., pp. 28-51. 

6 Loc. tit., p. 28 : Incipit liber qui secundum Arabes vocatur algebra et almucabala, et 
apud nos liber restauracionis nominatur, et fuit translatus a magistro Giurardo cremonense 
in toleto de arabico in latinum. 

42 



THE ARABIC TEXT AND THE TRANSLATIONS 43 

The Libri text varies essentially in phraseology and construction 
from that by Robert. The Arabic is closely followed up to the 
long list of problems, " Various Questions." Even here all the 
problems with the exception of two * are given in the Latin by 
Gerard, but not absolutely in the order in which they occur in the 
Arabic. The slight changes in the sequence of problems may 
well have been the fault of the particular Arabic manuscript which 
Gerard used, if it is not due to some transcriber of Gerard's work. 
One problem which is omitted is not very clear in the Arabic, but 
the second omission is a problem of the same type as others which 
are given. Some other slight omissions are made in the Latin 
text, and the longest of these corresponds to the passage in our 
text p. 84, line 25 to p. 86, line 2. Another omission in the Libri 
text corresponds to our text, page 74, line 25, quod . . . reperies. 
The Libri text also frequently omits the common invocation to 
the Deity which is so often interjected by Arabic writers. 

The Latin translation by Robert of Chester is not as faithful 
nor as correct as the text ascribed to Gerard of Cremona, pub- 
lished by Libri. Omissions, transpositions, and additions to the 
text are so numerous that it does not seem desirable to list them 
all. No evidence exists, however, that Robert's text is based 
upon another Arabic original than that of the Libri text. The 
text proper, as opposed to the illustrative problems, follows the 
general lines of the Arabic original. The longest omission is the 
section dealing largely with the operations upon the square root 
of 200, which is illustrated, in the Arabic and in the Libri text, 
by geometrical figures with corresponding demonstrations. 2 

A sentence is left out on page 98 of our text, line 6, after the 
word aequiparatur. This sentence Rosen translates, ' Compute 
in this manner every multiplication of the roots, whether the 
multiplication be more or less than two.' Lines 9-1 1, Natura . . . 

1 Rosen's translation, p. 48, line 15 to p. 50, line 5, and p. 53, lines 12-20. Neither of 
these problems is given by Robert of Chester, nor does either appear in the Boncompagni 
version. The first problem reads : " If some one say : ' I have purchased two measures of 
wheat or barley, each of them at a certain price ; I afterwards added the expenses, and the 
sum was equal to the difference of the two prices, added to the difference of the measures.' " 
The second reads : " Three-fourths of the fifth of a square are equal to four-fifths of its 
root." 

2 Rosen, loc. cit., p. 27, lines 5-18, and p. 31, line 11, to the bottom of p. 34; Libri, 
loc. cit., p. 269, lines 2-12, and p. 271, line 16-p. 274, line 14. The Libri version omits 
the statement of one problem, as stated by Rosen, p. 27, lines 14-16, but the geometrical 
explanation is complete. 



44 INTRODUCTION 

fractionifais, on the same page of our text, seems to be an addition 
by Robert. The introduction of the passage, ' On Mercantile 
Transactions,' pp. 120-124, is not at all carefully translated by 
Robert, who retains poor transliterations of four technical expres- 
sions used in the Arabic. The four expressions in question refer 
to the four terms of a proportion in which when three are given 
the fourth is determined. If a given quantity of goods is sold at 
a fixed or set price, then the price of any other quantity of the 
same goods, or the amount of goods to be obtained for a given 
sum of money, is determined by a proportion in which the three 
given quantities enter. The unit of measure, or quantity sold at 
a fixed price, is termed by Robert Almusarar, instead of al- 
musa-ir, and the fixed price Alszarar, instead of al-sVr ; the 
quantity of goods desired is Almuthemen, instead of al-muthamman 
and the amount to be expended for goods is termed Althemen, 
instead of al-thaman. Magul, which is used by Robert for the 
unknown term in a proportion, would be in modern transliteration 
al-maqul. 

Robert of Chester does not present the complete list of prob- 
lems which occur in the Arabic text of Al-Khowarizmi's algebra, 
but only a selection of about one-half of the total number. Upon 
what basis this selection was made does not appear, except that 
typical problems are chosen, and the repetitions which are found 
in the Arabic and the Libri text are eliminated. In the foot- 
notes to our English version we have indicated the problems 
which have been omitted by our author. 

The translation of the text and solutions of the problems which 
are given present peculiarities entirely similar to those which 
have been noted in the preceding discussion of the Latin text by 
Robert. A noteworthy omission is made both by Robert of Chester 
and by the translator of the version published by Libri. This con- 
cerns the fifth problem of the set of six which illustrate in order 
each of the six types of quadratic equations. After the solution 
of the problem to the point to which our text 1 carries the prob- 
lem, the Arabic, as translated by Rosen, adds : ' Or, if you please, 
you may add the root of four to the moiety of the roots ; the sum 
is seven, which is likewise one of the parts. This is one of the 
problems which may be solved by addition and subtraction.' 

1 Page 108, lines 1— 13. 



CHAPTER VI 

Preface and Additions Found in the Arabic Text of 
Al-Khowarizmi's Algebra 

The Arabic text of Al-Khowarizmi's algebra published by Rosen 
contains an author's preface which is not found either in the trans- 
lation published by Libri, or in that by Robert of Chester. As 
this reveals his conception of the purpose of the algebra, as well 
as some of the causes which led him to undertake the work, we 
present it here in the translation by Rosen. 1 Such prefaces in 
Arabic works usually, just as this one, contained invocations to 
the Deity and to Mohammed his prophet : in consequence the 
Christian translators, who were commonly connected with the 
Church, were wont to leave them out. A summary of the sections 
in the Arabic text which appear in neither of the Latin transla- 
tions is also given since the Arabic-English work by Rosen is not 
widely available, and since these additions show that Al-Kho- 
warizmi had grasped the possibility of the application of the algebra 
to geometry and trigonometry. This application is frequently 
neglected to-day by teachers of elementary algebra. 

The Author's Preface 

" In the Name of God, gracious and merciful ! " 

" This work was written by Mohammed ben Musa, of Khowarezm. He com- 
mences it thus : 

" Praised be God for his bounty towards those who deserve it by their virtuous 
acts : in performing which, as by him prescribed to his adoring creatures, we ex- 
press our thanks, and render ourselves worthy of the continuance (of his mercy), and 
preserve ourselves from change : acknowledging his might, bending before his 
power, and revering his greatness ! He sent Mohammed (on whom may the bless- 
ing of God repose !) with the mission of a prophet, long after any messenger from 
above had appeared, when justice had fallen into neglect, and when the true way of 
life was sought for in vain. Through him he cured of blindness, and saved through 
him from perdition, and increased through him what before was small, and collected 
through him what before was scattered. Praised be God our Lord ! and may his 
glory increase, and may all his names be hallowed ■ — besides whom there is no God ; 
and may his benediction rest on Mohammed the prophet and on his descendants ! 

1 Rosen, The Algebra of Mohammed ben Musa, pp. 1-4. 

45 



46 INTRODUCTION 

" The learned in times which have passed away, and among nations which have 
ceased to exist, were constantly employed in writing books on the several depart- 
ments of science and on the various branches of knowledge, bearing in mind those 
that were to come after them, and hoping for a reward proportionate to their ability, 
and trusting that their endeavors would meet with acknowledgement, attention, and 
remembrance — content as they were even with a small degree of praise ; small, if 
compared with the pains which they had undergone, and the difficulties which they 
had encountered in revealing the secrets and obscurities of science. 

" Some applied themselves to obtain information which was not known before them, 
and left it to posterity : others commented upon the difficulties in the works left by 
their predecessors, and defined the best method (of study), or rendered the access 
(to science) easier or placed it more within reach ; others again discovered mistakes 
in preceding works, and arranged that which was confused, or adjusted what was 
irregular, and corrected the faults of their fellow-laborers, without arrogance towards 
them, or taking pride in what they did themselves. 

" That fondness for science, by which God has distinguished the Imam al Mamun, 
the Commander of the Faithful (besides the caliphat which He has vouchsafed unto 
him by lawful succession, in the robe of which He has invested him, and with the 
honours of which He has adorned him), that affability and condescension which he 
shows to the learned, that promptitude with which he protects and supports them in 
the elucidation of obscurities and in the removal of difficulties, — has encouraged 1 
me to compose a short work on Calculating by (the rules of) Completion and Re- 
duction, confining it to what is easiest and most useful in arithmetic, such as men 
constantly require in cases of inheritance, legacies, partition, law-suits, and trade, 
and in all their dealings with one another, or where the measuring of lands, the dig- 
ging of canals, geometrical computation, and other objects of various sorts and kinds 
are concerned — relying on the goodness of my intention therein, and hoping that 
the learned will reward it, by obtaining (for me) through their prayers the excellence 
of the Divine mercy : in requital of which, may the choicest blessings and the 
abundant bounty of God be theirs ! My confidence rests with God, in this as in 
everything, and in Him I put my trust. He is the Lord of the Sublime Throne. 
May His blessing descend upon all the prophets and heavenly messengers ! " 

The Arabic version differs from the Latin translations which 
have come down to us, in giving an extended discussion of inheri- 
tance problems and also in discussing geometrical measurements. 
In the English translation by Rosen the inheritance problems, in- 
volving largely legal questions rather than algebraical ones, occupy 
79 pages as opposed to 70 for the algebra proper. The mensuration 
problems take some sixteen pages of text. The formulas are 

1 Several writers have asserted that the work of Al-Khowarizmi was written at the 
request of the caliph. The text shows that this is not Al-Khowarizmi's statement of 
the case. 

See Woepcke, Extrait dn Fakhri, p. 2 ; A. Marre, Le Messahat de Mohammed ben 
Moussa al Khdrezml, extrait de son algebre, in Annali di matematica, Vol. VII, first series, 
Rome, 1865, pp. 269-280. 



PREFACE AND ADDITIONS FOUND IN THE ARABIC TEXT 



47 



given for the area of a square and triangle. Three formulas are 
given for the circumference of a circle and the writer evidently 
recognizes them all as approximations- The formulas are: 



(i) 

(2) 

(3) 



c = \/iod 2 
62832 d 



20000 



The area of a circle is given as A = d 2 - \ d 2 — \ of \ d 2 . Other 
simple areas and volumes are discussed. Application of the 
algebra is found in two problems. One of these deals with finding 
the altitude of a triangle of which the sides are given ; the other 
with inscribing a square in a given triangle. 

As the problems, on finding the altitude of a triangle, being 
given the lengths of the sides, and on inscribing in an isosceles 
triangle a square, show that Al-Khowarizmi had an appreciation 
of the possibilities of the algebra, I present one of the problems, 
following Rosen's translation. 

" If some one says : 'There is a triangular piece of land, two of its sides having 
10 yards each, and the basis 12 ; what must be the length of one side of a quadrate 
situated within such a triangle?' the solution is this. At first you ascertain the 
height of the triangle, by multiplying the moiety of the basis, (which is six) by itself, 
and subtracting the product, which is thirty-six, from one of the two short sides 
multiplied by itself, which is one-hundred ; the remainder is sixty-four ; take the 
root from this ; it is eight. This is the height of the triangle. Its area is, therefore, 
forty-eight yards : such being the product of the height multiplied by the moiety of 
the basis, which is six. Now we assume that one side of the quadrate inquired for 
is thing. We multiply it by itself; thus it becomes a square, which we keep in mind. 
We know that there must remain two triangles on the two sides of the quadrate, and 
one above it. The two triangles on both sides of it are equal to each other : both 
having the same height and being rectangular. You find their area by multiplying 
thing by six less half a thing, which gives six things less half a square. This is the 
area of both the triangles on the two sides of the quadrate together. The area of 
the upper triangle will be found by multiplying 
eight less thing, which is the height, by half 
one thing. The product is four things less 
half a square. This altogether is equal to 
the area of the quadrate plus that of the three 
triangles : or, ten things are equal to forty- 
eight, which is the area of the great triangle. 
One thing from this is four yards and four- 
fifths of a yard ; and this is the length of any 
side of the quadrate. Here is the figure : " 




48 INTRODUCTION 

The inheritance problems occupy a large part of the original 
work ; the inclusion of one of these problems here will perhaps not 
be amiss. Only the first of the problems is given since the follow- 
ing problems are of the same general nature, involving other legal 
peculiarities. 

"A man dies, leaving two sons behind him, and bequeathing one-third of his 
capital to a stranger. He leaves ten dirhems of property and a claim of ten dirhems 
upon one of the sons. 

" Computation : You call the sum which is taken out of the debt thing. Add this 
to the capital which is ten dirhems. The sum is ten and thing. Subtract one-third 
of this, since he has bequeathed one-third of his property, that is, three dirhems and 
one-third of thing. The remainder is six dirhems (and two-thirds) and two-thirds 
of thing. Divide this between the two sons. The portion of each of them is three 
dirhems and one-third plus one-third of thing. This is equal to the thing which was 
sought for. Reduce it, by removing one-third from thing, on account of the other 
third of thing. There remain two-thirds of thing, equal to three dirhems and one- 
third. It is then only required that you complete the thing, by adding to it as much 
as one-half of the same ; accordingly, you add to three and one-third as much as 
one-half of them : This gives five dirhems, which is the thing that is taken out of 
the debts." 

The legal point involved in the problem given is that a son who 
owes to the estate of his father an amount greater than the son's 
portion of the estate, retains, in any event, the whole sum which he 
owes. Part is regarded as his share of the estate, and the re- 
mainder as a gift from the father. The above problem would have 
given exactly the same numerical results for any debt from five 
dirhems up ; however, if there were a claim of four dirhems against 
one of the sons, instead of ten, the debtor son would have received 
in cash -§ of one dirhem, the other son four and f dirhems, and the 
stranger four and -f dirhems. 

In algebraical symbolism, the equation is -§-(10 + 3:) = ix 
whence .r = 5 ; io+^ris the total estate left, and x is the share 
of each son. 






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CHAPTER VII 

Manuscripts of Robert of Chester's Translation of Al- 

Khowarizmi's Algebra 

i. the extant manuscripts 

Steinschneider 1 was the first in recent times to call attention 
to the translation of Al-Khowarizmi's algebra made by Robert 
of Chester. He suggested the desirability of publishing this text, 
referring to the manuscript in Vienna. To this same manuscript 
Curtze 2 later, and independently of Steinschneider, directed atten- 
tion and also suggested the desirability of the publication of the 
work. Wappler/ 1 in 1887, found a second copy of the algebra in 
a manuscript in Dresden, while some time later David Eugene 
Smith acquired for the Columbia University Library a manu- 
script from the hand of Johann Scheybl which contains a third 
transcription of the algebra. 

In addition to these manuscript copies of the text, a fragment 
of the translation was published by Adrien Romain of Louvain in 
1599, in a work bearing the title Commentaire sur Valgebre de Ma- 
humedben Musa el Chowarezmi. Unfortunately but a fragment of 
this published work had been preserved to modern times, and that 
precious fragment was doubtless destroyed with other and rarer 
books and manuscripts in the recent destruction of the University 
at Louvain. This work of Romain's was mentioned in a work of 
1643, published at Louvain, as being found in the Library there. 
Henri Bosmans, S. J., of Brussels has given a description 4 of the 

1 Steinschneider, Zeitschrift d. deutsehen morgenland. Gesellschaft, Vol. XXV (1871), 
p. 104; Zeitschrift f. Mathematik,V o\. XVI (i87i),pp. 392-393; Bibliotheca Mathematica, 
third series. Vol. I (1900), pp. 273-274; Sitzungsbericht d. Akad. d. Wissenschaften in 
Wien, Phil. hist. KL, Vol. CXLIX (1904), p. 72. 

2 Curtze, Centralblatt fur Bibliothekswesen, Vol. XVI (1899), p. 289. 

3 Wappler, Zur Geschichte der deutsehen Algebra im 13. Jahrhundert, Programm 
(Zwickau, 1887), pp. 1-2. 

4 Bosmans, Le fragment du commentaire d' Adrien Romain sur Valgebre de Mahumed 
ben Musa el-Chowarezmi, Annates de la Societe scienlifique de Bruxelles, Vol. XXX, part 
II (1906). 

49 



50 INTRODUCTION 

work, mentioning the fact noted by Romain that the latter had 
obtained an excellent manuscript copy of the algebra from his 
friend Thaddeus Hagec of Prague. I am indeed fortunate, 
through the courtesy of Professor Bosmans and of Professor B. 
Lefebvre of Louvain, to be able to include this fragment in my 
collation. Another fragment in Ms. was found after the textual 
notes were in type ; this brief portion from Codex Dresdensis C. 
8o m , of the fifteenth or sixteenth century, is included later in this 
chapter. 

Yet another fragment of the algebra was published by Wappler, 1 
who did not, however, ascribe the passage to Robert of Chester. 
The failure to connect it directly with the algebra in the same 
manuscript elsewhere mentioned by Wappler was due in part to 
the fact that the section in question, which is found p. 120, 1. 21 to p. 
124, 1. 16 of our text, is not in its proper place in the Dresden 
manuscript. The same paragraphs are found also in a manuscript 
of the University Library at Leipzig, Codex Lips. 1470. I 
am indebted for information concerning this manuscript to the 
courtesy of Director Boysen and Dr. Helssig, of the Univer- 
sity Library in Leipzig. It appears that this manuscript is 
almost entirely from the hand of Magister Virgilius Wellendorfer 
of Leipzig, written during his student days, between the years 
1 48 1 and 1487. The passage referred to was in all probability 
copied from the Dresden Codex C. 80. Apparently Wellendorfer 
had the intention of copying the algebra entire, for on folio 478 s 
we find the title, Textus algabre edidit Mahume Most Jilius. To 
the title he added a brief note, Sed utitur aliis nominibus . . . 
substantia et dragma, radicis . . . algorithmi. These words 
clearly indicate some acquaintance with our text, but the text 
itself is not found in the manuscript. 

The Scheybl Ms. (C), written in 1550, was evidently intended 
for publication. In printing Robert of Chester's text I have 
thought it best to follow the Ms. which was prepared by Scheybl. 
Although it contains some errors, and slight additions by 
Scheybl, these are quite easily distinguished. The advantage of 
following Scheybl's careful revision seemed obvious ; particularly 
the chapter divisions and sub-titles, many of which he supplied, 

1 Wappler, Zur Geschichte der Mathematik, Zeitschrift f. Math, und Physik, Vol. XLV 
(1900), Hist. lit. Abth., pp. 55-56. 



MANUSCRIPTS OF ROBERT OF CHESTER'S TRANSLATION 51 

make the text easier to follow. That the reader may have before 
him all the evidence in regard to the text, the readings of the 
other Mss. are recorded in the critical notes. These may seem 
needlessly full on account of the inclusion of many apparently 
unimportant variants; yet in any attempt to determine the parent- 
age of the Mss. such evidence frequently possesses a significance 
which is not at first sight apparent. 

The Dresden Codex C. 8o m , of which I have photographic 
copies through the courtesy of the director of the Dresden library, 
contains a fragment of Robert of Chester's translation of the alge- 
bra of Al-Khowarizmi. The passage is introductory to a work on 
arithmetic ; the latter contains portions of the algorism by Sacro- 
bosco, and of the algorism in verse by Alexander de Villa Dei, 
and also parts of the commentary on the former by Petrus de 
Dacia (1291), together with other comment and further exposi- 
tion. The " Rules corresponding to the rules of algebra," which 
we have reproduced on page 126 from the Vienna Ms. are also in- 
cluded with other algebraic material in this mathematical Ms. 
Here these rules 1 follow closely the Vienna text, whereas similar 
rules 2 in Codex Dresden C. 80, fol. 35 i a , do not. 

The fragment from Codex Dresden C. 8o m follows: 

" Incipit liber restauracionis numeri quern edidit machumed filius moysi algau- 
riszmi quare dixit machumued. 

" Laus deo creatori qui homini contulit scientiam inveniendi vim numerorum. 
Considerans enim omne id quo indigent homines ex numero, inveni id totum esse 
numerum. Et nil aliud esse numerum nisi quod ex vnitatibus componitur. Vnitas 
ergo est qua vnaquaque res dicitur vna et vnitas in omni numero reperitur. Inveni 
autem omnem numerum essentialiter ita dispositum vt omnis numerus vnitatem 
excedat vsque ad 10. Decenus quoque numerus ad modum vnitatis disponitur, 
vnde et duplicatur et triplicatur quemadmodum factum est ex vnitate. Fiuntque 
ex eius duplicatione 20, ex eius triplicatione 30. Et sic multiplicando decenum 
numerum ad centenum peruenitur. Proinde centenus numerus duplicatur et triplica- 
tur, etc. ad modum numeri deceni. Et sic centenum numerum duplicando et 
triplicando, etc. Millenus excrescat numerus. Ad hunc ergo modum millenus 
numerus ad modos numerorum, vsque ad infinitam numeri investigationem 
conuertitur." 

In this passage the sentence, " Vnitas ergo est qua vnaquaque res 
dicitur vna," is interjected from Sacrobosco's algorism ; it is a 
translation of Euclid's definition of a unit. The work which im- 

1 Published by Wappler, Programm (Zwickau, 1887), note 1, page 14. 

2 Published by Wappler, loc. cit., pp. 13-14. 



52 INTRODUCTION 

mediately follows concerns arithmetic proper, and the remainder 
of the material in the Ms. is also mathematical. 

The Vienna Ms. (V) is assigned in the catalogue of Mss. of the 
Vienna library to the fourteenth century, and the character of the 
writing agrees with this dating. In the fifteenth century Peur- 
bach and Regiomontanus, diligent students of Arabic mathe- 
matics, were connected with the University of Vienna, which was 
then a mathematical centre. This copy itself may have been 
acquired by Regiomontanus for the library at Vienna; other Mss. 
from the hand of Regiomontanus are preserved in Vienna. 

The Dresden Codex C. 80 (D) was written in the latter part of 
the fifteenth century. Wappler 1 states that Johann Widmann 
of Eger, whose activity at the University of Leipzig falls at the 
end of the fifteenth century, transcribed certain portions of the 
Dresden Codex, but the writer of our text is not known. Adam 
Riese, in the beginning of the sixteenth century, also used this 
Ms. The algebra of our text begins on folio 340 s and terminates 
in the middle of folio 348 b ; the section relating to commercial 
problems is not found in its proper place, but appears in the 
Ms. on folio 30 1 a . For the collation of this latter section I have 
used the printed text, mentioned above, by Wappler. 

II. THE VIENNA MANUSCRIPT (v) 

Codex Vindobonensis 4770 (Rec. 3246) XIV. 339.8 (V). 

Fol. i a -i2 b , Liber restauratio?iis et oppositionis numeri ; our 
text. 

Fol. I3 a ~40 a , M. Jordanus Nemorarius, De numeris datis, incor- 
rectly designated in the catalogue as the Tractatus arithmeticus 
by the same author. Many manuscript copies are extant, and the 
text was published by Treutlein in Vol. II, Abhandlungen zur 
Geschichte der Mathematik (Leipzig, 1879), pp. 135-166, but from 
an incomplete manuscript of the work. The necessary additions 
are given by Curtze, Commentar zu dem " Tractatus de Numeris 
Datis " des Jordanus Nemorarius, Zeitschrift fur Mathematik und 
Pkysik, Hist. -lit.- Abtk., Vol. XXXVI, pp. 1-23, 41-63, 81-95, 
and 1 21-138. 

Fol. 4<D b -44 b , blank. 

1 Wappler, loc. at., Programm, p. 9. 



MANUSCRIPTS OF ROBERT OF CHESTER'S TRANSLATION 53 

Fol. 45 a ~5o b , Tractatus geometricus cumfiguris. Begins, " Punc- 
tum est cuius pars non est . . ." and ends, " equaliter distantes 
fuerint constituti." 

Fol. 5i a -i73 b , blank. 

Fol. I74 a ~324 b , Carmen quadripartitum de matematica cum 
commentario subnexo. This is the Quadripartitum numerorum 
by Johannis de Muris, of which many manuscript copies are ex- 
tant. The work as a whole has never been published. Two 
chapters of the second book which relate to practical arithmetic 
have been published by Nagl, Abhandlungen zur Geschichte der 
Mathematik, Vol. V (Leipzig, 188), pp. 135-146. I have made a 
study of the third book and I have given selections from the 
metrical portion of the work, as well as a few passages from the 
third book and from other parts of the work, Bibliotheca Mathe- 
matical third series, Vol. XIII (1913), pp. 99-114. 

Fol. 325 a -32 7 b , blank. 

Fol. 328 a ~337 a , Tractatus de ponderibus (in fine mutilus). This 
treatise begins, " Marcha est limitata ponderis . . ." and ends, " ad 
tertium altare et deinceps." This is also the work of Jordanus 
Nemorarius, and has not been edited. 

Fol. 337 b -338 b , blank. 

Fol. 339 a , Notabile de algorismo proportionum. 

III. THE DRESDEN MANUSCRIPT (d) 

Codex Dresdensis C. 80. Fifteenth century. 416 pages, num- 
bered 1-225, 227-417. Paper. By various hands. Folio, bound 
in parchment (D). 

Page 1, Vnum dat finger brucke duo. . . . etc. Evidently on 
finger-reckoning. 

Page i b ~5 b . Large portions of the algorism of Sacrobosco. 

Page 6 a . Divisio numeri. 

Page 6 b ~9 b , Schachirica mercatorum computatio, etc. Various 
fragments. 

Pages 10, 20-23, 84-128, 145-153, i73- : 7 6 > 186-190, 198-200, 
220-233, 246-257, 269-279, 327-339, and 381-384 are blank. 

Page n-19, Arithmetic of Johannis de Muris. This is a frag- 
ment of the Arithmetica communis of de Muris which was printed 
in 15 15 at Vienna, and again, as Arithmeticae speculativae Libri 
duo, at Mainz in 1538. 



54 INTRODUCTION 

Page 24-7 i b , Arithmetic of Boethius, of which there are nu- 
merous editions, with a critical edition by Friedlein (Leipzig, 
1867). 

Pages 72-83- Excerpts from the arithmetic of Boethius. 

Pages 129-134. Regule de Alegorismo. The algorism (of 
integers) published by Boncompagni {Trattati d'arithmetica, 
Rome, 1857, pp. 25-49). 

Pages i35-i42 b . Sequitur de phisicis. Probably the discus- 
sion of fractions in the preceding algorism. 

Pages I42 b -i44. Carmen de ponder ibus. 

Begins, Pondera postremis Veterum memorata libellis ; ends, 
Argentum argento liquidis cum mergitur Vndis. 

Pages 154-157. Incipit liber de Sarracenico et de Limitibus et 
cetera Orelibacio de Abacis. Latino. Arabico. Sarracenico et 
de Limitibus. 

Begins, Quoniam ad Raciones quasdam presentis libelli. 

Pages 1 5 7 b - 166. De diuersitate fraccionum capitulum prijnum. 
De additione et duplatione. . . . De radicum extractione capitu- 
lum 14. 

Begins, Quoniam in precedentibus frequenter contingit; ends, 
Et haec de computacione fraccionum sufficiant. 

Pages 167-172. Incipit Canon Magistri Johannis de Muris 
super tabula tabularum que dicitur proporcionum. 

Canon by John of Meurs, edited by John of Gmunden in 1433. 

Pages 1 77— i85 b . Jordanus Nemorarius de minuciis libri duo. 

See Enestrom, Das Bruchrechnen des Jordanus A r emorarius, 
Bibliotheca Mathematical third series, Vol. XIV, pp. 41-54. 

Pages 1 91— I97 b . Qtiid sit proporcio capitulum primum. De 
3 cl Manerie proporcionis capitulum secundum . . . De diuisione 
proporcionum cap. 10. Quantitatem aliquant me7isurare. 

Pages 20i a -2o6 a , Algorithmus proportionum, by Nicholas 
Oresme. Published by Curtze, Programm (Thorn, 1868). 

Pages 2o6 b -2i9 b , 2 34-245 b . {De proporcionibus.) 

Pages 2 58-268 b . Regule super rithmachiam. On the ancient 
game called Rythmomachia of which there are three standard 
treatises published; see Smith, Ra ra Arithmetical. 271, and a 
description of the game by Smith and Eaton, American Mathe- 
matical Monthly, Vol. XVIII (191 1), pp. 73-80. 

Pages 28o a -285 b , Algorithmus minuciarum by Johannes de 



MANUSCRIPTS OF ROBERT OF CHESTER'S TRANSLATION 55 

Lineriis (fourteenth century). Printed at Padua, 1483, and 
Venice, 1540. 

Pages 286 a -29i a , 292 b -3oo b , 3°3 b -3°5 a > 3o6*-3i$ h . 

Various fragments, including portions of the Quadripartitum 
numerorum by John of Meurs ; see Wappler, Programm, pp. 7, 

31-32. 

Pages 2C)i b -292 a , Algoritkmus de duplici differentia. 

Page 30 1 a . The passage of our text relating to commercial 
transactions, in the handwriting of Johann Widmann of Eger. 

Pages 30i b -303 a , Mathematical lecture by Gottfried de Wolack, 
written 1467 or 1468. 

Pages 3i6 a -323 b . De numeris datis, by Jordanus Nemorarius. 

Pages 33i a -334 b . Pro regularum Algabre. 

Pages 340 a -348 b . Our text. 

Pages 349 a -365 b . Latin algebra, published by Wappler, Pro- 
gramm (Zwickau, 1887), pp. 11-30. 

Pages 366-36 7 b . Cautelae Magistri Campani ex libro de Al- 
gebra siue de Cossa et Censu. 

Pages 368-378 b . An algebra in German, somewhat similar to 
the preceding Latin algebra ; see Wappler, Programm, and Ab- 
handl. z. Geschichte d. Math., Vol. IX, pp. 539-540, where it is 
discussed. 

Pages 379-380, various. 

Pages 385-397 b . De mensuratione terrarum et corporum, trans- 
lated by Gerard of Cremona. Not published. Includes algebraic 
problems. 

Pages 397 b ~4o6. Liber augmenti et diminutionis ; published 
by Libri, Histoire des sciences mathematiques en Italie, Vol. I, 
(Paris, 1838), pp. 304-371. 

Pages 406*^-407, 409-417. Various. 

The above description is based upon the Katalog der Hand- 
schriften der Konigl. Offentlichen Bibliothek zu Dresden (Leipzig, 
1882) by Schnorr von Carolsfeld, and upon the articles by 
Wappler, as cited above. 

IV. THE COLUMBIA UNIVERSITY MANUSCRIPT (c) 

Codex Universitatis Columbiae; Columbia University Library 
Manuscript, X 512, Sch. 2, Q. 308 pages. (C.) 

Pages 1-68, Breuis ac dilucida regularum Algebrae descriptio 



56 INTRODUCTION 

autore Joanne Scheubelio, in inclyta Tubingensi accidentia Enclidis 
professore ordinario. This is a briefer treatment of the algebra 
than that published by Scheybl in 1550, as a preface to the first 
six books of Euclid, Euclidis Megarensis, Philosophi Mathematici 
excellentissimi, sex libri priores. . . . Algebrae porro Regulae, 
propter mimerorum exempla, passim propositionibus adiccta, his 
libris praemissae sunt, eaedemque demonstratae (Basle, 1 550). This 
algebra was published, separate from the Euclid, in Paris, in 1 55 1. 
The text of the algebra in the Columbia manuscript has not been 
published. 

Pages 69-70, blank. 

Pages 71-122, Liber Algebrae et Almucabola, continens demon- 
strationes aequationum regularum Algebrae. Our text. 

Pages 123-157, Addita quaedam pro dec laratione Algebrae, by 
Scheybl. This is explanatory of the preceding. It is printed on 
pages 128 to 156 of this book. 

Page 158, blank. 

Pages 159-308, Liber Jordani Nemorarii de datis in quatuor 
partes digestus. This is not the complete text of the De numeris 
datis, but contains the statement of the problems and their solu- 
tions according to the rules of algebra. In these solutions Scheybl 
introduces the use of n for number, co. for cosa, for the first power 
of the unknown, and a symbol which is very similar to the square 
root sign for the second power of the unknown, or substantia. 

V. THE RELATIONS OF THE MANUSCRIPTS 

The determination of the relation between the Vienna and 
Dresden Mss., as well as the relation of the Scheybl Ms. and the 
two fragments to each of these, is based not only upon a study of 
particular words and phrases but in large measure upon the omis- 
sions made by the various scribes in copying. The main test 
used is, appropriately enough for these mathematical Mss., an 
arithmetical one. 1 In connection with the omissions we may 
observe, as noted by Havet, 2 that a copyist passes easily by error 
from a given ending or word to a similar ending or word which 

1 A. C. Clark, The Primitive Text of the Gospel and Acts (Oxford, 1914), pp. i-vii and 
1-10. 

2 L. Havet, Manuel de critique verbale appliquk aux textes latins (Paris, 191 1), pp. 
130, 200. 



MANUSCRIPTS OF ROBERT OF CHESTER'S TRANSLATION 57 

occurs later in the contiguous text ; this type of error is the most 
frequent one made in copying Mss. The "jump from like to like " 
is particularly prone to occur when the parallel passages recur in 
similar parts of neighboring lines ; in this event the omission 
approximates the length of a line. Another frequent error in 
copying is to omit entirely one or more lines or to repeat a whole 
line. An examination of the lengths of all such passages leads 
to a fairly definite notion of the length of line of the parent man- 
uscript. It need hardly be stated that any omission is more 
readily made when the appearance, at least, of sense is preserved 
after the omission. In our mathematical text the recurrence of 
similar words and phrases is frequent, and in many places the 
omission of a line is possible with the preservation of a measure 
of meaning. 

In Mss. which antedate the tenth century, the length of any 
omission is determined with comparative ease as a certain num- 
ber of letters, for few abbreviations, and those of a standard type, 
were used. However, in Mss. of the type of the Dresden and 
Vienna Mss., with which we are primarily concerned, the length 
in letters of any omission is by no means a fixed and definite 
quantity. The abbreviations used by the copyists of the twelfth 
to the fifteenth century commonly varied, not only from page to 
page, but even from line to line. On the first page of the Vienna 
Ms. (Plate I), where the conscious effort would be made, probably, 
to be uniform in notation, the copyist wrote fbus in line 18, and 
tribus in line 26, while on the following page he wrote tbq, 
and elsewhere he writes 3 b 9. Another possible form is trib<^ 
although this does not appear in our Ms. This word then could 
count either for 3, 4, 5, or 6 letters. Similarly on this same first 
page (line 11) duplicatione is written duplica oe , while triplicatione, 
which immediately follows, is written fplicatione. In the Dresden 
Ms. there are entirely similar variations. Thus in line 5 of fol. 
340 15 (Plate II) 5 a appears for quinta, in line 7 qnteiox quintae, 
quinario in line 15, and quzqz and quqz in lines 29 and 32 for 
quinque; in line 37 substanciam appears in full and in the next 
line as subam. 

In counting the number of letters in any omission I have 
assumed the common abbreviations used in the contiguous pas- 
sages of the Vienna Ms. ; for as that is quite certainly the oldest 



58 INTRODUCTION 

of the Mss. which we are examining, it is in all probability more 
nearly like the parent Ms. whose existence is established with 
considerable certainty by our study. On the first page of the 
Vienna Ms. (Plate I) the line varies in length from 35 to 48 
letters, with an average of 40^ letters, while on the first half of 
fol. 340 b of the Dresden Ms. the line varies from 40 to 52 letters 
with an average of 45^ letters. These facts give an indication of 
the latitude in variation which we may assume in the line length 
of the parent Ms. 

The omissions of the Vienna Ms. will first be investigated as 
that is the oldest of our texts. In the text of page 84, lines 7-9, 
the parent Ms. evidently read : ■ 

Sed linea b h similis est linee g d. Nam quoniam linea g 1 et 
linea h e in quantitate habentur consimiles quoniam linea g 1 
similis est linee d e. 

The Vienna scribe passed from the first similis est liiiee to the 
second. In the text of the same page, lines 12-13: 

Area igitur quam linee t e, e a circumdant similis est aree quam 
linee m I, I d circumdant. Area igitur . . . 

The scribe dropped from the one Area igitur to the other. 

In the text given in lines 8-9 of the footnote to lines 1-12, 
page 88 : 

. . . similis est aree quam c t, 1 1 similis quam / e circumdant. 
Area ergo n z similis est aree . . . the scribe of the Vienna Ms. 
passed from the first similis est aree to the second. On page 1 12, 
text of lines 1-4 : 

. . . absque xx rebus. Rem quoque in re multiplicata, et erit 
substantia. Hec insimul iunge et erunt c et due substantie 
absque xx rebus. The omission can be conceived of here as of 
the text between absque and absque, or xx and xx, or between 
rebus and rebus, and similarly in the preceding illustrations. So 
in line 18, page 116 (footnote): in duo et 4 a et erunt xx et 4 ta , et 
multiplica v radices in duo et 4 ta et erunt 1 1 res et 4 ta . . . . the 
text is omitted between any word of the first in duo et 4* et erunt 
and the same word where this phrase recurs. 

These five passages are in length, roughly, 45, 43, 38, 54 to 60, 
and 41 letters, respectively. They indicate a line length of about 
40 letters in the parent Ms. One omission which is not from like 
to like appears to be that of a complete line in the text of page 104, 



MANUSCRIPTS OF ROBERT OF CHESTER'S TRANSLATION 59 

lines 22-24: The length of this omission is 45 letters. On 
page no, in line 27, there is an omission by the Vienna scribe of 
some 36 letters, but as the passage is not at all clear in the 
Dresden Ms., having unamquamque where antequam unamquam- 
que should have been written, this may have been a deliberate 
omission. The omission in the text of page 100, line 19, which is 
short (about 20 letters) may also have been deliberate as the 
meaning is not clear. Another short omission is found in the 
text to page 78, line 12. 

As these omissions in the Vienna Ms. correspond to portions 
of the Arabic text, and as all are found in the Dresden Ms., they 
establish the fact that the Vienna Ms. is not the ancestor of the 
Dresden Ms. ; similarly the omissions of the Dresden Ms. are 
found in the Vienna Ms., proving that the Dresden could not be 
the parent Ms. of the Vienna Ms., entirely apart from the fact that 
the Vienna is quite definitely older. 

In the Dresden Ms. some eleven omissions require examina- 
tion. Of these eight are instances where the scribe has passed 
from one word or phrase to a similar word or phrase recurring in 
the adjacent text. Such omissions are, as we have noted, quite 
likely to be about the length of the line of the parent Ms., but may 
be materially shorter or longer. Five of these omissions also sug- 
gest a line of about forty letters in the parent Ms. Four of these 
omissions are rectified by marginal additions made by a second 
hand, but this does not change their value as indicating the length 
of line in the parent. Four of these omissions are indicated in the 
footnote to line 6, page 92. The first from fuerint to fuerint; the 
second from sine re to sine re ; the third f rom procreant to pro- 
creant, and so also the fourth although here a 10 intervenes. The 
lengths in letters are 37, 43, 16, and 38, respectively. In line 7 
on the same page is another short omission, from like to like, in 
the expression, et eius sextam in dragma et eius sextam ; the 
Dresden Ms. omits the last five words. In the first line on the 
same page we have the evident omission of a full line, no recur- 
ring word appearing. 

A much longer omission than any yet mentioned is found on 
page 94, lines 34-36: 

. . . perueniet una substantia abiecta. Et si dixerit 10 sine re 
in re, dicas 10 in re 10 res procreant, et sine re in re substantiam 



60 INTRODUCTION 

generat diminutivam. Hoc igitur ad 10 res perueniet abiecta 
substantia. 

The scribe passed from perueniet to perueniet two lines below. 
Similar omissions, in length about 16 and 25 letters, are found in 
the text, page 102, line 4, and page 40, line 19. The omission of 
A quibus 21 demptis after producentur 25 (page 1 10, line 8), is not 
easily explained, but the omission in the text of page 96, line 17, 
appears to be that of a complete line of 38 letters. 

The single illustration of the repetition of a full line or more 
in either Ms. occurs in the text on page 114, line 29, where the 
scribe runs back from the coequantes at the end of one sentence 
(line 29), to the coequantes at the end of the preceding sentence ; 
the length is 54 letters. A shorter repetition occurs in the text 
of page 116, line 6. In the text of page 84, lines 12-13: 

Area igitur quam linee t e, e a circumdant similis est aree quam 
linee m /, Id circumdant. Area igitur t a similis est aree m d. . . . 
we have noted that the Vienna scribe passed from the one Area 
igitur to the second ; the Dresden scribe writes m after the first 
similis est aree, passing to the second, but he corrects the error. 
The length here is 43 letters. Since one Ms. omits and the other 
starts to omit this line, it was possibly omitted in the body of the 
text of the parent, and supplied in the margin. 

The Vienna manuscript contains on the first page two marginal 
additions, somewhat in the nature of titles (see Plate I). To 
these there correspond similar additions in the Dresden manu- 
script, but while the latter continues with numerous other margi- 
nal additions the Vienna manuscript does not follow that practise. 
Among the particularly noteworthy marginal additions by the 
first hand in the Dresden manuscript are the geometrical figures 
to be used in connection with the geometrical demonstrations of 
the solutions of quadratic equations. The Vienna text contains 
no such figures. But the Dresden figures are evidently not de- 
rived directly from the figures in the Arabic text ; they give every 
evidence of being constructed by the writer of the Dresden manu- 
script upon the basis of the geometrical explanation given in the 
text. The lettering does not vary greatly from the Arabic, and 
two of the figures are left quite incomplete. 

The second hand has made in the Dresden text several marginal 
additions based, in general, upon the Vienna manuscript, and once 






MANUSCRIPTS OF ROBERT OF CHESTER'S TRANSLATION 61 

in agreement with Scheybl's reading as opposed to the Vienna 
reading. Thus on page 66 of our text the marginal notes to words 
in lines 17 and 18 alii centenum and alii decern, and on page 68 to 
a word in line 2, alii coniunctis. Four rather long marginal cor- 
rections are made, also apparently based upon the Vienna manu- 
script, in the text to page 92, line 1, in the text as given in note 
6 on the same page, in the text given on page 94, lines 34-36, 
and on page 96, line 17. 

We have found then in each of these two manuscripts seven 
definite indications of a parent manuscript with a line length of 
between 36 and 54 letters, not very different from the line length 
in these Mss. themselves. But aside from the line length of the 
parent there are other indications that the two Mss. have a com- 
mon parent. In the text of page 86, line 15, both Mss. read ex 
duobus et qtiarta in se ipsis where the sense requires ex unitate et 
medietate in se ipsis ; this error was evidently in the parent Ms., 
and possibly in the autograph. Similarly in the text of page 106, 
lines 11 and 14, both Mss. read plainly 36, and that twice, whereas 
49 is the correct numerical result here. The scribe of the Dresden 
Ms. commonly (5 times) writes hiis and the other scribe his, but 
in the text of page 76, line 15, this procedure is reversed; the in- 
dication is that hiis was the form employed in the parent. The 
Dresden scribe writes addicias and the Vienna scribe adicias, ex- 
cept in the text of page 76, line 20 where the latter also writes 
addicias ; probably this form was used in the parent Ms. In the 
text of page 102, line 26, the Vienna Ms. reads 10 sine re, and the 
Dresden Ms. rem, while the sense requires 10 res sine substantia. 
Either the translation was incorrect in the parent, or, more prob- 
ably, the passage was illegible. In the text of page 112, line 22, 
the Vienna Ms. reads 2000, 500, 50 et 4 a , for 2550 et 4 a ; the 
Dresden Ms. writes the expression in words. This corresponds 
to a direct translation from the Arabic, for most early mathemati- 
cal Mss. in Arabic follow the practise of writing numbers in full 
numeral words, and not in Hindu-Arabic notation. 

The discussion of Scheybl's text is somewhat more complicated 
than that of the two preceding, for Scheybl follows sometimes the 
one, sometimes the other, and frequently neither, of the two older 
texts. As we have indicated above, there is probability that the 
present Vienna text may have been in the library of the Univer- 



62 INTRODUCTION 

sity at Vienna when Scheybl was a student there. Further, we 
know that the Dresden Codex was used by Johann Widmann of 
Eger towards the end of the fifteenth century when Widmann was 
lecturing on algebra in the University of Leipzig; and a little 
later this same manuscript was used by Adam Riese. Although 
Scheybl was connected as teacher and student with the Univer- 
sity of Tubingen, his first work 1 was published at Leipzig. 
Scheybl may have been familiar, then, with both of our manu- 
scripts, or with the parent. Scheybl's manuscript was prepared 
with care for the printer, and the few omissions do not throw any 
light upon his source. 

Another difficulty in connection with the Scheybl text is the 
fact that he took many liberties with the text. I have noted in 
many places where Scheybl wrote the word which is given in the 
Vienna and Dresden Mss., and then deleted to substitute a word 
of his own. Thus, in the text of page 78, line 4, the words aequa- 
lium and scilicet both, though at first written, were crossed out by 
Scheybl, and on page 80 similar deleted words in the text of lines 
3, 8, and 17 correspond also to words in the Vienna and Dresden 
Mss. In the text of page 76, line 15, of page 82, line 15, and page 
100, line 19, the deleted word follows the Dresden Ms. and not the 
Vienna, while the reverse is true in the text of page 82, line 8, and 
page 102, line 12. The agreements of Scheybl's Ms. with the 
Vienna, as opposed to the Dresden readings, number about 125, 
whereas, the reverse agreements with the Dresden Ms. number 
about 70; these include phrases as well as single words. Notably, 
the two concluding paragraphs of the Vienna Ms. with the date 
and place of the translation, appear in Scheybl's Ms. and not in 
the Dresden Ms. 

In the text of page 68, line 6, after colligitur Scheybl omits a 
passage of some fifty letters ending in coiiiungitur ; while this may 
be an omission from like to like, yet it may also have been deliber- 
ately left out as the statement is a repetition, found in the Arabic, 
of a passage which precedes (lines 1-2). In the text of page 72, 
line 6, fuerint ut sunt due uel 3 s uel plures seu pauciores fuerint 
Scheybl passes from one fuerint to the second, an omission of 
some 36 letters. The omission of about ninety letters after exten- 

1 De numeris et diver sis rationibus seu regulis computatiomun opusculuvi (Leipzig, 
1545)- 



MANUSCRIPTS OF ROBERT OF CHESTER'S TRANSLATION 63 

ditur in the text of page 96, line 23, appears to be deliberate, as 
the meaning of the passage is not clear. So, also, the omission in 
the text of page 116, line 13, may have been deliberate, as the mul- 
tiplication is a simple repetition of work which precedes. 

Another difficulty in any exact determination of the genesis of 
Scheybl's text is the fact that he had access to a copy of the alge- 
bra of Al-Khowarizmi in the translation which we have designated 
as the Libri text. The evidence of this familiarity is found in the 
Addita, written by Scheybl, which are printed on pages 128-156 of 
this work, for herein are contained portions of the algebra which 
were not translated by Robert of Chester, notably the problems 
involving the square root of two hundred (pages 142-144). 

Only a fragment remains of the Romain version, constituting 
about 24 lines of our text. In this brief space there are some 
twelve agreements with the Dresden and Vienna Mss., and varia- 
tions from the Scheybl text. However, one agreement with the 
Scheybl manuscript shows either familiarity with Scheybl's work, 
or a common source other than the Vienna and Dresden Mss. 
The title " Liber Algebrae et Almucabola, de quaestionibus arith- 
meticis et geometricis " appears only in the Romain fragment and in 
Scheybl's text; and somewhat similarly the word " creatori " after 
" Laus deo " in line 10, page 66, is common to the Romain version 
and the other Mss. except the Vienna Ms. 

The fragment of our text in Codex Dresden C. 8o m , which we 
have reproduced above, follows exactly none of the other texts. 
Line omissions do not occur, but the spellings and transpositions 
agree sometimes with the Vienna, sometimes with the Dresden 
readings; in one instance wz'/for nihil, the agreement is with the 
Romain fragment as opposed to the Vienna and Dresden C. 80 
readings. This fragment, then, appears also to be based on a 
parent of the extant Mss. 




<b 



i 



■ 



O 

d 









H 



CO 
■-> 

o 



— *' 



ABBREVIATIONS 

MANUSCRIPTS 

C = Codex Universitatis Columbiae R = Fragmentum Romain 

D = Codex Dresdensis V = Codex Vindobonensis 

After the note to line 6 of the second page of the Latin text, the notes without any letter 
indicate the concurrence of V and D. 

OTHER ABBREVIATIONS 



add. vel 


+ = additum 






ras. 


= erasura 


corr. 


= correctum 




relict. 


= relictum 


del. 


= deleta 






spat. 


= spatium 


man. 


= manus 






superscr. 


= superscriptum 


marg. 


= margine 






Tab. 


= Tabula 


n. 


= nota 






text. 


= textus 


om. 


= omissum 






tit. 


= titulum 


quaest. 


= quaestio 


1 = 


numerus 
radix vel res 
substantia 
et 


vac. 


= vacuum 



65 



LIBER ALGEBRAE ET ALMUCABOLA 

CONTINENS DEMONSTRATIONES AEQUATIONUM REGULARUM 

ALGEBRAE 

Ab incerto authore olim arabice conscriptus atque deinde a Roberto Cestrensi, in 
5 ciuitate Secobiensi anno 1183, vt fertur, latino sermoni donatus. 

LIBER ALGEBRAE ET ALMUCABOLA 

Liber Algebrae et Almucabola, de quaestionibus arithmeticis et geometricis. 
In nomine dei pii et misericordis incipit liber Restaurationis et Oppositionis 
numeri quern aedidit Mahomet, filius Mosi Algaurizin. Dixit Mahomet, Laus deo 

10 creatori, qui homini contulit scientiam inueniendi vim numerorum. Considerans 
enim omneid quo indigent homines, ex numeris componi, inueni illud totum esse 
numerum, et inueni nihil aliud esse numerum, nisi quod ex vnitatibus componitur. 
Vnitas ergo in omni numero reperitur. Inueni autem omnem numerum ita dis- 
positum, vt omnis numerus vnitatem excedit vsque ad decern, denarius quoque 

is numerus ad modum vnitatis disponitur, vnde et duplicatur et triplicatur, quem- 
admodum factum cum vnitate. Fiuntque ex eius duplicatione, 20 : et tripli- 
catione, 30. Et sic multiplicando denarium numerum, ad centenarium peruenitur. 
Ita centenarius numerus duplicatur et triplicatur, sicut denarius numerus. Et 
sic centenarium numerum duplicando et triplicando etc. millenarius excrescit 

20 numerus. Ad hunc modum numerum millenarium secundum ordinem numerorum 
multiplicando, vsque ad infinitam numeri inuestigationem peruenitur. 

Postea inueni numerum restaurationis et oppositionis his tribus modis esse 

1-6. om. VDR. 16. est ex pro cum VDR. eius om. V. et 

7. Liber . . . geometricis om. VD ; + Prae- + ex VDR. 

fatio R. 17. duplicando VDR. centum DR + alii 

8. et 1 om. D. instauracionis D. centenum in marg. D man. 2; centenum V el sic 

9. Mahumed filius moysi algaurizim V; ubique. 

Machumed filius moysi algaurizm D; Mahumed 18. Ita: Proinde VD ; Post modum R. cen- 

filius Moysis algaorizim R. Mahomet 2 : Ma- tenus D. et om. V. sicut denarius numerus : 

humed VR ; machumed D. ad modum 10" (= decimi) V; ad modum numeri 

10. creatori om. V. decimi D et in marg. man. 2 alii decern ; ad modum 
n. ex numero VDR. componi om. VDR. decenarii numeri R. 

id pro illud V. 19. decenum pro centenarium VD et corr. in 

12. exlun (?) sed del. pro et D. nil R. centenum D man. 2. triplicando multiplicando 

13. omnem om. D. ita + essentialiter VD ; duplicando V. etc. om. VDR. millenus VD. 
-f- necessario R. 20. hunc + ergo VDR. millenus numerus 

14. vnitatem corr. ex vnitas D man. 2. ad modos VD ; millenarius numerus ad modos R. 
excedat VDR. decenus pro denarius jere ubique 21. multiplicando om. VDR. infiniti V 
V ; decimus ubique D ; decenarius ubique R. numeri om. V. conuertitur pro peruenitur VDR 

15. ad in marg. D man. 2 pro ex del. tripli- 22. hiis D. 



catur + et D. 



66 



THE BOOK OF ALGEBRA AND ALMUCABOLA 1 

Containing Demonstrations of the Rules of the 
Equations of Algebra 

Written some time ago in Arabic by an unknown author and afterwards, according 
to tradition in 1183, 2 put into Latin by Robert of Chester in the city of Segovia. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 

The Book of Algebra and Almucabola, concerning arithmetical and geomet- 
rical problems. 

In the name of God, tender and compassionate, begins the book of Resto- 
ration and Opposition of number put forth by Mohammed Al-Khowarizmi, 
the son of Moses. 3 Mohammed said, Praise God the creator who has be- 
stowed upon man the power to discover the significance of numbers. Indeed, 
reflecting that all things which men need require computation, I discovered 
that all things involve number and I discovered that number is nothing 
other than that which is composed of units. Unity therefore is implied in 
every number. Moreover I discovered all numbers to be so arranged that 
they proceed from unity up to ten. The number ten is treated in the same 
manner as the unit, and for this reason doubled and tripled just as in the case 
of unity. Out of its duplication arises 20, and from its triplication 30. And 
so multiplying the number ten you arrive at one-hundred. Again the num- 
ber one-hundred is doubled and tripled like the number ten. So by doub- 
ling and tripling etc. the number one-hundred grows to one-thousand. 
In this way multiplying the number one-thousand according to the various 
denominations of numbers you come even to the investigation of number 
to infinity. 

Furthermore I discovered that the numbers of restoration and opposition 

1 Algebra and almucabola are transliterations of Arabic words meaning 'the restoration,' or 
' making whole,' and ' the opposition,' or ' balancing.' The first refers to the transference of nega- 
tive terms and the second to the combination of like terms which occur in both members or to 
the combination of like terms in the same member. For a discussion of the terms algebra and 
almucabola, see Karpinski, Algebra, in Modem Language Notes, Vol. XXVII (1913), p. 93. Al- 
Karkhi included these two operations under algebra and the simple equating of the two members 
as almucabola, but Woepcke adds (Extrait du Fakhri, p. 64) that this is contrary to the common 
usage. The title al-jebr w almuqabala is still used in Arabic. The Arabic verb stem jbr, from 
which algebra is derived, means ' to restore.' So in Spain and Portugal a surgeon was called an 
algebrista. See also note 3, p. 107. 

2 The date is given in the Spanish Era; 1145 a.d., according to our reckoning. 

3 Mohammed ibn Musa, Al-Khowarizmi. The word algorism is derived from his patronymic ; 
the spelling and use in the Latin (see p. 76, line 18), indicate the process of evolution, although 
the term came into use through Al-Khowarizmi 's arithmetic and not his algebra. 

67 



68 



LIBER ALGEBRAE ET ALMUCABOLA 



i inuentum, scilicet radicibus, substantiis et numeris. Solus numerus tamen neque 
radicibus neque substantiis vlla proportione coniunctus est. Earum igitur 
radix est omnis res ex vnitatibus cum se ipsa multiplicata aut omnis numerus 
supra vnitatem cum se ipso multiplicatus : aut quod infra vnitatem diminutum 

s cum se ipso multiplicatum reperitur. Substantia vero est omne illud quod ex 
multiplicatione radicis cum se ipsa colligitur. Ex his igitur tribus modis semper 
duo sunt sibi inuicem coaequantia, sicut diceres 



10 



Substantiae radices coaequant 
Substantiae numeros coaequant, et 
Radices numeros coaequant. 



De substantiis radices coaequantibus . Ca [put] pri [mum]. 

Substantiae quae radices coaequant sunt, si dicas, 

Substantia quinque coaequatur radicibus. 

Radix igitur substantiae sunt 5, et 25 ipsam componunt substantiam, quae 
15 videlicet suis quinque aequatur radicibus. Et etiam si dicas, 

Tertia pars substantiae quatuor aequatur radicibus: 

Radix igitur substantiae sunt 12, et 144 ipsam demonstrant substantiam. Et 
etiam ad similitudinem, 

Quinque substantiarum 10 radices coaequantium. Vna igitur substantia dua- 
20 bus radicibus aequiparatur, et radix substantiae sunt 2 — et substantiam quater- 
narius ostendit numerus. 

Eodem namque modo, hoc quod ex substantiis excreuerit, aut minus ea fuerit, 
ad vnam conuertitur substantiam. Et similiter facies cum eo quod cum ipsis ex 
radicibus fuerit. 
25 De substantiis numeros coaequantibus. Ca [put] II. 

Substantiae vero numeros coaequantes hoc modo proponuntur. 

Substantia nouenario coaequatur numero. Nouenarius igitur numerus men- 
surat substantiam cuius vnam radicem ternarius ostendit numerus. Eodem 
modo iuxta multitudinem et paucitatem substantiarum ipsae substantiae ad vnius 



1-2. id est pro scilicet VR ; et in D. Solis 
numeris VR. neque bis supra versum D 

man. 2, pro in del. bis; nee bis R. con- 

iungitur DR el in marg. alii coniunctis D man. 2 ; 
coniunctis V. In marg. QuidTi^V; Radix D. 

3. in se pro cum se fere ubique VDR. ipsa 
om. V ; ipsam R. 

4. ipsa pro ipso fere ubique D ; ipsum R passim. 
5-6. vero om. D. ex radicis in se ipsa (ipsa 

om. V; ipsam R) multiplicacione VDR. In 
marg. Substantia VD. Add. post colligitur, 

Solus siquidem (quidem R) numerus neque (nee R) 
radicibus neque (nee R) substantiis ulla propor- 
cione coniungitur VDR. ergo pro igitur passim 
VD. semper om. VD. 

7. semet pro sibi. adinuicem V. sicuti. 

8. coaequant + et. 

9-10. et radices numeros coaequant om. D. 
11. Titulum om. 



coequantur 



12. Sed et substantie. sunt ( + est D) quasi 
diceres VD, et C sed del.; diceres pro dicas fere 
ubique. In marg. D 

13. quinque: suis 5. 

14. 35 V. ££ assimilantur 

15. coequare pro 
aequare passim, quasi 
diceres. 

17. ergo V. 

22. ergo pro namque. 
et in marg. man. 2 alii hoc. 



3 



his D 



hoc om. V; 
eis pro ea. 
23. conuertas V; conuertatur D. facias de. 

25. Titulum om. 

26. In substantiis. coequantibus. pro- 
ponitur D. In marg. ) assimilantur <j> D. 

27. equatur V. numerus om. V. 

28. cuius scilicet. Eodem + ergo. 

29. modo hoc (hie D) est. iuxta plurali- 
tatem uel. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 69 

are composed of these three kinds : namely, roots, squares 1 and numbers. 

However number alone is connected neither with roots nor with squares by 

any ratio. Of these then the root is anything composed of units which can 

be multiplied by itself, or any number greater than unity multiplied by 

itself : or that which is found to be diminished below unity when multiplied 

by itself. The square is that which results from the multiplication of a 

root by itself. 

Of these three forms, then, two may be equal to each other, as for 

example : 

Squares equal to roots, 

Squares equal to numbers, and 

Roots equal to numbers. 2 

CHAPTER I 

Concerning squares equal to roots 3 

The following is an example of squares equal to roots : a square is equal 
to 5 roots. The root of the square then is 5, and 25 forms its square which, 
of course, equals five of its roots. 4 

Another example : the third part of a square equals four roots. Then 
the root of the square is 12 and 144 designates its square. 5 And similarly, 
five squares equal 10 roots. Therefore one square equals two roots and the 
root of the square is 2. Four represents the square. 6 

In the same manner then that which involves more than one square, or 
is less than one, is reduced to one square. Likewise you perform the same 
operation upon the roots which accompany the squares. 

CHAPTER II 

Concerning squares equal to numbers 3 

Squares equal to numbers are illustrated in the following manner : a 
square is equal to nine. Then nine measures the square of which three 
represents one root. 7 

Whether there are many or few squares they will have to be reduced in 
the same manner to the form of one square. That is to say, if there 

1 Literally ' substances,' being a translation of the Arabic word mat, used for the second power 
of the unknown. Gerard of Cremona used census, which has a similar meaning. 

2 These are the three types designated as ' simple ' by Omar al-Khayyami, Al-Karkhi, and 
Leonard of Pisa. They correspond in modern algebraic notation to the following : 

ax 2 = bx, ax 2 = n, and bx = n. 

3 These and the following chapter headings were doubtless supplied by Scheybl. 

4 x 2 = 5 x, x = 5, x 2 = 25. 6 5 x 2 = 4 x, x = 12, x 2 = 144. 
6 5 x = 10, x = 2, x 2 = 4. 7 x 2 = 9, x = 3. 



70 LIBER ALGEBRAE ET ALMUCABOLA 

i substantiae similitudinem erunt tractandae. Hoc est, si substantiae duae vel tres 

vel quatuor, siue etiam plures fuerint, earum cum suis radicibus coaequatio, sicut 

vnius cum sua radice, quaerenda est. Si vero minus vna fuerit, hoc est, si tertia 

vel quarta vel quinta pars substantiae vel radicis proposita fuerit, eodem modo ea 

s tractetur, vt si dicas, 

Quinque substantiae 80 coaequantur. Vna igitur substantia quintae parti 
numeri 80 coaequatur, quam videlicet 16 componunt. Et etiam si dicas, 

Medietas substantiae 18 coaequatur. Haec igitur substantia 36 coaequatur. 
Hoc modo omnes substantiae, quotquot in vnum coniunctae, seu quae ab aliis 
10 diminutae fuerint, ad vnam conuertentur substantiam. Hoc idem cum numeris 
etiam, qui cum substantiis fuerint, agendum est. 
De radicibus numeros coaequantibus . Ca [put] III. 
Radices quae numeros coaequant sunt, si dicas, 

Radix ternario aequatur numero. Huius igitur radicis substantiam nouenarius 
*s habet numerus. Et etiam si dicas, 

Quatuor radices vigeno coaequantur numero. Vna igitur radix substantiae 
huius quinario coaequatur numero. Et etiam si dicas, 

Media radix denario coaequatur numero. Tota igitur radix vigeno aequatur 
numero, cuius videlicet substantiam 400 demonstrant. 
20 Radices igitur et substantiae et numeri solum, quemadmodum diximus, dis- 
tinguuntur. Vnde et ex his tribus modis quos iam praemisimus, tria oriuntur 
genera tripliciter distincta ; vt 

Substantia et radices numeros coaequant 
Substa[n]tia et numeri radices coaequant, et 
25 Radices et numeri substantiam coaequant. 

De substantiis et radicibus numeros coaequantibus. Ca [put] IIII. 

Substantiae vero et radices quae numeros coaequant, sunt, si dicas, 

Substantia et 10 radices 39 coaequantur drachmis. Huius igitur artis inuesti- 

gatio talis est : die, quae est substantia, cui si similitudinem decern suarum radicum 

30 adiunxeris, ad 39 tota haec collectio protendatur. Modus hanc artem inueniendi 

est, vt radices iam pronunciatas per medium diuidas, sed radices in hac inter- 

rogatione sunt 10, accipe igitur 5, et iis cum se ipsis multiplicatis producuntur 25 ; 

1. id est pro Hoc est V passim . si om. D. 20. soli. secundum quod iam diximus. 

2. seu quatuor seu. radicibus + queratur. 21. modis om. 22. Et pro vt. 
cquacio V. 23. Substantie. 

4. vel'ora. D. hoc modo tractetur. Sicut diceres. 24. Et substantie. radices et coequant D. 

7. numeri om. et 2 om. D. 

9. substantias. coniunctae + fuerint. 25. substantias. 

10. conuertas. quod et de eo quod cum ipsis 26. Titulum om. V ; Substantias et radicibus 
ex numeris fuerit pro Hoc idem . . . fuerint. numeri coequantibus. D. 

11. est om. D. 12. Titulum om. 27. numerum coequantes. sicut. 

13. Sed et radices. sicut diceres. 28. dragmatibus pro drachmis; dragma pro 

14. coequatur pro equatur. drachma ubique. 
16. equantur pro coaequantur. In marg. 29. Die + ergo. 

y T'- z , 30. xxxvni D. 

T^cyfiui (p v. ^ 2 eas v ea D p ro .j s ipsa p ro ipsis v 

18. equatur pro coaequatur. coequatur pro multiplica et fient 25. facio siue sum pro pro- 
aequatur. duco ubique. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 71 

are two or three or four squares, or even more, the equation formed by 
them with their roots is to be reduced to the form of one square with its 
root. Further if there be less than one square, that is if a third or a fourth 
or a fifth part of a square or root is proposed, this is treated in the same 
manner. 1 

For example, five squares equal 80. Therefore one square equals the fifth 
part of the number 80 which, of course, is 16. 2 Or, to take another example, 
half of a square equals 18. This square therefore equals 36. 3 In like manner 
all squares, however many, are reduced to one square, or what is less than 
one is reduced to one square. The same operation must be performed 
upon the numbers which accompany the squares. 

CHAPTER III 

Concerning roots equal to numbers 

The following is an example of roots equal to numbers : a root is equal to 
3. Therefore nine is the square of this root. 4 

Another example : four roots equal 20. Therefore one root of this square is 
5. 5 Still another example : half a root is equal to ten. The whole root 
therefore equals 20, of which, of course, 400 represents the square. 6 

Therefore roots and squares and pure numbers are, as we have shown, 
distinguished from one another. Whence also from these three kinds which 
we have just explained, three distinct types of equations 7 are formed in- 
volving three elements, as 

A square and roots equal to numbers, 

A square and numbers equal to roots, and 

Roots and numbers equal to a square. 8 

CHAPTER IV 

Concerning squares and roots equal to numbers 

The following is an example of squares and roots equal to numbers : a 
square and 10 roots are equal to 39 units. The question therefore in this 
type of equation is about as follows : what is the square which combined 
with ten of its roots will give a sum total of 39 ? The manner of solving 
this type of equation is to take one-half of the roots just mentioned. Now 
the roots in the problem before us are 10. Therefore take 5, which multi- 
plied by itself gives 25, an amount which you add to 39, giving 64. 

1 Our modern expression " to complete the square," used in algebra, originally meant to make 
the coefficient of x 2 equal to unity, i.e. make one whole square. 

2 5 x 2 = 80 ; x 2 = 16. 3 i x 2 = 18 ; x 2 = 36. i x = 3 ; x 2 = 0. 

s 4 x = 20 ; x = 5 ; x 2 = 25. 6 I x = 10 ; x = 20 ; x 2 = 400. 7 ' Types of equations ' = genera. 
8 Abu Kamil, Omar Al-Khayyami, Al-Karkhi, and Leonard designate these as ' composite ' 
types. In modern notation : ax 2 + bx = n ; ax 2 + n = bx; ax 2 = bx + n. 



72 LIBER ALGEBRAE ET ALMUCABOLA 

i quae omnia 39 adiicias, et veniunt 64. Huius igitur radice quadrata accepta, 
quae est 8, ab ea medietatem radicum 5 subtrahas, et manebunt 3. Ternarius 
igitur numerus huius substantiae vnam ostendit radicem, quae videlicet sub- 
stantia nouenario dinoscitur numero. Nouem igitur illam eomponunt substan- 

5 tiam. 

Similiter quotquot substantiae propositae fuerint, omnes ad vnam conuertas 
substantiam. Similiter quicquid cum eis ex numeris siue radicibus fuerit, id omne 
eo modo quo cum substantiis existi, conuertas. Huius autem conuersionis talis 
est modus, vt si dicas, 

10 Duae substantiae et 10 radices 48 drachmis coaequantur. 

Huius artis talis est inuestigatio, vt dicas, Quae sunt duae substantiae inuicem 
collectae, quibus si similitudo 10 radicum earum adiuncta fuerit, ad 48 tota exten- 
datur collectio. Nunc autem oportet, vt duas substantias ad vnam conuertas. 
Sed iam manifestum est, quoniam vna substantia medietatem duarum designat, 

15 igitur omnem rem in hac quaestione tibi propositam, ad medium conuertas, dicendo : 

Substantia et 5 radices 24 drachmis coaequantur. Modus huius rei talis est, vt 

dicas, Quae est substantia, cui si quinque suas radices adiunxeris, ad 24 excrescat. 

Nunc etiam oportet, vt ad regulam supra datam animum conuertas, et diuidas 

radices per medium et veniunt 2 et vnius medietas, iis cum se ipsis multiplicatis, 

20 producuntur 6 et 1/4, illis 24 adiicias, veniunt 30 et vnius 1/4. Postea huius ag- 
gregati radicem quadratam accipias, quam scilicet 5 et vnius medietas eomponunt, 
ex qua medietatem radicum, 2 1/2, subtrahas et manebunt 3, quae vnam radicem 
substantiae exprimunt, quam substantiam nouenarius componit numerus. Et si 
diceretur, 

25 Medietas substantiae et quinque radices 28 coaequantur drachmis. 

Huius questionis talis est modus, vt dicas, Quae est substantia, cuius medietati 
si quinque suas radices adiunxeris, tota summa ad 28 excrescat, ita tamen vt 
substantia quae prius diminuta fuerat, perfecta compleatur. Igitur huius sub- 
stantiae medietas cum radicibus secum pronunciatis, duplicanda est; veniunt 

30 autem, 

I. quas (+ suis D) super 39 adicias (addicias et 13. ut ad unam substantiam duas conuertas V. 
sic ubique D) et fient 64. Huius ergo collectionis. 15. et hoc (hie D) est ut dicas pro dicendo. 
quadrata om., etsicubique. assumpta pro accepta. 16. substantiam pro substantia. 28 pro 24 V. 

2. quae est : id est V ; est D. id est quinque coequant V. 
pro 5 et sic passim. diminuas pro subtrahas pas- 17. sui pro suas D. 

sim. remanebunt : remanere pro manere ubique. 18. dictam pro datam. diuide. 

In marg.T), figurac ; vide Tab. II. 19. erunt pro veniunt. eas + ergo pro iis. 

3. scilicet D. in seipsa D. multiplica. 

4. noscitur. iam pro illam. 20. et fient 6 et 1/4 (numeri pro 1/4 bis D) quas 

6. fuerint + ut sunt due uel (uel om. D) 3 s videlicet super 24 adicias fientque (fiantque D). 
uel plures seu (siue D) pauciores fuerint. summe pro aggregati. 

7. quotquot V; quidquid D. seu V. ad 21. accipies V. componit D. 

id cui (cuius D) substantiam conuertisti pro id 22. qua + videlicet. radicis substantiam pro 

omne . . . existi. radicem substantiae. 

8. ergo pro autem. 24. diceret. 

9. si om. 26. est 1 om. D. medietati + cui supra versum 

II. Huius + autem D. est hec pro talis est. D man. 2. 

ut si dicas. 27. rx pro radices V. sumas et suas supra 

12. 10 rx. + unius V; x radicum + unius D. versum D man. 2. 25 pro 28 V. quod pro vt. 
In tnarg. 2 ^? ~\* |OV* equantur 24. D. 29. fietque pro veniunt autem. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 73 

Having taken then the square root of this which is 8, subtract from it the 
half of the roots, 5, leaving 3. The number three therefore represents one 
root of this square, which itself, of course, is 9. Nine therefore gives that 
square. 1 

Similarly however many squares are proposed all are to be reduced to 
one square. Similarly also you may reduce whatever numbers or roots 
accompany them in the same way in which you have reduced the squares. 

The following is an example of this reduction : two squares and ten 
roots equal 48 units. 2 The question therefore in this type of equation is 
something like this : what are the two squares which when combined are 
such that if ten roots of them are added, the sum total equals 48 ? First 
of all it is necessary that the two squares be reduced to one. But 
since one square is the half of two, it is at once evident that you should 
divide by two all the given terms in this problem. This gives a square 
and 5 roots equal to 24 units. The meaning of this is about as follows : 
what is the square which amounts to 24 when you add to it 5 of its roots ? 
At the outset it is necessary, recalling the rule above given, that you take 
one-half of the roots. This gives two and one-half, which multiplied by 
itself gives 6|. Add this to 24, giving 305. Take then of this total the 
square root, which is, of course, 5^. From this subtract half of the 
roots, 2 1, leaving 3, which expresses one root of the square, which itself 
is 9. 

Another possible example : half a square and five roots are equal to 28 
units. 3 The import of this problem is something like this : what is the 
square which is such that when to its half you add five of its roots the sum 
total amounts to 28 ? Now however it is necessary that the square, which 
here is given as less than a whole square, should be completed. 4 Therefore the 
half of this square together with the roots which accompany it must be 
doubled. We have then, a square and 10 roots equal to 56 units. There- 

1 ** + io* = 39; 1 of 10 is 5, 5 2 is 25, 25 + 39 = 64. 
V64 = 8; 8- 5 = 3. x = s;x 2 = g. 



For the general type x 2 + bx = n, the solution is x = y(- j + » ; the negative value of 

the square root is neglected, as that would give a negative root of the equation. 

2 2 x 2 + 10 x = 48, reducing to x 2 + 5 x = 24; \ of 5 is 2\, (2§) 2 = 6|, 24 + 65=305, 
V3T1-2I =3. 

b ft 

The general type ax 2 + bx = n is reduced to the preceding by division, giving x 2 + - x = -> 

a a 



\l b \ 2 n b 
and the solution is, as before, x = \[ — I + - ~ — ' 

' \2 al a 2 a 



3 2 x 2 + 5 x = 28, reducing to x 2 + 10 x = 56. x = V 5 2 + 56 - 5, or x = 4. Note that 
the value of x 2 is not given here as it usually is. 

4 Attention is called to the force of the expression, " completing the square," as here used with 
the meaning to make the coefficient of the second power of the unknown quantity equal to 
unity or making one whole square. See also page 81, footnote 1. 



74 LIBER ALGEBRAE ET ALMUCABOLA 

i Substantia et 10 radices 56 drachmis aequales. 

Diuide igitur radices per medium, et veniunt 5, quibus cum seipsis multiplicatis, 
producuntur 25 ; ilia adde 56, et colliguntur 81 ; huius collecti radicem quadra- 
tarn accipias, quam nouenarius componit numerus, atque ex ea medietatem radi- 
s cum 5 subtrahas et manebunt 4, substantiae radix. 

Hoc modo cum omnibus substantiis quotquot ipsae fuerint, cum radicibus 
item et drachmis agendum est. 

De substantiis et numeris radices coaequantibus . Ca [put] V. 
Propositio huius rei talis est, vt dicas, 
10 Substantia et 21 drachmae 10 radicibus coaequantur. 

Ad hoc inuestigandum talis datur regula, vt dicas, Quae est substantia, cui 
si 21 drachmas adiunxeris, tota summa simul decern ipsius substantiae radices 
exhibeat. Huius quaestionis solutio hoc modo concipitur, vt radices primum per 
medium diuidas, et veniunt in hoc casu 5, haec cum seipsis multiplicata, produ- 
15 cuntur 25. Ex illis 21 drachmas, quas paulo ante cum substantiis commemoraui- 
mus, subtrahas, et manebunt 4, horum radicem quadratam accipias, vt sunt 2, 
quae ex medietate radicum 5 diminuas, et manebunt 3, vnam radicem huius sub- 
stantiae constituentia, quam scilicet substantiam nouenarius complet numerus. 
Quod si libuerit, poteris ipsa 2, quae a medietate radicum iam diminuisti, medietati 
20 radicum 5 scilicet addere, et veniunt 7 ; quae vnam substantiae radicem demon- 
strant quam substantiam 49 adimplent. Cum igitur aliquod huius capitis exem- 
plum tibi propositum fuerit, ipsius modum cum adiectione, quemadmodum dictum 
est, inuestiga, quern si cum adiectione non inueneris, procul dubio cum diminu- 
tione reperies. Hoc enim caput solum adiectione simul et diminutione indiget 
25 quod in aliis capitibus praemissis minime reperies. 

Sciendum est etiam, quando radices iuxta hoc caput mediaueris, et medietatem 
deinde cum seipsa multiplicaueris, si quod ex multiplicatione tollitur vel pro- 
creatur, minus fuerit drachmis cum substantia pronunciatis : quaestio tibi 
proposita nulla erit. At si drachmis aequale fuerit vel procreatur vna radix 

1. drachmis om. equiparantes pro aequales. 16. id est pro vt sunt. 

2. erunt 5 et postea eas in seipsis multiplica 18. nouenus pro nouenarius. 

fientque 25. adde igitur eas super 56 eruntque (que 19. Et si volueris. poteris om. ipsi medi- 
care. D) 81 pro veniunt 5 . . . colliguntur 81. etati et (id est D) 5 pro medietati radicum 5. 

3. ergo summe pro collecti. 20. addicias et fient 7 ; addicias corr. in adicias 

4. et pro atque. 5. substantiae radix om. V. quae vnam : Hec igitur (ergo D) 7 unam. 

6. Hoc + ergo. fuerint + et. 21. quam + scilicet. adimplet D. aliquid. 

7. et numeris secum pronunciatis pro item et capitis: capitulum pro caput ubique. exemplum 
drachmis. om. 

8. Titulum om.V ; radicem pro radices D. 22. addicione D et sic passim. secundum quod. 

9. autem huius artis pro huius rei. talis om. 23. quam D. 24. Hie D. enim om. 
~ . -, 1 ~- n 25. tribus pro aliis. prefer hunc (hoc superscr. 
S> T V > A. D. d man _ 2 ) solum in quibus radices mediantur 



In marg. 



11. hoc + ergo. pro praemissis. 

12. similis pro simul. radicibus procreetur. 26. est om. quoniam quando. eas pro me- 

13. Huius + autem. hac regula pro hoc modo. dietatem. 

vt om. Primum ergo radices. 27. deinde om. seipsasV; seipsis D. col- 

14. diuides D. et fient. in hoc casu om. ligitur V. vel procreatur om. et sic infra. 
14-16. eas ergo in se multiplica et erunt 25. Ex 28. pronunciatis + fuerit. quaestio + que. 

his (hiis D) ergo 21 minue (diminuas D) que cum 29. facta fuerat adnullata est, ac si ipsa simul 

substantia iam pretaxauimus pro haec cum . . . dragmatibus fuerit pro proposita . . . vel procrea- 

subtrahas. tur. facta C scd del. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 75 

fore take one-half of the roots, giving 5, which multiplied by itself produces 
25. Add this to 56, making 81. Extract the square root of this total, which 
gives 9, and from this subtract half of the roots, 5, leaving 4 as the root of the 
square. 

In this manner you should perform the same operation upon all squares, 
however many of them there are, and also upon the roots and the units. 

CHAPTER V 

Concerning squares and numbers equal to roots 

The following is an illustration of this type : a square and 2 1 units 
equal 10 roots. 1 The rule for the investigation of this type of equation 
is as follows: what is the square which is such that when you add 21 
units the sum total equals 10 roots of that square? The solution of this 
type of problem is obtained in the following manner. You take first one- 
half of the roots, giving in this instance 5, which multiplied by itself gives 
25. From 25 subtract the 21 units to which we have just referred in 
connection with the squares. This gives 4, of which you extract the square 
root, which is 2. From the half of the roots, or 5, you take 2 away, and 3 
remains, constituting one root of this square which itself is, of course, q. 2 

If you wish you may add to the half of the roots, namely 5, the same 2 
which you have just subtracted from the half of the roots. This give 7, 
which stands for one root of the square, and 49 completes the square. 3 There- 
fore when any problem of this type is proposed to you, try the solution of it 
by addition as we have said. If you do not solve it by addition, without 
doubt you will find it by subtraction. And indeed this type alone requires 
both addition and subtraction, and this you do not find at all in the preceding 
types. 4 

You ought to understand also that when you take the half of the roots 
in this form of equation and then multiply the half by itself, if that which 
proceeds or results from the multiplication is less than the units above- 
mentioned as accompanying the square, you have no equation. 5 If equal 

1 x 2 +21 = ioj;. For this type of equation both solutions are presented since both roots are 
positive. A negative number would not be accepted as a solution by the Arabs of this time, 
nor indeed were they fully accepted until the time of Descartes. 

2 For the general type, x 2 + n = bx the solution isx=-±A/(-] -n, and both positive and 



negative values of the radical give positive solutions of the equation proposed. In this problem 
we have x 2 + 21 = 10 x; J of 10 is 5, 5 2 is 25, 25 — 21 = 4. 

V4 = 2 ; 5 — 2 = 3, one root ; 5 + 2 = 7, the other root. 

3 Another use of the expression, " completing the square." 

4 The Vienna MS. defines, 'in which the roots are halved.' 

5 This corresponds to the condition, b 2 — 4 ac < o, in the equation ax 2 +bx + c = o ; 
in this event the roots are imaginary. 



76 LIBER ALGEBRAE ET ALMUCABOLA 

i substantiae simul etiam medietas radicum, quae cum substantia sunt, pro- 
nunciatur, adiectione simul et diminutione abiectis. Quicquid igitur duarum 
substantiarum, aut plus, aut minus substantia propositum fuerit, ad vnam 
conuertas substantiam, sicut in primo capite praediximus. 
5 De radicibus et numeris substantiam coaequantibus. Ca [put] VI. 
In hoc capite sic proponitur, 

Tres radices et quatuor ex numeris coaequantur substantiae. 

Ad hoc inuestigandum talis datur regula, quod scilicet radices per medium 

diuidas et venit vnum et alterius medietas, hoc deinde cum seipsis multiplices, et 

10 producuntur 2 1/4, his 4 ex numeris adiicias, et veniunt 6 1/4, huius postea radicem 

quadratam accipias, hoc est 2 1/2. Atque earn tandem medietati radicum, vni 

scilicet et dimidio, adiicias, et veniunt 4, quae vnam substantiae radicem com- 

ponunt, quam deinde substantiam numerus 16 adimplet. Quicquid igitur plus 

siue minus substantia tibi propositum fuerit, ad vnam conuertas substantiam. 

is Ex his igitur modis, de quibus in principio libri mentionem fecimus, sunt tres 

priores, in quibus radices non mediantur, in tribus vero posterioribus vel residuis 

mediantur radices prout superius liquet. 

DIXIT ALGAURIZIN 

Sex sunt modi de quibus, quantum ad numeros pertinet, sufficienter diximus. 
20 Nunc vero oportet, vt quod per numeros proposuimus, ex geometrica idem verum 
esse demonstremus. Nostra igitur prima propositio talis est, 
Substantia et 10 radices 39 coaequantur drachmis. 

Huius probatio est, vt quadratum cuius latera ignorantur, proponamus. 

Hoc autem quadratum quod loco substantiae ponimus, et eius radicem scire volu- 

25 mus atque designare. Sit igitur quadratum a b cuius vnumquodque latus vnam 

ostendit radicem. lam manifestum est, quoniam, quando aliquod eius latus cum 

1. similis erit medietati. sunt om. pro- 14. fuerat D. 

nuncianturV. 15. Hiis (His D) ergo 6 modis de; His sed del. 

2. Quidquid. et Ex his in marg. C. 

3. substantiarum, aut minus substantia seu (siue 16. primi pro priores. mediant bis D. 
D) maius ; seu pro siue ubique V. posterioribus vel om. 

4. iam diximus. <yg -a ± •, 18. + Nunc vero oportet quod numero propo- 

5. Titulum om. V. In marg. \ ^^~ ' 1 D. suimus geometrice idem verum esse probemus 

6. hoc + autem. primum sic proponuntur. ante Dixit D man. 2. alguarizimV; alghuarizim 

7. Tres + autem. ex numero uni. D. 

8. Huius igitur pro Ad hoc inuestigandum. ig. Sex + autem. numerum. In marg. 
quantus pro quod scilicet. De 4 prima D. 

9. et fiet una radix et. igitur pro deinde. 20. numero pro per numeros. ex om. geo- 
multiplicata, fientque duo et 4 a . Hec igitur (ergo metrice. 

D) super 4 adicias et erunt 6 et 4 a . 21. probemus. 

10. 6 1/2 C. ergo summe pro postea. 22. 4 pro 10, 29 pro 39 V; xxx et nouem 

11. id est pro hoc est. duo et medium pro 2 1/2 D. 

et sic saepius. 23. rumbus pro quadratum ubique. Rumbum 

11-12. quam super medietatem radicum adicias in text D et rumbum in marg. man. 2. 
id est super unum et alterius medietatem fientque 24. Hie igitur rumbus substantiam quam pro 

(que om. D) 4 pro Atque ... 4. Hoc . . . ponimus. cuius radices V. 

12. quae + scilicet D. 25. atque om. designet (designetur D). et 

13. 16 ergo substantiam adimplent pro quam .. . ipse est pro Sit igitur. unumquotque V. vnam 
adimplet. QuotquotV; Quidquid D. maius + eius. 

pro plus. 26. ostendet. Et iam. aliquid. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 77 

to the units, it follows that a root of the square will be the same as the half 
of the roots which accompany the square, without either addition or dim- 
inution. 1 Whenever a problem is proposed that involves two squares, or 
more or less than a single square, reduce to one square just as we have 
indicated in the first chapter. 



CHAPTER VI 

Concerning roots and numbers equal to a square 

An example of this type is proposed as follows : three roots and the 
number four are equal to a square. 2 The rule for the investigation of this 
kind of problem is, you see, that you take half of the roots, giving one and 
one-half ; this you multiply by itself, producing 2 \. To 2j add 4, giving 
6|, of which you then take the square root, that is, 2§. To 2\ you now add 
the half of the roots, or if, giving 4, which indicates one root of the square. 
Then 16 completes the square. 3 Now also whatever is proposed to you 
either more or less than a square, reduce to one square. 

Now of the types of equations which we mentioned in the beginning of 
this book, the first three are such that the roots are not halved, while in 
the following or remaining three, the roots are halved, as appears above. 

GEOMETRICAL DEMONSTRATIONS 

We have said enough, says Al-Khowarizmi, so far as numbers are 
concerned, about the six types of equations. Now, however, it is necessary 
that we should demonstrate geometrically the truth a 
of the same problems which we have explained in 
numbers. Therefore our first proposition is this, 
that a square and 10 roots equal 39 units. 

The proof is that we construct a square of un- 
known sides, and let this square figure represent the 
square (second power of the unknown) which together ~~ & 

with its root you wish to find. Let the square, FlG - i-— This figure ap- 

,i i , r , . , . . pears only in the Co- 

then, be a b, of which any side represents one root. lumbia manuscript. 

1 Condition for equal roots, b 2 — 4 ac = o. 

2 3* + 4 = x 2 ; §of 3 is i|, (i|) 2 = 2J, i\ + ^ = 6\, Vol = A, 2£ + i§ = 4, the root. 

3 The solution of the general type bx + n = ax 2 , reduced by division to - x + - = x 2 , is 

; a a 

' ft 

-\ 1 , and only the positive value of the radical is taken, since the negative 

a 2a 

value would give a negative root of the proposed equation. 



-ih 



78 LIBER ALGEBRAE ET ALMUCABOLA 

i numero numerorum multiplicauerimus, tunc hoc quod ex multiplicatione colligitur 
erit numerus radicum radici ipsius numeri aequalis. Quoniam ergo decern radices 
cum substantia pronunciantur, quartam igitur partem numeri decern accipimus, 
atque vnicuique lateri quadrati aream aequedistantium laterum applicamus quarum 

5 longitudinem quidem longitudo quadrati primo descripti, latitudinem vero duo et 
dimidium, quae sunt numeri 10 quarta pars, demonstrant. Quatuor igitur areae 
laterum aequedistantium primo quadrato a b applicantur. Quarum singularum 
longitudo longitudini vnius radicis quadrati a b aequalis erit, latitudo etiam sin- 
gularum 2 et medium, vt iam dictum est, demonstrat. Sunt autem hae areae 

ioc d ef. Ex hoc igitur quod diximus, sit area laterum inaequalium, quae similiter 
ponuntur ignota, in cuius videlicet quatuor angulis quorum vniuscuiusque quan- 
titas areae, quam 2 et dimidium cum duobus et dimidio multiplicata perficiunt, 
imperfectionem maioris seu totius areae ostendunt. Vnde fit vt circunduc- 
tionem areae maioris, cum adiectione duorum et dimidii cum duobus et dimidio 

is quatuor vicibus multiplicatorum compleamus, generat autem haec tota multipli- 
catio 25. Et iam manifestum est, quoniam primum quadratum, quod substantiam 
significat, et quatuor areae ipsum quadratum circundantes, 39 perficiunt, quibus 
quando 25, hoc est quatuor minora quadrata, quae scilicet super quatuor 
angulos quadrati a b ponuntur, adiecerimus, quadratum maius, G H vocatum, 

20 circunductione complebitur. Vnde etiam haec tota numeri summa vsque ad 
64 excrescet, cuius summae radicem octonarius obtinet numerus, quo etiam vnum 
eius latus compleri probatur. Igitur vbi ex numero octonario quartam partem 
numeri denarii, sicut in extremitatibus quadrati maioris G H ponuntur, subtraxeri- 
mus bis, 3 ex ipsius latere manebunt. Quinque ergo ex octo subtractis, 3 manere 

25 necesse est ; quae simul vni lateri quadrati primi, quod est a b, aequiparantur. 
Haec igitur 3 quadrati vnam radicem, hoc est vnam radicem substantiae 

1. tollitur D. 11. id est in cuius unoquoque angulo pro quorum 

2. consimilis pro aequalis. Quando igitur V. vniuscuiusque. quantum D. 

3. substantia primum 10 (xiii D) radices pro- 12. cum duobus et dimidio om. V; in duo 
posuimus 4 am partem denarii id est (numeri pro id et medium quantitas D. multiplicata om. D. 
est D) 2 et medium accipiemus (accipiens D) pro circumdant pro perficiunt. 

substantia . . . accipimus. 13. seu totius om. areae + idem V. in pro vtD 

4. At (ac D) eciam unicuique 4 0r laterum rumbi 14. in pro cum 2 . 

primi aream equalium laterum applicabimus ; aequa- 15. ut pro generat autem; perficit sed del. C. 

Hum sed del. C. quarum + scilicet ; scilicet sed 16. 25 + pariat. primus rumbus, qui el sic 

del. C. passim. 

5. quidem om. unius lateri rumbi primi 17. significat : signare pro significare ubique. 
demonstrent. Earum (+ earum in marg. D man. 2) quadratum om. perficiunt (faciunt D) + nume- 
vero latitudinem pro quadrati . . . vero. rum. 

6. medium pro dimidium ubique. numerum 18. hoc est quatuor: et 4 V; id est iiii D. 
obtineant que videlicet 4 am partem denarii numeri minora om. quatuor 2 om. 

pro quae . . . demonstrant. 19. adiciemus. id est rumbus {om. D) e c pro 

7. equalium pro aequedistantium. ab: id est G H vocatum. 

a b. omni pro singularum. 20. Vnde etiam + et V ; +adD. 

8. une radicum V. manet consimilis pro 21. unius octonarius. qui. 
aequaliserit. quarumque latitudinem pro latitudo 22. Quando igitur pro Igitur vbi. similitudi- 
etiam singularum. nem 4 e partis denarii numeri simul in pro quartam 

9. 2 om. D. demonstrant. Ethee(ow.D) . . . sicut in. 

sunt. . 23. qui est e c pro G H et sic poslea. 

10. g d h c et passim. sint D, sit D man. 2. 25. contra similiter pro quae simul. 
aree pro area D. equalium pro inaequalium. 26. vnam + huius. id est. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 



79 



a 


c 




f 




d 




e 


b 



Fig. 



2. — From the Columbia manu- 
script. 





9 




c 


b a 


,! 




I, 





Fig. 3. — From 
the Dresden 
manuscript. 



When we multiply any side of this by a number (of numbers) J it is evident 

that that which results from the multiplication will be a number of roots 

equal to the root of the same number (of the 

square). Since then ten roots were proposed 

with the square, we take a fourth part of the 

number ten and apply to each side of the 

square an area of equidistant sides, of which 

the length should be the same as the length 

of the square first described and the breadth 

2§, which is a fourth part of 10. Therefore 

four areas of equidistant sides are applied to 

the first square, a b. Of each of these the 

length is the length of one root of the square 

a b and also the breadth of each is 2\ , as we 

have just said. These now are the areas, c, d, e, f. There- 
fore it follows from what we have said that there will be four 
areas having sides of unequal length, which also are regarded 
as unknown. The size of the areas in each of the four 
corners, which is found by multiplying 2\ by 2§, completes 
that which is lacking in the larger or whole area. Whence 

it is that we complete the drawing of the larger area by the addition of 

the four products, each 2§ by 2§ ; the whole q 

of this multiplication gives 25. 

And now it is evident that the first square 

figure, which represents the square of the 

unknown (x 2 ), and the four surrounding areas 

(10 x) make 39. When we add 25 to this, 

that is, the four smaller squares which indeed 

are placed at the four angles of the square 

a b, the drawing of the larger square, called 

G H, is completed. Whence also the sum 

total of this is 64, of which 8 is the root, 

and by this is designated one side of the 

completed figure. Therefore when we sub- 
tract from eight twice the fourth part of 10, which is placed at the 

extremities of the larger square G H, there will remain but 3. Ffve 

being subtracted from 8, 3 necessarily remains, which is equal to one 

side of the first square a b. 2 

This three then expresses one root of the square figure, that is, one 

root of the proposed square of the unknown, and 9 the square itself. 





c 




/ 


a 

b 


d 




e 





Fig. 4. — Final completed figure, 
the Columbia manuscript. 



H 

From 



1 Evidently meaning a pure number. 

2 The proportions of the figures are not correct to scale. 



80 LIBER ALGEBRAE ET ALMUCABOLA 

i propositae : nouenarius deinde numerus ipsam substantiam exprimit. Ergo 
numerum denarium mediamus, et alteram eius medietatem cum seipsa multi- 
plicamus, deinde totum multiplicationis productum numero 39 adiicimus, vt 
maioris quadrati G H circunductio compleatur. Nam eius quatuor angulorum 

s diminutio totam huius quadrati circunductionem imperfectam reddebat. Mani- 
festum enim est, quod quarta pars omnis numeri cum suo aequali, ac deinde cum 
quatuor multiplicata, eandem perficiat numerum, quern medietas numeri cum 
seipsa multiplicata, perficit. Igitur si radicum medietas cum seipsa multiplicetur, 
huius multiplicationis summa, multiplicationem quartae partis cum seipsa ac 

10 deinde cum quatuor multiplicatae, sufficienter euacuet, adaequabit vel delebit. 

Ad hoc etiam idem demonstrandum, altera datur formula, quae talis est. Quad- 

rato a b substantiam significante aequalitatem decern radicum addimus ; has radices 

deinde per medium diuidamus, venient 5, ex quibus duas areas ad duo latera 

quadrati a b constituamus, hae autem vocentur a g et b d, et erit vtriusque vtra- 

15 que latitudo vni lateri quadrati a b aequalis ; vtramque denique longitudinem 
numerus quinarius adimplebit. Superest iam, vt ex multiplicatione 5 cum 5, 
quae medietatem radicum quas ad duo latera quadrati prioris substantiam signi- 
ficantis, addidimus, quadratum faciamus. Vnde iam manifestum est, quod duae 
areae, quae supra duo latera ponuntur, et quae 10 radices substantiae significant, 

2osimul cum quadrato priori, quod est substantia, 39 ex numero adimpleant. 
Manifestum etiam, quod area maioris seu totius quadrati per multiplicationem 
5 cum 5 perficiatur. Hoc ergo quadratum perficiatur, atque ad perfectionem 
eius numerus 25 ad priora 39 adiiciatur : tota igitur haec summa vsque ad 64 excres- 
cet. Nunc summae huius radicem quadratam, quae vnum latus quadrati maioris 

25 designat, accipiamus, atque inde aequalitatem eius quod ei addidimus, hoc est 5, 

1. proposito obtinent, quam videlicet substan- 14-16. constituas, proponimus et ipse sunt aree 
tiam nouenarius adimplet numerus (+ si D man. 2). b d et b g (et b g add. in marg. D man. 2), quare 
Igitur. simul (similes sed del. D) latitudines (latitudinem D) 

2. unam pro alteram eius. in seipsam. mul- uni lateri rumbi a b habentur consimiles. Earum 
tiplicauimus V ; multiplicauerimus D. vero (earum Nam D) longitudines quinarius com- 

3. et pro deinde. totam multiplicationis plet numerus, qui medietatem 10 radicum super duo 
summam super (sunt D) 39 addidimus ; summam latera rumbi a b adiectarum adimplet. 

39 C sed del. 16. Restat igitur pro Superest iam. vt + 

4. compleretur. Nam + et. rumbum. 

6. quam pro quod saepius. consimili ducta, 17. radicum : 10 radicum designant. super 
deinde in 4 V; consimili a e ductum in 4 or D. pro ad; sup C sed del. primi D; extremi V. 

7. eundem perficiet. quem + si V; + et si substantiam significantis om. 

D. medietas + unum D. numeri om. 18. addidimus qui et substantiam signet faci- 

8. semel duceretur pro multiplicata ; semel C amus. Et iam. est om. V. 
sed del. perficeret. si om. D. medietas + 20. primo. adimplent. 

semel. ducatur pro multiplicetur; ducere pro 21. Manifestum + est. quam ex perfectione 

multiplicare saepius. aree pro quod area. seu totius om. ex mul- 

10. iii D, sed 4 superscr. man. 2. adaequabit tiplicatione. 
vel delebit om. 22. 5 2 + que (et D) sunt 25 nondum. per- 
il, demonstrandum: comprobandum V; pro- ficitur. et pro atque. perfectam pro perfectionem. 
bandum D. Sit (que est D) a b ipsam signans 23. rumbi maioris qui est rumbus a h super 39 
substantiam (+ ad D) cuius similitudinem 10 pro adiciatur pro eius . . . adiiciatur. TotoergoD. 
quae talis est . . . decern. vsque om. V. 

12. radices. 10 ergo (igitur D) pro has. 24. Huius ergo summe. 

13. deinde om. et fient pro venient. quibus 25. accipias et ex ea similitudinem eius quod ei 
+ similiter. super pro ad. addidisti id est 5. 



9 


a 

b 


h 


d 



THE BOOK OF ALGEBRA AND ALMUCABOLA 81 

Hence we take half of ten and multiply this by itself. We then add the 
whole product of the multiplication to 39, that the drawing of the larger 
square G H may be completed ; * for the lack of the four corners rendered 
incomplete the drawing of the whole of this square. Now it is evident 
that the fourth part of any number multiplied by itself and then multi- 
plied by four gives the same number as half of the number multiplied 
by itself. 2 Therefore if half of the roots is multiplied by itself, the sum 
total of this multiplication will wipe out, equal or cancel 3 the multiplication 
of the fourth part by itself and then by four. 

Another method 4 also of demonstrating the same is given in this manner : 
to the square a b representing the square of the unknown we add ten roots 
and then take half of these roots, giving 5. From this 
we construct two areas added to two sides of the square 
figure a b. These again are called a g and b d. The 
breadth of each is equal to the breadth of one side of 
the square a b and each length is equal to 5. We 
have now to complete the square by the product of 5 
and 5, which, representing the half of the roots, we 
add to the two sides of the first square figure, which figure. From^the 
represents the second power of the unknown. Whence Dresden manu- 

... script. 

it now appears that the two areas which we joined to 

the two sides, representing ten roots, together with the first 
square, representing x 2 , 6 equals 39. Furthermore it is 
evident that the area of the larger or whole square is 
formed by the addition of the product of 5 by 5. This 

Fig. 6. — From . , . . . . 

the Dresden square is completed and for its completion 25 is added to 39. 

manuscript. ^e sum total is 64. Now we take the square root of 

this, representing one side of the larger square and then we subtract 
from it the equal of that which we added, namely 5. Three remains, 

1 This corresponds to our algebraic process of completing the square. The correspondence 
of the geometrical procedure to the terminology and the methods employed in algebra makes it 
highly desirable to present the geometrical and algebraical discussions together to students of 
elementary mathematics. 

■♦©'-(!)' 

3 The words adaequabit vel delebit are doubtless added by Scheybl to explain the force of 
euacuet. 

4 A method slightly different from either of these is given by Abu Kamil and also in the 
Boncompagni version of Al-Khowarizmi's algebra, ascribed to Gerard of Cremona. This con- 
sists in applying to one side of the square a rectangle with its length equal to 10, while the other 
dimension is the same as that of the square. The two together represent x 2 + 10 x, or 39. Bi- 
sect the side whose length is 10. Now by Euclid II, 6, the square on half the side 10 plus the 
side of the original square, (x + $) 2 , equals the whole rectangle (39) plus the square 25 of half 
the side 10. The rest of the demonstration is similar to that here given. 

5 1 use x 2 for substantia here and in the following demonstrations. 



9 


a 

b 


h 


d 



82 LIBER ALGEBRAE ET ALMUCABOLA 

i subtrahamus, et manebunt 3 ; quae latus quadrati a b hoc est vnam radicem sub- 
stantiae propositae complere probantur. Tria igitur huius substantiae sunt radix ; 
et substantia nouem. 

De substantia et drachmis res coaequantibus. 

5 Substantia et 21 drachmae 10 rebus aequiparantur. 

Propositio haec seu quaestio in capite quinto proposita fuit, cuius hie demon- 
stratio docetur. Quadratum igitur a b, quod latera habet ignota, substantiam 
pono, atque ei parallelogrammum rectangulum, cuius vtraque latitudo vni lateri 
quadrati a b aequalis sit, cuiusque longitudo vtraque rerum seu radicum medie- 

10 tatem referat, applico. Deinde vero huius rectanguli summam 21 ex numeris 
constituo, qui numerus cum ipsa substantia propositus est. Haec autem area vel 
rectangulum b g inscribitur , cuius simul latitudo g d dinoscitur , longitudo ergo duarum 
arearum inuicem coniunctarum, in h d terminatur. Et iam manifestum est quod 
haec longitudo denarium obtinet numerum, quoniam omnis area quadrilatera et 

15 rectorum angulorum, ex multiplicatione sui lateris cum vnitate semel, vnam 

obtinet radicem : et si cum binario numero, duae eiusdem areae nascuntur radices. 

Quoniam igitur primo sic proposuit, vna substantia et 21 drachmae, 10 radi- 

cibus aequiparantur, manifestum est, quod longitudo lateris h d in denario termi- 

netur numero, quia latus h b vnam substantiae radicem obtinet, latus igitur h d 

20 super punctum e per medium diuide. Vnde et linea e h lineae e d net aequalis, 
atque ducta ex puncto e linea perpendiculari e t : haec eadem perpendicularis 
ipsi h a lineae aequalis erit. Lineae ergo e t portionem, quae ipsaque d e linea 
breuior est, in rectum adiicio e c, et fiet linea t c aequalis / g lineae, vnde quadratum 
/ /, quod ex multiplicatione medietatis radicum cum se ipsa multiplicatae, id est ex 

25 multiplicatione quinarii cum quinario (in hoc casu) colligitur, nobis eueniet. Et 
iam manifestum etiam est, quoniam area b g 21, quae substantiae addidimus, in se 
obtinet, ex a ea igitur b g per lineam t c, quae est vnum latus areae / / extrahamus 
atque ex eadem area b g aream b t minuamus, deinde vero super lineam e c quae 

I. subtrahas. quae + simul. qui pro hoc. 15. angulorum equalium pro rectorum angu- 
vnam radicem om. substantia complere pro- lorum. cum om. D. unum pro vnitate V. 
batur. + Hoc igitur (ergo D) unam substantie semel om.V ; semel + deducta D. 
(substantiam D) radicem adimplet (adimplent D). 16. binario + ducatur. 

3. et om. V. et nouem D. 17. primum. drachmae om. 

4. Tituliim om. V; 21 dragmatibus 4 res D. 18. est om. D. 

5. Substantia + vero. 19. quoniam pro quia. a b V; iD pro h b. 

6. Propositio . . . docetur om. 20. diuido. et linee e d D. fiet similis V; 

7. habeat. similes sunt D. 

8. cui V; tibi D; cui C sed del. pro atque ei. 21-23. atque ducta ex puncto e linea ... in 
aream laterum equalium equidistancium V ; aream rectum adiicio e c om. Add. Sed et iam mani- 
lateris equalium D pro parallelogrammum rectangu- festum est quam linea e c similis sit h b. In linea 
lum. vtraque om. ergoc / simile residui d e super e I adiciam ut et aree 

9. similis; similis pro aequalis saepius. eiusque circumduccie (circumduccione D) adimpleatur. 

V. vtraque om. ad (om. D) quamlibet hanc 24. qui V ; est D pro quod. in seipsam V; 

(om. D) quantitatem pro rerum . . . referat. om. D. 

10. Summa igitur huius aree 21 ex numero. 25. in hoc casu om. colligitur (colligi D) + 

II. qui + simul est. eciam pro autem. vel et fiunt 25. nobis eueniet om. D. 
rectangulum om. 26. a g. quae + in simul. 

12. a g. inscribi D. d in g d om. C. 27. obtineat. igitur om. D. a g pro b g. 
cognoscitur V. linea pro per lineam. 

13. iam om. D. quam sua pro quod haec. 28. at V, ac D pro atque. eadem om. at 

14. obtineat. 4 te V; quadrata D. V, ac D pro deinde vero. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 



83 



which proves to be one side of the square a b, that is, one root of the pro- 
posed x 2 . Therefore three is the root of this x 2 , and x 2 is 9. 




Fig. 



7. — Incomplete figure. From the Columbia 
manuscript, where it twice appears. 



b z 
h a 




c g 

e y d 



Fig. 8. 2 — From 
the Dresden 
manuscript. 



Concerning a Square and Units Equal to Unknown Quantities 

A square and 21 units are equal to ten unknowns. This proposition or 
problem was proposed in the fifth chapter and here a geometrical 
demonstration is presented. 

Suppose that the square a b, a 
having unknown sides, represents 
x 2 and apply to it a rectangu- 
lar parallelogram of which the 
breadth is equal to one side of 
the square a b and the length is 
any quantity you please. 1 Then 
the numerical value of this rec- 
tangle is 21, which number accompanies the same x 2 . Moreover this area 
or rectangle is called b g, of which one side is g d and the 
length of the two areas together is finally h d. And it 
is now evident that this length represents 10, since every 
quadrilateral having right angles (every square) gives for 
the product of one of its sides by unity one root, and if 
multiplied by two gives two roots of its area. 
Therefore since the problem was given, x 2 and 21 units equal 10 roots, it 
is evident that the length of the side k d is 10, for the side h b designates one 
root of x 2 . Therefore bisect the side h d at the point e 3 so that the line e h 
is equal to the line e d. From the point e draw the perpendicular e t. This 
perpendicular equals h a. Add to the prolongation of the line e t a part e c 
equal to the amount by which it is less than d e and then / c will equal t g. 
Whence we arrive at the square / / which is the product of half of the roots 
multiplied by itself, that is the product, in this instance, of 5 and 5. More- 
over we know that the area b g which we add to x 2 amounts to 2 1 . Therefore 
we cut across the area b g with the line / c, which is one side of the area / /, 
and thus decrease the area b g by the amount of the area b t. Then we form 

1 Scheybl's text is incorrect here. 

2 If the lettering of this figure is made to conform to that of our text the demonstration will be 
seen to be not materially different; it is based more directly on Euclid II, 5. This proposition, 
following Heath, The Thirteen Books of Euclid's Elements, reads as follows : " If a straight line 
be cut into equal and unequal segments, the rectangle contained by the unequal segments of the 
whole together with the square on the straight line between the points of section is equal to the 
square on the half." This is one of the propositions of Euclid which connect very directly with 
the geometrical solution of the quadratic equation. 

3 The completed figure (Fig. 9) appears on p. 85. The lettering of Fig. 7 does not correspond 
to that of the completed figure. 



84 LIBER ALGEBRAE ET ALMUCABOLA 

i est in quo linea I e c lineam a h in quantitate deuincit, quadratum enmc ponamus. 
Vnde et iam manifestum, cum linea / c aequalis sit linea / c, nam ipsae in quadrato 
1 1 aequales protenduntur, similiter et linea e c lineae m c aequalis, quia ipsae quad- 
ratum e m aequali dimensione circundant : aequalium igitur ab aequalibus lineis 

s subtractione facta, linea / c lineae / m aequalis relinquetur quod est notandum. 

Rursus manifestum est, quoniam linea g d aequalis est lineae a h, cum ipsae in 

latitudine areae h g aequali dimensione tenduntur, sed linea a h aequalis est lineae 

h b cum ipsae in vno quadrato appareant. Item quoniam linea g I aequalis est 

lineae d e, cum ipsae in vno quadrato reperiantur aequales, sed linea d e aequalis 

10 est h c, cum ipsae decern radices per medium diuidant ; linea igitur d I residuum 
lineae g I, lineae e b ex linea e h residuae aequalis erit : atque tandem, cum linea 
t e lineae I m ex superiori demonstratione, non sit inaequalis, area quam linea / e 
et e b circundant, comprehensae sub / m et d I lineis areae aequalis erit. Area 
igitur / b aequalis est m d areae. Et iam manifestum est, quoniam quadratum 

is 1 1 25 in se continet, cum ergo ex eodem quadrato / 1 areas d t et m d, quae videlicet 
duabus areis g e et / b, 20 et vnum in se continentibus, sunt aequales, subtraxerimus, 
quadratum n c nobis manebit, qui simul numerum, qui est inter 25 et 21, obtinet. 
Et hie numerus est quaternarius, cuius videlicet radicem duo designant, quae 
latus e c adimplent. Hoc autem latus aequale est lineae e b, quoniam e c aequale 

20 est d I lateri, cum ipsa in latitudine areae d c aequalia protendantur. Iam mani- 
festum est, quoniam d I aequalis est lineae e b, quando igitur e b quae sunt duo, 
ex e h, quae sunt radicum medietas, quam quinarius ostendit numerus, abstuleri- 
mus, linea b h ternarium ostendens numerum restabit. Ternarius igitur numerus 
radicem primae demonstrat substantiae. 

25 Quod si contra lineam e c lineae e h, quaemedietatem radicum continet, addideri- 
mus, colligentur 7, quae lineam n h ostendunt. Et tunc radix substantiae maior 

1. uincit. e et n m c faciamus pro enmc te et e b circundant om. V; area igitur quam linee 
ponamus. t e, e a circumdant D. 

2. Sed pro Vnde. manifestum est, quam et 13. comprehensae sub . . . aequalis erit om. V; 
linea c t similis. nam + et. similis est aree quam linee m 1,1 d circumdant. D. 

3. similis est. quoniam et pro quia. 14. / a pro t b. area pro quadratum. 

4. diuisio pro dimensio ubique. aequalium 15. contineat. area pro eodem quadrato. 
. . . facta om. areas + areas D. et om. 

5. linea -f igitur V; + ergo D. restat con- 16. e t, eg 21 V; at, e g 20 et unum D. 
similis. quod est notandum om. consimiles V ; similes D. 

6. Et iam eciam pro Rursus. quoniam + et. 17. similitudinem pro simul V. numeri. 
similis sit et sic saepius. b h pro a h et sic infra. 19. et hoc (hie D). autem om. simile est 
quoniam pro cum. ipsae + est V. e a V ; similis est e a D pro aequale est lineae e b. 

8. d e V ; g dD pro hb. cum ipsae . . . lineae similis pro aequale. 2 

d e om. V; Nam quoniam linea g I et linea h e in 20. lateri om. Nam et ipse pro cum ipsa 

quantitate habentur consimiles quoniam linea g I aequalia om. continentur. Et iam. 

similis est linee d e D. 21. d vel ( ?) pro dlD lineae om. e a pro e b. 

9. nam pro cum. reperiuntur. aequales om. quando igitur e b om. V; Cum e a D. 

sed om. lineaque. 22. radicis. quaternarius D el quinarius in 

10. Nam eciam et pro cum. diuidunt. dl marg.man. 2. 

residuum lineae g I, lineae om. 23. designans pro ostendens. 

11. e a pro e b. residua. erit + linee d I ex 25. Vel pro Quod D. contra om. linea pro 
linea g I residue. Sed et pro atque tandem, lineam D. e n (e h D) super lineam c h (e h 
cum. D) addiderimus. quae + solum V ; quae + 

12. ex superiori demonstratione om. iam semel D. 

fuerat consimilis pro non sit inaequalis. area quam 26. et {om. V) fiunt 7. a or pro maior D. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 



85 







e n 


h b 




■ 



d 



m 



I 



Fig. q. — Completed figure. 



From the Columbia manuscript, where it appears 
twice. 



the square en m c upon the line e c, which is of the length by which the line 
tec exceeds the line a h. Whence, since / c equals / c, being found in the 
square 1 1, and similarly since e c equals m c, these being equal dimensions of 
the square e m, and further equal lines being subtracted from equal lines, 
it is evident that t e is left equal to / m. This is to be noted. 

Again it is evident that the line g d is equal to a h, since they represent in 
breadth equal dimensions of the area h g, and the line a h equals h b as they 
appear in one 

square. Also a, — — , f- ,g 

since the line g I 
is equal to d e, 
being found in 
the same square, 
and d e is equal 
to he, each being 
the half of ten 
roots, therefore 
the line d I, the 
residuum of the 
line g I, is equal 
to e b, the resi- 
duum of the line e h. And so, as the line / e, by the above demonstration, 
is not unequal to the line I m, the area which is included by the lines / e and 
e b is equal to the area comprehended by the lines I m and d I. Therefore 
the area t b equals the area m d. The square / / equals 25. Therefore 
when we subtract from this same square / / the areas d t and m d, which are 
of course equal to the two areas g e and t b, containing 21, it is evident that 
we have left the square n c, which amounts to the difference between 25 
and 21. This number is four, of which the root is two, and this gives the 
Moreover e c equals d /, since each represents the breadth of the 
Since d I equals e b, it is evident that when e b, which is two, is 

taken from e h, which is half 
of the roots, or five, three 
remains for the line b h. 
Therefore three is the root 
of the first x 2 . 

On the contrary if we add 
the line e c to the line e h, rep- 
resenting half of the roots, 
we get 7 which is n h. And 
so the root of the square is 

Fig. 10. — Unlettered figure. From the Columbia manuscript, greater than (the TOOt of) the 



line e c. 
area d c 



86 LIBER ALGEBRAE ET ALMUCABOLA 

i priore substantia erit, quia videlicet super ipsam 21 addidimus. Et similiter ipsa 
suis 10 radicibus aequalis net, quod demonstrare voluimus. 
De radicibus et numeris substantiam coaequantibus. 
Tres radices et 4 ex numeris coaequant substantiam. 

5 Quadratum igitur cuius latera ignota ponuntur propono, quod sit a b c d, atque 
hoc quadratum tribus radicibus et quatuor ex numeris, vt diximus, aequale con- 
stituo. Quoniam autem manifestum est, quod si vnum latus omnis quadrati 
semel in vnitatem ducamus vna radix eiusdem quadrati necessario nascatur ex 
quadrato igitur abed per lineam ef aream af resecemus, atque vnum huius areae 

10 latus numerum ternarium, id est numerum radicum, significare constituamus. 
Sit autem hoc latus linea a e. Nobis igitur manifestum quoniam area e c numero 
quaternario, quern supra radices adiecimus, adimpletur, igitur super punctum g 
latus a e tres radices signans, per lineam in duo media diuidamus, ex quorum vno 
quadratum, quod est g k I e faciamus, quadratum inquam, quod ex multiplicatione 

15 medietatis radicum in seipsa ductae perficitur, hoc est ex vnitate et medietate cum 
seipsis multiplicatis producitur. 

Deinde lineam k m quae sit lineae e d aequalis, lineae g k adiiciamus, fietque 
linea g m aequalis lineae g d, vt quadratum, quod est g 0, inde nascatur. Et iam 
manifestum est, quoniam linea a d aequalis est lineae e f, sed g d aequalis est e n 

20 cum ipsae in quadrato g tenduntur aequales, linea igitur g a lineae w/manebit 
aequalis. Quia vero g a aequalis est lineae g e cum ipsae radices medient, id est per 
medium diuidant, linea insuper g e lineae k I aequalis, nam ipsae in latitudine areae 
era continentur aequales: linea igitur & / lineae nf aequalis erit. Rursus manifestum 
est, quoniam linea d g aequalis est lineae e n cum ipsae in quadrato g tendantur 

25 aequales. Sed linea g e aequalis est lineae e I cum ipsae quadratum g I aequali 

1. ista pro priore. erit om. quando pro quia. 15. ductae: deducte V; deductam D. per- 
addidimus, fiet ipsa substantia 10 suis consimilis ficitur om. 2 b u8 e t 4 ay ; duobus et 4 ta D pro 
radicibus et hoc est (tamen D) quod explanare vnitate et medietate. 

voluimus. 16. deductis perficitur. 

2. Add. Sequitur huius alterius partis pro ad- 17. 1 1 pro k m. quae sit om. a h pro e d. 
ditione figura geometrica C. consimilem. e t pro g k. et ffietque D. 

3. Titulum om. V ; De tribus radicibus et 4 or ex 18. a e pro g m. el pro g d. area e m pro 
numero D. go. 

4. Tres + autem. numero D. 19. a g pro ad. h z, sed h e (a e D). h n 

5. ponuntur + substantiam. sit + rumbus a pro en. 

d. Sed et totum (+ habet D) pro atque hoc. 20. quoniam pro cum. rumbo em. eg 

6. numero. que prediximus. equalem. pro g a. n c V ; z n D. remansit consimilis 
constituto D. pro manebit aequalis. 

7. Et iam pro Quoniam autem. 21. Sed et linea pro Quia vero. e g pro g a. 

8. simul D. duxerimus V; duximus D. h e pro g e. Nam et ^ro cum. mediant. 

eius V. nascetur. Aream igitur (ergo D) h d 22. diuidunt. quoque h e (ergo h a sed a corr . 

ex area a d resecemus (repetemus D sed resecemus ex e man. 1 D) similis est linee t c pro insuper . . . 

superscr. man. 2) et unum aliquod (aliud D) pro ex aequalis. 
quadrato . . . huius areae. 23. h I pro em. n z pro hi. c I pro n f. 

10. signans constituamus. Et iam pro Rursus. 

11. Erit quoque (que D) hoc latus (+ simile D) 24. a h pro d g. lineae om. m n pro e n. 
z d . area h b. 4 or ex numero que. quoniam in latitudine aree a z proponuntur pro cum 

12. adimpleat. e pro g. ipsae . . . tendantur. 

13. ehgV; hg~D. per lineam om. qui- 25. Sed + et. a e pro g e. quoniam pro 
bus pro quorum vno. cum. rumbum e m V ; rumbos eos D pro qua- 

14. area e c pro g k I e. Eritque hec area que dratum g 0. 
pro quadratum inquam, quod. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 



87 




Fig. 11. 
Incomplete figures. 



Fig. 12. 
From the Columbia manuscript. 



first square. Of course when you add 21 to it the sum is equal likewise to 
ten of its roots which we desired to demonstrate. 1 

Concerning Roots and Numbers Equal to a Square 

Three roots and four are equal to x 2 . 

I suppose a square, which is a b c d, of which the sides are unknown ; this 
square, as we have said, 
equals three roots and four in 
number. If one side of any 
square is multiplied by unity 
you necessarily obtain one 
root of the same square. 
Therefore we cut off from the 
square abed the area a f by ^ 
the line e f and one side of 
this area we take to be three, 
constituting the number of 

the roots ; let this side be the line a e. Now it is clear to us that the area 
e c amounts to the four which is added to the roots. Hence we bisect the 
side a e, representing three roots, at the point g. Upon this half construct 
a square, which is g k I e. I say that this square is made by the multiplica- 
tion of half of the roots by itself, that is, produced by one and one-half 
multiplied by itself. Then to the line g k we add the line k m, which is equal 

I, to the line e d. The line g m equals the line g 
d, thus forming a square which is g 0. Since 
a d is equal to e f and g d equal to e n, as 
they occur in the square g 0, it is evident 
-,to that g a is left equal to n f. The line g a is 

equal to the line g e, being half of the roots, 
that is, bisecting the roots, and it is further 
true that g e equals k I, for each measures 
the breadth of the area e m. Therefore k I 
equals n f. Again since d g equals e n, 
being in the square g 0, and g e equals e I, as 



a 



n 



f 



d 

Fig. 



o 
13. — Completed figure. From the they measure the same dimension of the 

Columbia manuscript. gquare g ^ [t . g evi(knt ^ ^ ^ g ^ . g 



1 This paragraph is not found in the Libri version but appears in the Arabic as published by 
Rosen. The translation follows the Vienna version. The figure to be used in the geometrical 
demonstration to obtain by addition the second root of the given quadratic equation appears at 
the bottom of the preceding page. (Fig. 10.) The Boncompagni version (loc. cit. p. 35) varies 
by letting the middle point fall first within the side of the first square, and secondly without : 
Cum itaque dividitur per medium linea b e ad punctum 2, cadet ergo inter puncta g e aut b g : 
sit hoc prius inter puncta b g. 



88 LIBER ALGEBRAE ET ALMUCABOLA 

i dimensione circundant ; linea igitur e d lineae / n aequalis manebit. Quia vero 
e d lineae aequalis est lineae n o, cum ipsae in latitudine areae proponantur 
aequales ; et linea igitur n o lineae / n aequalis erit, atque tandem cum linea k I 
lineae n f ex superiore demonstratione non sit inaequalis, area quam lineae k I et 
sin circundant, comprehensae sub nfetno lineis areae aequalis erit. Area 
igitur k n aequalis est n c areae. Et iam manifestum, cum area e c quatuor ex 
numeris, quae supra tres radices addidimus, in se contineat, duae areae e o et k n 
vni areae e c, quae quatuor ex numeris in se continet, in quantitate aequales 
fiunt. Manifestum est igitur nobis, quadratum g o ex multiplicatione medietatis 

ioradicum, id est i^ ex numeris, cum suo consimili, et adiectione numeri quatuor, 
cui duae areae e o et k n aequales sunt, compleri. Est autem totum hoc 
quadratum numero senario et vni quartae aequale, atque eius radicem duo et 
medium designant, quae in latere d g continentur. Restat igitur nobis ex latere 
quadrati primi, quod est area abed quae totam substantiam significat, radicum 

15 medietas, quae vnum et medium in se continet, latus etiam g a. Cum ergo linea 
d g, quae est latus quadrati g 0, continentis in se id quod ex multiplicatione 
medietatis radicum cum seipsa colligitur, cuiusque adiectio sunt quatuor, quae 
diximus. Et hoc totum sex et vnam quartam in se continet, quarum radicem 2\ 
super lineam g a quae est medietas trium radicum, vnum et medium in se conti- 

20 nens, addimus. Tota igitur haec summa, ad quaternarium excrescet numerum, 
quae est linea a d ; item radix substantiae atque insuper etiam quadratum a c. 
Tota autem substantia in 16 terminatur. Et hoc est quod exponere voluimus. 

Haec igitur geometrica compendiose diximus vt ea quae alioquin oculis mentis 
difficultate quadam concipiuntur, his geometrice perspectis, ad eadem intelligenda 

25 facilior huius disciplinae aditus paretur. 

MAHOMET ALGAURIZIN DE ADDITIS ET DIMINUTIS 

Et inueni (inquit Mahomet Algoarizin) omnem numerum restaurationis et 

1-12. diuisione (+ diuisione D et diuisione su- 13. a e et sic infra, 

perscr. D man. 2) circumdant. Et linea h e similis 14. a d. quae + et. 

est linee e t. Nam et ipse rumbum e c equali Ion- 15. duo pro vnum. que simul lineam e g 

gitudine circumdant. Remanet eciam linea a h si- perficiunt (perfit D). Cum ergo lineam a e que est 

milis linee / /, linea ergo t I similis sit (est D) linee radix rumbi e m que hoc pro latus ... id. 

mn. Sed et linea c I linee n z iam fuerat consimilis. 17. radicem D. in suo consimili + deducte. 

Area igitur quam mn, nz circumdant similis est aree colligitur + continent. 

(+ quam c t, 1 1 similis quam / e circumdant. Area 18. obtinet. quare radix duo sunt et medium 

ergo n z similis est aree D ; n z pro n t del.) c I. Et (+ que si V). 

iam manifestum est quoniam area a z 4 ex numero 19. e g. in se om. 

que sunt (super D) 3 radices addidimus in se obti- 20. addiderimus. igitur om. 

neat due igitur aree an etc I vni aree a % que 4 ex nu- 21. a g que est radix substantie que est area a d. 

mero in se continet in quantitate fiunt consimiles. 22. igitur V; ergo D pro autem. 16 + ut 

Manifestum ergo est nobis quoniam rumbus e m subiecta docet descripcio. 

qui (numeri D) ex multiplicatione radicum in suo 23. geometrice. que oculis mentis quasi qua- 

consimili deducte que scilicet duobus (duo habet D) dam difficultate concipiuntur, perspectis his (hiis D) 

et medium colligitur totum adimpleat (adimplet D) geometrice figuris, ad. 

id est duo et j que simul rumbum e c perficiunt 25. faciliorem disciplinem D. prebeant adi- 

cuius videlicet adiectio sunt 4 ex numero que due turn ; aditur paratur C. Pagina fere tota vac. V. 

aree a n, c I adimplent. Fietque (Sitque D) hoc 26. Mahumed Algorismus V; Mauhumed al- 

totum senario numero et vnius 4 e coequale (co- gaorisim D. Titulum om. C. 

equales D), cuius simul radicem (radicis D) duo et. 27. Mahumed Alguarizmi (algaorismi D). 



b 
z 


a 
n c h 








x t e 
d 9 



THE BOOK OF ALGEBRA AND ALMUCABOLA 89 

left equal to I n. Further since e d equals n 0, as they measure the breadth 
of the same area, therefore n equals / n. Now as from the above demon- 
stration k I is not unequal to n f, it follows that the area comprehended by 
k I and / n equals the area included by the lines n f and no. m 
Therefore the area k n equals the area n c. The area e c 
amounts to the number four which we added to the three 
roots, and it is evident that the two areas e and k n are 
equal in quantity to the one area e c, containing four in 
number. It is clear to us that the square g consists of the 
product of one-half of the roots, i.e. if, by itself, with the 
addition of the two areas e and k n. Thus the sum total 

• -i • i ^ IG- I 4- — From 

of this square is 6j and the root of it is given by 2\ , which the Vienna 
is contained by the side d g. We have left of the side of manuscri P t - 1 
the first square, which is the area a b, representing x 2 , the half of the roots 
amounting to if which is the side g a. The line d g is a side of the square 
g 0, containing the product of one-half of the roots by itself with the addi- 
tion of four, as we have said. This total amounts to 6j. We add the root 
of this, 2\ , to the line g a which is one-half of the three roots, amounting 
to i|. Hence this sum total reaches four, which is the line a d. This is 
the root of x 2 and also further of the square a c. The whole of x 2 is finally 
16. This is what we desired to explain. 

We have now explained these things concisely by geometry in order that 
what is necessary for an understanding of this branch of study might be 
made easier. The things which with some difficulty are conceived of by the 
eye of the mind are made clear by geometrical figures. 2 

Positives and Negatives 

And I found, says Mohammed Al-Khowarizmi, that all problems of resto- 
ration and opposition are included in the six chapters which we have set 
forth in the beginning of this book. 3 

1 The writer of the Vienna manuscript took no pains to make the proportions of his figures 
correct. Thus in this figure a b d g is intended to represent a square, while e is supposed to be 
the middle point of the line h g. Further e c is to be a square and likewise e m, and also the 
rectangle n t is intended to be equal to the rectangle z m. Moreover the proof in the Vienna 
manuscript is not consistent with the lettering of the figure, showing that the copyist did not 
succeed in following closely the argument of the text. Similarly the figures in the Boncompagni 
version do not have the correct proportions. 

For the proof based directly on Euclid II. 6, see page 133. The Boncompagni version and 
Abu Kamil make explicit reference to the propositions of Euclid. 

2 This paragraph is not found in either the Libri or the Arabic versions ; nor in the Boncom- 
pagni text. 

3 The evident meaning of this passage is that all problems leading to equations of the first or 
second degree can be solved by the methods set forth in the preceding text. 



9 o LIBER ALGEBRAE ET ALMUCABOLA 

i oppositionis in sex capitibus, quae in principio huius libri praemisimus, contineri. 

Nunc porro, quomodo res vel radices, quando vel solae vel cum illis numeri fuerint, 

aut quando ex eis numeri extracti, seu cum ipsae ex numeris extractae fuerint, ad 

inuicem multiplicentur, vei quomodo ad inuicem iungantur, vel ex aliis dimi- 

5 nuantur, deinceps dicendum est. 

In primis ergo sciendum est, quod numerus cum numero multiplicari non possit, 
nisi cum numerus multiplicandus toties sumatur, quoties in numero cum quo 
ipse multiplicatur, vnitas reperitur. Cum ergo nodi numerorum et cum illis 
aliquot vnitates propositae fuerint, aut si vnitates ab illis subtractae fuerint, tunc 
io multiplicatio quater repetenda erit; hoc est nodi primo cum nodis, vnitates deinde 
cum nodis, et nodi cum vnitatibus, ac tandem vnitates cum vnitatibus multipli- 
candae erunt. Cum itaque vnitates quae cum nodis pronunciantur, omnes 
adiectae siue omnes diminutae fuerint, quarta multiplicatio erit addenda. Quod 
si quaedam earum fuerint adiectae, quaedam vero diminutae ; quarta multipli- 
15 catio erit minuenda. 

Similitudo talis est, 10 et duo cum 10 et vno multiplicanda sunt. Multiplica 
ergo 10 cum 10, et producuntur 100 ; deinde 2 cum 10, et procreantur 20 addenda. 
Similiter 10 cum vno, et procreantur 10 addenda, et duo cum vnitate, et produ- 
cuntur 2 addenda. Tota igitur huius multiplicationis summa in 132 terminatur. 
20 Et hoc est quod diximus, quando vnitates quae cum nodis pronunciantur, omnes 
fuerint adiectae. 

At quando 10 sine 2 cum 10 sine vno multiplicare volueris, dicas 10 cum 10 

generant 100, et duo diminuta cum 10 procreant 20 diminuenda. Item 10 cum 

vno procreant 10 diminuta. Hoc autem totum 70 complectitur. Sed duo di- 

25 minuta cum vno diminuto, duo procreant addenda. Tota ergo haec summa in 72 

terminatur. Et hoc est quod diximus, quando omnes diminutae fuerint. 

Si autem 10 et 2 cum 10 sine vno multiplicare volueris : dicas 10 cum 10 100, 

1. capitulo D. 

2. Nunc ergo dicendum (addendum D) est. 
id est radices quando sole fuerint uel quando cum. 

3. uel pro aut V. numeros extraxerint. 
quando pro cum. 

4. et qualiter adinuicem iungantur, et qualiter producunt 132 
quidam (qui D) ex aliis diminuantur pro vel ' . . . 20. qui. 

est. 22. 10 sine uno in 10 sine uno (+ in 10 sine 

6. est om. V. quam. in numerum. uno V). 

7. cum : om. duplicandusD. tociens dupli- 23-25. et unum diminutum in 10 deductum, 
cetur quociens. in {om. D) quern. 10 procreant diminutiua. Item unum diminu- 

8. multiplicatio D. modi D saepins ; C add. : turn (diminutiuum D) in 10 deductum, 10 generat 
seu incerta et obscura cognitio. diminutiua. Hoc ergo totum 80 (81, et 80 C 

10-12. id est nodos in (+ prima D) nodis et uni- sed del.) amplectitur. Sed et unum diminutum 
tates in nodis (om. D) et iterum unitates in nodis in uno diminutiuo unum procreant adiectiuum. 
et unitates cum unitatibus. Cum ergo. 26. C add.: vt sequitur, 

13. adiectiuus pro adiectus siue addendus el 10 sine 2 
saepius. su» pro siue D. diminutiuus pro cum 10 sine 1 multipli. 
diminutus siue diminuendus et saepius. 

14. alie V ; alia D pro quaedam 2 . multiplico 
V ; multiplicato D. 

16-19. Dicam ergo 10 in 10, 100 procreant, et 
unum in 10, 10 generat adiectiua et 2 in 10, 20 pro- 
creant adiectiua, et duo in 1 duo generant adiectiua. 



19. 


132 creantur 


D. 




C add. : vt 


sequens 


habet calculus. 


10 


et 


2 








cum 


10 
100 


et 


1 

:o 
10 


multiplicata 
1 





100 mi. 


20 


mi. 


10 


Summa prod. 


70 


plus 


1 




7i 


cas pro Dicas D. 


fiunt 100 V 



THE BOOK OF ALGEBRA AND ALMUCABOLA 91 

Now further the method is to be explained by which you multiply un- 
known quantities or roots, either when alone, or when numbers are joined to 
them or subtracted from them, or when they are subtracted from numbers ; 
also in what manner they are added to, or, in turn, subtracted from, each 
other. 

In the first place you should understand that the only way to multiply a 
number by a number is to take the number to be multiplied as many times 
as there are units in the number by which it is to be multiplied. 1 When 
therefore the nodes 2 of numbers are proposed either with some units or if 
units are subtracted from them, then the multiplication is fourfold, i.e., first 
the nodes are multiplied by the nodes, then the units by the nodes and the 
nodes by the units, and finally the units by the units. When therefore 
the units which accompany the nodes are both added or both subtracted 
the fourth product is to be added. But if one is added and the other sub- 
tracted then the fourth product is to be subtracted. 

A problem of this kind is given by the following : 10 and 2 are to be multi- 
plied by 10 and 1. Hence multiply 10 by 10, giving 100; then 2 by 10, 
giving 20 to be added ; likewise 10 by 1, giving 10 to be added. Two by 1 
gives 2 to be added. The sum total of this multiplication is finally 132. 3 
And this illustrates what we have said in respect to the type in which the 
units which accompany the nodes are both to be added. 4 

But when you wish to multiply 10 less 2 by 10 less 1, you say 10 by 10 
gives 100; 2 to be subtracted by 10 gives 20 to be subtracted; also 10 
by 1 gives 10 to be subtracted. This total, then, amounts to 70. But 
negative 2 multiplied by negative one gives positive 2. Therefore the sum 
total is finally J2. 5 This illustrates what we have said when both (bino- 
mials) involve negatives. 

Moreover if you wish to multiply 10 and 2 by 10 less 1, you say 10 by 

1 To this definition Al-Khowarizmi refers in his arithmetic (Tratiati, I, p. 10). 

2 Scheybl adds that this is a word of uncertain and obscure meaning. The Arabic word 'nqtid 
is connected with the verb meaning "to knot," referring to tying knots on a string to indicate 
numbers. The Libri and Boncompagni texts use articuli, while Rosen translates' greater numbers.' 
See also F. Woepcke, in Journal Asiatique, Vol. I (6), 1863, p. 276. 

3 The Columbia manuscript continues with the following addition by Scheybl : 

The calculation is as follows : 10 + 2 

10 + 1 

100 20 

10 1 [written by mistake for 2] 

132 

4 This is one of the early attempts at a discussion of the multiplication of binominals, including 
(x + a) times (x + b), (x -a) times (x - b), and (.v - a) times (x + b). 

5 Scheybl's text contains by error 81 for 72. The problem as given in the Arabic and Libri 
versions is (10- 1) by (10— 1) with the product 81. This undoubtedly was given by Robert 
of Chester, and is so recorded in the Vienna and Dresden MSS. Scheybl evidently varied from 
the text before him, but' neglected to make necessary changes in the numerical computation. 



9 2 



LIBER ALGEBRAE ET ALMUCABOLA 



i et 2 adiecta cum 10 multiplicata 20 generant addenda. Item 10 cum vno diminuto 
multiplicata, 10 procreant diminuenda. Haec autem summa vsque ad 100 et 10 
protenditur. Sed 2 adiecta cum vno diminuto multiplicata, 2 procreant di- 
minuenda. Vnde tota multiplicationis summa ad 108 extenditur. Et hoc est 
s quod etiam diximus, quando quaedam earum fuerint adiectae quaedam vero 
diminutae. 

Similiter in fractionibus, si dicas : drachma et eius sexta cum drachma et eius 

sexta. Dicas, drachma cum drachma drachma[m], et drachmae sexta cum 

drachma drachmae sextam procreat. Item sexta cum drachma sextam procreat, 

10 et sexta cum sexta sextam 'sextae, id est tricesimam sextam drachmae procreat. 

Erit autem hoc tctum, drachma ^ et ^ drachmae. 

Si eodem modo drachmafm] sine sexta cum drachma sine sexta multiplicares, 
tantum net quantum si f cum suo aequali multiplicares. Vnde et haec multi- 
plicatio ad 25 partes ex tricesimis sextis partibus vnius drachmae extendetur, id 
is ad f et I sextae. 

Modus autem multiplicationis est, vt drachmam cum drachma multiplices, 
et producetur drachma; deinde sine sexta cum drachma, sextam procreat dimi- 
nuendam ; item drachmam cum sine sexta, et producetur vna sexta diminuenda. 
Duae .igitur tertiae vnius drachmae supersunt. Et sine sexta cum sine sexta, 
20 sextam sextae generat addendam. Tota igitur haec summa ad § et sextam 
sextae extenditur. 



1. et (om. V) unum diminutum in 10 multiplica- 
tum, 10 generat diminutiua, 2 quoque (ergo D) 
adiectiua in 10 deducta, 20 procreant adiectiua; 
duo . . . adiectiua 2 in marg. D man. 2. 

3. 10 pro 2 2 C. Hec igitur. 

4. Vnde + et. 180. C add. Haec autem se- 
quenti calculo patent : 10 et 2 

10 sine 1 





100 


plus 
mi. 


20 
10 








Summa 


pro. 


no 








Minus 


2 






Et om.; Est C. 






108 






6. diminutiue 


et add. 


Hoc 


igitur 


modo 


res 


inuicem multiplicantur. 


Quando 


cum 


ipsis 



numeri fuerint (+ et quando ipse sine numeris 
fuerint. Et quando numeri sine ipsis propositi 
fuerint V). Et si tibi propositum fuerit 10 sine re 
(+ in 10 deducta quantum constituunt. Dicas 
10 in 10, 100 et sine re V; + et expositio rei est 
re in 10 deducta. Dicas 10 in 10, 100 et sine re 
in marg. D man. 2) in 10, 10 generat (generant D) 
radices diminutiuas. Dicas ergo quod tota hec 
summa usque ad 100 extenditur 10 rebus abiectis. 
Et si dixerit 10 et res in 10 deducta quantum pro- 
creant. Dicas 10 in 10 centum, et res adiectiua 
decies deducta 10 res generat adiectiuas. Tota 
igitur (ergo D) hec summa ad 100 et 10 res exten- 
ditur. Si autem dixerit 10 et res in suo consimili 
quantum multiplicata faciunt. Dicas 10 in 10, 
100 procreant (+ et 10 in re 10 res procreant V; 
+ res D). Item 10 in re, 10 res procreant et res 



in re substantiam generat adiectiuam. Ergo 
tota hec summa ad 100 ex numero et 20 res et 
substantiam extenditur adiectiuam. Si autem 
sic (si D) proponat, 10 sine re in 10 sine re quantum 
faciunt. Dicas 10 in 10, 100 et sine re in 10, 
res procreant 10 (+ diminutiuas et sine re in 10, 
10 similiter diminutiuas res procreant V) et sine 
re in (+ x D) sine re substantiam generat (generant 
D) adiectiuam. Erit ergo hoc totum 100 et 
substantia 20 rebus abiectis ; Vide infra pag. 94. 

7. Eodem modo si dragma (dragmam D) et 
eius sextam (+ in dragma et eius 6 tam V) duxeris, 
quantum net. 

8. et om. dragma in 6 tam , sextam dragmatis. 

9. forma pro sexta D. 

10. et pro cum D. Jg (xxxvi D) dragmatis 
generat adiectiuam. Erit ergo hoc totum dragma 
et i vel § (+ dragma D) et sextam (due D) 6 te . 

12. duxeris. 16. huius pro autem. ut dragma. 

17. fietque. et pro deinde. in dragma + 
multiplicatum. 

18. et sine 6 ta (om. D) in dragma, sextam pro- 
creat similiter diminutiuam. 

19. vnius om. in sine sexta + deductum. 
21. C add.: Sequitur calculus. 

drachma sine sexta 
drach. sine sex. 



drach. 



sine & 
sine I 



manent § . Accedit I de J. 
Vnde multiplicationis productum tandem ad § plus 
A sese extendat. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 93 

10, 100, and positive 2 multiplied by 10 gives positive 20. Also 10 multi- 
plied by negative 1 gives negative 10. This sum, moreover, amounts to 
no. But positive 2 multiplied by negative 1 gives negative 2. Whence 
the sum total of this multiplication equals 108. 1 And this illustrates the 
type of process when units are to be added and others to be subtracted. 2 

Likewise in the case of fractions, if the problem is a unit and one-sixth 
(to be multiplied) by a unit and one-sixth. You say, unit by unit gives 
unit ; and one-sixth of a unit by a unit gives one-sixth of a unit. Also one- 
sixth by a unit gives one-sixth and one-sixth by a sixth gives one-sixth of a 
sixth, i.e. one thirty-sixth. The total will be a unit and ^ and ^ of a unit. 

In the same manner, if you multiply a unit less one-sixth by a unit less 
one-sixth, the product will be the same as § multiplied by its equal. Whence 
this product equals 25 thirty-sixths of one unit, i.e. § and ^ of one-sixth. 

Now the method of this multiplication is that you multiply unit by unit, 
giving unit; then negative one-sixth by unit, giving negative one-sixth; 
then you multiply a unit by negative one-sixth, giving one-sixth negative. 
Therefore two-thirds of one unit remain. And negative one-sixth multi- 
plied by negative one-sixth produces one-sixth of one-sixth positive. The 
sum total therefore amounts to § and one-sixth of one-sixth. 3 

1 Scheybl adds : Moreover this is evident by the following calculation : 

10+2 
10 — 1 



100 + 20 
— 10 



no 
— 2 
108 



Similarly to the example which immediately precedes this Scheybl adds : as follows : 



10 


— 2 


10 


— I 


100 


— 20 




— IO 



70 

+ I [2] 

71 [72] 

2 Evidently considerable interchange of text was made at this point by Scheybl. The passage 
inserted in the footnote to line 6 should be compared with the 28 lines of Scheybl's text on 
page 94, which are not found at that point in the Dresden and Vienna manuscripts. 

3 Scheybl adds : The calculation follows : 

unit — 5 

unit — I 

unit — \ 
1 



giving f 



Add § of I 
Whence the product of this multiplication finally amounts to § plus £$. 



94 LIBER ALGEBRAE ET ALMUCABOLA 

i Sequuntur nunc similes nodorum multiplicationes, per res seu radices et numeros 
expositae. Eodem modo res inter se multiplicantur, quando cum ipsis numeri_ 
vel ipsae sine numeris fuerint, vel quando sine ipsis numeri propositi fuerint. 
Dicendo multiplicetur i res et 10 cum i re et 10, dicas igitur res cum re substan- 
s tiam, et res cum 10 multiplicata 10 res generat. Item 10 cum re 10 res, et 10 cum 
10 multiplicata ioo generant. Erit ergo totum i substantia 20 res et 100 ex 
numeris. 

Et si dicas, multiplicetur 1 res cum 1 re : dicas, res cum re multiplicata pro- 
ducit substantiam. Atque tantum quidem est multiplicationis productum. 

10 Similiter res sine 10 cum re sine 10 : die, res cum re substantiam producit ; 
sine 10 vero cum re 10 res producit diminuendas. Item res cum re 10 res diminu- 
endas ; sine 10 vero cum sine 10 100 ex numeris addendas procreant. Vnde 
totum multiplicationis productum ad 1 substantiam sine 20 radicibus, additis 
vero ex numeris 100, sese extendit. 

is Vel etiam si dicas, 10 cum 10 ; item 10 sine re cum 10 sine re : die 10 cum 10 
multiplicata procreant 100. Atque tantum est productum multiplicationis 
prions. Die deinde 10 cum 10 centum addenda ; sine re vero cum 10, 10 res 
generat diminuendas. Item die 10 cum sine re 10 res diminuendas ; sine re vero 
cum sine re, substantiam procreat addendam. Vnde totum multiplicationis 

20 posterioris productum ad 100 absque 20 rebus, vna substantia vero adiecta, sese 
extendit. 

Si autem quaesieris 10 et res cum suo aequali multiplicata, quantum producunt ? 
Die 10 cum 10, 100 ; et res cum 10, 10 res procreat. Item 10 cum re, 10 res ; et 
res cum re, substantiam generat. Tota autem haec multiplicatio ad 100 ex 

25 numero, 20 res et vnam substantiam sese extendet. 

Quod si sic quaesieris decern sine re cum 10, vel decern et res cum 10 : producet 
multiplicatio prior 100 ex numeris absque 10 rebus, posterior vero 100 ex numeris 
et 10 res. 

Si autem dixerit aliquis, decern sine re cum 10 et re multiplicata quantum 

3ofaciunt? Dicas 10 cum 10, 100 drachmas; et sine re cum 10, 10 res procreat 
diminuendas. Item 10 cum re, res 10 generant addendas ; et sine re cum re, 
substantiam procreat diminuendam. Hoc ergo totum ad 100 drachmas proueniet, 
vna substantia abiecta. 

Et si dixerit, decern sine re cum re : die 10 cum re, 10 res procreant, et sine re 

35 cum re, substantiam generat diminuendam. Hoc ergo ad 10 res perueniet abiecta 
substantia. 



1-28. om. Vide pag. 92, n. 6. 






29. 


aliquis om. 


7. C add.: vt sequitur, 






3°- 


Ducas D. drachmas om. 


1 res et 10 






31- 


et res in 10, 10 res. generat. 


cum 1 re. et 10 






32. 


procreant D. Hoc om. D. perueniet 


1 sub. et 10 res 






33- 


vna om. D. C add. : Sequitur calculus. 


10 res et 100 


N 




10 sine re. 


Summa pro. 1 sub. 20 res 100 


ex 


nu. 




cum 10 et re. 


14. C add.: Sequitur calculus. 








100 sine 10 reb. 


1 res sine 10 








10 res sine substantia 



100 sine substantia 



1 sub. sine 10 re. 34-36. Et . . . substantia in marg. D man. 2. 

sine 10 re. plus 100 34. dicas pro die saepe. 

1 sub. sine 10(20) res plus 100 35. perueniat. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 95 

Similar multiplications of nodes, 1 illustrated by things or roots and 
numbers, follow. In the same manner the unknowns 2 are multiplied by 
themselves, either when numbers added to them, or when numbers are sub- 
tracted from them, or when they are to be subtracted from numbers. For 
example, to multiply x + 10 by x + 10, you proceed thus : x by x, x 2 , and x 
multiplied by 10 gives 10 x ; also 10 by x, 10 x, and 10 multiplied by 10 gives 
100. The sum total is then x 2 , 20 x, and ioo. 3 

Another example : multiply x by x. You say that x by x gives x 2 , and 
this is the product of the multiplication. 

Similarly, x — 10 by x — 10 : xbyx gives x 2 and negative iobyx gives 
negative 10 x. Also negative 4 10 by x gives 10 x negative, and negative 10 
by negative 10 gives positive 100. Whence the total product of this multi- 
plication amounts to x 2 less 20 x, with 100 to be added. 3 

Or also if you multiply 10 by 10, and again 10 — x by 10 — x : 10 multi- 
plied by 10 gives 100 ; so much is the product of the first multiplication. 
Then, 10 multiplied by 10 gives positive 100 ; negative x by 10, 10 x nega- 
tive ; also 10 by negative x gives negative iox; negative x by negative x 
gives positive x 2 . Whence the total product of the second multiplication 
extends to 100 less 20 x, with x 2 to be added. 

Again, if you try to find the product of 10 + x multiplied by its equal you 
proceed thus : 10 by 10, 100 ; and x by 10 gives 10 x ; also 10 by x, 10 x, and 
xbyx gives x 2 . The total product amounts to 100, 20X and x 2 . 

Now if you try to find the product of either 10 — x by 10 or 10 +xby 
10, the first product is 100 — iox and the other 100 + 10 x. 

Further some one may ask, how much is the product of 10 — x by 10 + x ? 
You proceed thus : 10 by 10, 100 units, and negative x by 10 gives negative 
iox; also 10 by x gives 10 x positive, and negative x by x gives negative x 2 . 
This total then equals 100 units less x 2 . 3 

Another problem : 10 — x by x. You proceed thus : 10 by x gives iox, 
and negative x by x gives x 2 to be subtracted. This then equals 10 x less x 2 . 

1 See p. 91, footnote 2. 

2 Res (literally ' thing ') is used in such a technical sense that it seems better to translate 
by 'unknown,' as in this instance, or by x as in much of the following work. 

3 Scheybl adds to this, and to two problems below, the following calculation forms which ap- 
proach the modern symbolism for the product of binomials. Scheybl prefaces with the words, 
' as follows,' or ' the calculation follows ' : 

x + 10 x — 10 10 — x 

X + 10 X — 10 10 + X 

X 2 + IOX X 2 — IOX IOO — 10 X 

10 x + 100 TV — 10 .v + 100 + 10 x — x 2 



X-, 20 X, IOO X- — 20 X + IOO IOO — X 2 

Scheybl uses 100 A T here for the number 100, following the notation employed by him in his printed 
works on algebra as well as in the algebra text which is found in the same manuscript with this 
version of Robert of Chester. 

4 In the Latin text (1. n) Scheybl has res cum re instead of sine 10 cum re. 



96 LIBER ALGEBRAE ET ALMUCABOLA 

i Si autem dixerit, decern et res cum re sine 10, quantum procreant? Die 10 

cum re multiplicata, 10 res generant ; et res cum re, substantiam generat adden- 

dam. Item 10 cum sine 10, ioo drachmas procreant subtrahendas ; et res cum 

sine 10 multiplicata res 10 generat diminuendas. Dicas ergo quod haec tota 

s summa vsque ad vnam substantiam, ioo drachmis abiectis, sese extendat. 

Si autem quis dixerit, decern drachmae et rei medietas cum medietate drachmae, 
quinque rebus abiectis, multiplicatae, quantum procreant ? Dicas, decern cum 
medietate drachmae multiplicata, 5 drachmas procreant, et medietas rei cum 
medietate drachmae, quartam rei procreat addendam. Item 10 cum sine 5 rebus 

10 multiplicata, 50 procreant res diminuendas. Vnde tota haec multiplicationis 
summa ad 5 drachmas, 49 rebus et tribus quartis vnius rei abiectis excrescet. 
Postea medietate rei cum sine 5 rebus multiplicata, duae substantiae et media 
producentur diminuendae. Tota igitur multiplicationis summa ad 5 drachmas, 
duabus substantiis et media nee non etiam 49 rebus et tribus rei quartis abiectis, 

15 excrescet. 

Et si dixerit, decern et res cum re absque 10 multiplicata, quantum faciunt ? 
Est quasi diceres, res et 10 cum re sine 10. Vnde sic respondeas : res cum re 
multiplicata, substantiam generat ; et 10 cum re, 10 radices generant addendas. 
Item res cum sine 10 multiplicata, 10 res procreat diminuendas. Vnde 10 res 

20 adiectae et 10 res diminutae seu ablatae, cum prius tantum tribuat quantum pos- 
terius aufert, negligunt ; et relinquitur substantia sola. Porro 10 cum sine 10, 
100 drachmas generant ex omni substantia diminuendas. Tota igitur haec multi- 
plicatio ad substantiam, 100 drachmis abiectis, extenditur. 

CAPUT ADIECTIONIS ET DIMINUTIONIS 

25 Sciendum est, quod omnis radix substantiae propositae est ignota ; duplicatur 
etiam et triplicatur et caet., atque ex ipsius duplicatione et triplicatione cum sua 
substantia talis nascitur numerus cuius videlicet vna radix duabus siue tribus 
radicibus suae substantiae aequiparatur. Quod totum euenire videtur iuxta 
multiplicationem numeri supra vnitatem naturaliter dispositi. Nam si radices 

1. Dicas res in 10. 13. diminutiua corr. in diminuendae C. 

2. generat. res in re + ducta (deducta D). 17. et pro Est D. diceret. sine 10 . . . 

3. et sine 10 in 10 + deducta. diminutiua. multiplicata in marg. D man. 2. 

sine (cum D) 10 in re. 18. re + multiplicata. 10 res. adiectiuas. 

4. substantia absque 100, illi (id D) cum quo 19. et sine re (x D) in re. diminutiuas. 
opposuisti equatur (coequatur D). Quod id circo 19-22. unde (+ re in 100 superscr. D man. 2) 
contingere videtur quia proiecisti 10 res diminutiuas adiectiua cum diminutiuis adnullantur (adnullatur 
cum 10 rebus (+ et D) adiectiuis. Unde eciam D) et remanet substantia, ac 10 in sine 10 100 ge- 
substantia absque 100 dragmatibus permansit nerantur ex omni substantia diminutiua. 

pro quod haec . . . extendat. 23. extenditur + Et quotquot (quidquid D) 

7-10. multiplicata, quantum procreant. Dicas fuerit in multiplicatione adiectum seu diminutum 

medietas dragmatis in 10 ducta dragmatibus, 5 semper in lance consimili apponitur id est si unum 

dragmata progenerat (progenerant D) et medietas fuerit adiectiuum alterum erit diminutiuum. 
dragmatis in rei medietate (medietatem D) deducta, 24. Titulum Radicum algorismus V ; om. D. 

quartam rei procreat (procreant D) adiectiue, et sine 25. quam V; quando D pro quod. seu V; 

10 rebus in 10 multiplicatum dragmatibus, 50 res siue D pro est 2 . ignote. 
procreat diminutiuas. Unde et. 26. aut pro etiam et. ut pro atque. 

11. 36 pro 49 bis. excrescit. 27. substantia + multiplicatione. nascatur. 

12. Postea multiplica medietatem dragmatis 28. seuV; siue D pro suae. equiparantur D. 
absque 5 rebus in medietatem rei adiectiue fientque 29. super sola unitate V ; super solam unitatem 
due substantie et medium diminutiue. D. Nam + et. radicem. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 97 

Yet another problem : how much is 10 + x by # — 10? 10 multiplied 
by x gives 10 x, and x by x gives positive x 2 ; also 10 by negative 10 gives 
100 units negative, and x multiplied by negative 10 gives negative 10 x. 
You can say, therefore, that this sum total amounts to x 2 less 100 units. 

If moreover some one asks what is the product of 10 units and one-half x 
multiplied by one-half a unit less 5 x, you proceed thus : 10 multiplied by 
one-half a unit gives 5 units and one-half of x by one-half a unit gives one- 
fourth x ; also 10 by negative 5 x gives negative 50 x. Whence the sum total 
of this multiplication amounts to 5 units, from which are to be subtracted 1 
49 x and f x. Then \ x multiplied by negative 5 x gives two and one-half 
x 2 negative. The sum total of the multiplication amounts to 5 units, with 
two and one-half x 2 , and 49 x and f x to be subtracted. 2 

Another problem : how much is 10 + x multiplied by x — 10 ? This 
is the same as x + 10 by x — 10. Whence you proceed in this manner : 
x multiplied by x gives x 2 , and 10 by x gives 10 positive roots ; also x multi- 
plied by negative 10 gives negative 10 x. Whence the 10 x to be added 
(positive) and the iox to be subtracted (negative), or taken away, cancel 
each other, since the first adds as much as the second takes away and x 2 
alone remains. Then 10 by negative 10 gives 100 units to be subtracted 
from x 2 . This total product therefore amounts to x 2 less 3 100 units. 

On Increasing and Diminishing 4 

The fact must be recognized that every root of any given square is 
unknown ; it is also doubled or tripled, etc. in such a way that by doubling 
and tripling it, by the multiplication of its square, a number is formed of 
which one root is equal to two or three roots of the given unknown square. 
All of this turns out to be like the multiplication of any number beyond 
unity, all in natural order. For if you wish to double the roots, you multiply 

1 Error made by Scheybl, who writes "added" instead of "subtracted." 

2 ( 10 + - j (| - 5 x) = 5 - 49! x - t.\ x 2 . 

3 Scheybl writes adiectis for abiectis. The problem is, of course, that x + 10 multiplied by 
x — 10 gives x 2 — 100. 

4 An algebraical work with this title is supposed to have been written by Al-Khowarizmi. 
The Libri and the Arabic versions follow with four problems which do not occur in Robert of 

Chester's translation. However, Scheybl takes up three of these problems in his additions, on 
pages 142-144 of this work. 

These problems, following Rosen, op. cii., p. 27, are as follows : " Know that the root of two 
hundred minus ten, added to twenty minus the root of two hundred, is just ten. The root of 
two hundred, minus ten, subtracted from twenty minus the root of two hundred, is thirty minus 
twice the root of two hundred; twice the root of two hundred is equal to the root of eight 
hundred. A hundred and a square minus twenty roots, added to fifty and ten roots minus two 
squares, is a hundred and fifty, minus a square and minus ten roots. A hundred and a square, 
minus twenty roots, diminished by fifty and ten roots minus two squares, is fifty dirhems and 
three squares minus thirty roots. I shall hereafter explain to you the reason of this by a figure, 
which will be annexed to this chapter." 



98 liber algebrae et almucabola 

i duplicare volueris, binarium cum binario multiplices et quod ex multiplicatione 
excreuerit, cum ipsius radicis substantia multiplices ; et is excrescet numerus, 
cuius vna radix duabus ipsius substantiae radicibus fiet aequalis. Quod si radicem 
triplicare volueris, ternarium cum ternario multiplices, et quod ex multiplicatione 
5 excreuerit, cum ipsius radicis substantia multiplices ; et is tibi nascetur numerus, 
cuius vna radix tribus radicibus primae substantiae aequiparatur. Si autem 
medietatem radicis habere volueris, oportet vt medietatem cum medietate, ac 
cum producto postea ipsam substantiam multiplices; et erit radix quae tollitur 
medietati radicis substantiae aequalis. Natura enim numeri hoc exigit, vt 

10 quemadmodum in numeris integris multiplicatur, ita etiam et in numeris dimi- 
nutis, hoc est in fractionibus. Eodem modo cum tertiis, cum quartis, atque omni 
eo quod ipso integro minus est, agendum erit. Similitudo autem multiplicationis 
huius talis est. 

Similitudo multiplicationis prima. Accipiamus exempli gratia radicem numeri 

is nouenarii, ac deinde multiplicemus. Quod si duplationem radicis numeri nouem 
habere volueris, dicas, bis duo procreant 4, que cum nouenario multiplicata, ad 
36 excrescet multitudo. Huius itaque multitudinis accipias radicem, id est 
numerum senarium, qui duabus radicibus numeri nouenarii, hoc est numeri 
ternarii duplo aequalis reperitur ; idem est enim numerum senarium semel 

20 accipere. 

Similitudo multiplicationis secunda. Si autem radicem numeri nouem tri- 
plicare volueris, ternarium cum ternario multiplices, et fient 9 ; quae si cum seip- 
sis, hoc est cum nouenario multiplicaueris, vsque ad 81 excrescet numerus, cuius 
vnam radicem nouenarius complet numerus, qui tribus radicibus nouenarii, hoc 

25 est numero nouem, videtur aequalis. 

Similitudo multiplicationis tertia. Sed si medietatem radicis saepe dicti 
numeri, habere volueris, medium cum medio multiplica, et fiet quarta, quam 
si cum 9 multiplicaueris, duo et quartam vnius perficies. Horum igitur 
radicem accipias, id est vnum et medium, quae medietatem radicis numeri 

30 nouenarii, hoc est medietatem numeri ternarii, adimplent. Nam vnum cum sui 

1. radicare pro duplicare D. binario in 18. senarium + accipe. unum pro numeri ' D. 
binario D. dicas V; ducas D pro multiplices. id est numero 3 ris . 

numerum ut (et D) pro et. ig. bis sumpto similis V; simul D pro duplo 

2. substantiam. talis pro is. aequalis. ternarium que est radix nouenarii, 
4. ducas (+ productum in V) substantiam ut bis accipere et numerum semel accipere senarium 

talis tibi nascatur pro multiplices . . . nascetur. V ; senarium semel accipere quod si numerum 

6. equiparantur D. ternarium que radix est numeri nouenarii, bis 

7. assumere volueris. et pro ac. accipere D pro senarium semel accipere. 

8. cum producto om. in pro ipsam. tol- 21. Similitudo multiplicationis secunda om 
litur + hoc est elicitur C. et sic infra. numeri nouem om. 

10. multiplicoD. diminutis + multiplicetur. 22. ducas numerum, et fiunt g que si (similiter D). 

n. hoc est in fractionibus om. Hoc ergo 23. hoc est cum nouenario om. cuius + vide- 

(quo- D) modo seu (siue D) in 3 is seu (siue D) licet. 

in 4 ls seu (siue D) in eo quod minus est. 24. qui scilicet tribus radicibus nouenarii numeri 

12-14. igitur huius multiplicacione prima talis videtur equalis (om. D). 

est ut pro autem . . . gratia. 26. Et pro Sed D. radicis om. V. saepe 

15. Et pro Quod D. duplicationem. nu- dicti numeri om. 

meri nouem habere om. 27. multiplicare pro habere; multiplicare C 

16. bis bini. ducta (ducitur D) + numero. sed del. fietque. quia pro quam. 

17. multiplicatio pro multitudo; multitudo 2g. quam pro quae. 

corr. ex multipli C. summe pro multitudinis. 30. medietatem om. numeri om. V. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 99 

2 by 2, and the product by the unknown square of the same root. The 
result will be a number of which one root will be equal to two roots of the 
given unknown square. 1 And if you wish to triple the root, you multiply 

3 and 3, and the product by the square of the root ; so you obtain a number 
of which one root is equal to three roots of the first square. 2 Moreover, if 
you wish to take one-half of a root, it is necessary to multiply one-half by 
one-half, and then the product by the square itself. The root which is 
taken will be one-half of the root of the given square. 3 Indeed the nature 
of numbers requires that just as integral numbers are multiplied so also are 
lesser numbers, i.e. fractions. You proceed then in the same manner with 
thirds, with fourths, and so with every number less than an integer ; illus- 
trations follow. 4 

First illustration 5 : Take the root of nine to be multiplied. If you wish 
to double the root of nine you proceed as follows : 2 by 2 gives 4, which you 
multiply by 9, giving 36. Take the root of this, i.e. 6, which is found to be 
two roots of nine, i.e. the double of three. For three, the root of nine, added 
to itself gives 6. 6 

Second illustration : If you wish to triple the root of 9, you multiply 3 
by 3, giving 9, which multiplied by itself, i.e. by 9, gives 81 . Of this number 
9 is the root, and this is seen to be equal to 3 roots of 9, i.e. 3 X3. 7 

Third illustration : If, however, you wish to take one-half of the root, 
multiply one-half by one-half, giving \, which when multiplied by 9 will 
give 2j. Take the root of this, i.e. i|, which is one-half of the root of 9, 

1 2 Vx = V2 2 • .V. 



2 1 Vr = V?2 



3 >.v = » 3- • X. 
31-%/ «/"J i " _ IX 

2 v X — 2 ' 2 ' *^ — *\ / _ ' 

\4 

4 This section begins in the Arabic with the four problems which we have given in footnote 
4 on the preceding page of the translation. The part which corresponds to this paragraph, 
following Rosen, pp. 27-28, is as follows: 'If you require to double the root of any known or 
unknown square (the meaning of its duplication being that you multiply it by two), then it 
will suffice to multiply two by two, and then by the square ; the root of the product is equal to 
twice the root of the original square. 

If you require to take it thrice, you multiply three by three, and then by the square; the 
root of the product is thrice the root of the original square. 

Compute in this manner every multiplication of the roots whether the multiplication be 
more or less than two. 

If you require to find the moiety of the root of the square, you need only multiply a half by 
a half, which is a quarter; and then this by the square: the root of the product will be half 
the root of the first square. 

Follow the same rule when you seek for a third, or a quarter of a root, or any larger or 
smaller quota of it, whatever may be the denominator or the numerator. Examples of this ..." 

5 In translating I omit some words added by Scheybl. 

6 2 Vg = V 4 . g = 6. The problems may appear trivial, but the reader should note that this 
is the first approach to an algebraic treatment in systematic form of surd quantities. Al-Kho- 
warizmi proceeds admirably from known to unknown. 

7 3 ^9 = ^3 ' 3 ' 9 = 9- 



IOO 



LIBER ALGEBRAE ET ALMUCABOLA 



i ipsius medio bis acceptum ternarium complet numerum. Secundum ergo hunc 
modum in huius modi multiplicationibus cum omnibus radicibus, quotquot 
integrae vel fractae fuerint, agendum erit. 

Modus diuidendi 

5 Si autem radicem numeri nouenarii in radicem quaternarii diuidere volueris. 
Diuide 9 in 4 et exeunt 2j, quorum radix, quae in vno et dimidio terminatur. 
vnam complet particulam. 

Si autem e contrario diuidere. volueris, id est radicem numeri quaternarii in 
radicem numeri nouenarii, diuide 4 in 9, et exeunt quatuor nonae vnius ; harum 
10 radicem, id est duas tercias, accipe. Et hoc est quod vni particulae scilicet con- 
tingit. 

Alius diuisionis modus 

Quod si radicem numeri nouenarii in radicem numeri quaternarii diuidere vo- 
lueris, ita tamen vt substantia in substantiam non diuidatur, radicem numeri 

15 nouenarii, quoties volueris, duplica vel collige, et scias cuius numeri 
numerus ex collectione proueniens radix habeatur. Hunc ergo modum numeri 
in 4, aut in alium numerum, in quern radicem primam diuidere voluisti diuide. 
Nam eius radix vni eueniet. Iuxta ergo hunc modum si tres radices vel quatuor, 
seu pauciores, seu medietatem radicis, aut minus, aut quotquot fuerint, numeri 

20 nouenarii diuidere volueris, cum omnibus iis agendum erit. Operare ergo secun- 
dum modum quern proposuimus, et rem ita se habere inuenies, si deus voluerit. 

Sequuntur nunc similes multiplicationes, per res seu radices et numeros expositae 

Sed si radicem numeri nouenarii cum radice numeri quaternarii multiplicare 
volueris, multiplicemus 9 cum 4 et producentur 36. Sume ergo horum radicem, 
25 id est 6. Et hoc est quod producitur ex radice numeri nouenarii cum radice numeri 
quaternarii multiplicata. Eodem modo si multiplicare volueris radicem numeri 
quinarii cum radice numeri denarii. Multiplica ergo 5 cum 10, et producentur 50, 
quorum radix substantiam, hoc est ipsum quod voluisti, significat. 



1. mediabis pro medio bis D. 

2. modi om. D. multiplicatione quotquot 
radices seu {om. D) integre seu diminute fuerint. 

4. Titulum om, et sic infra. 

5. radices pro radicem ' D. numeri om. V. 
numeri quaternarii D. diuide per vel diuide super 
pro diuide in saepe. 

6. Deinde pro Diuide D. fientque duo et 4 a . 
quarum D. quae + scilicet radix. 

8. econuerso. super pro in. 

o. numeri om. D. fientque. nouene D. 
tunc earum radicem assume id est duas unius accipe 
33s pro vnius . . . accipe. 

10. particulae scilicet om. 

13. super. 

14. tamen quod substantiam super. 

15. quotiens. duplica V ; multiplica duplica 
D. vel collige om. scito. 



16. ex duplicatione concretus radix habeatur. 
Per hunc. 

17. primo V; primum D. 

18. Nam + et. ergo om. D. aut plures 
pro vel quatuor. 

19. aut quotquot fuerint, numeri nouenarii om. 
V ; aut quod volueris, numeri nouenarii D ; quot- 
quot + volueris C sed del. 

20. cum omnibus iis om. est. OperacioD. 
secundum quod diximus et inuenies si deus voluerit. 

23. Sed + et. radicem om. D. numeri 2 



24. Multiplica. 

25. quod excreuit. 

26. multiplica D. 

27. numeri om. 



fientque 36. 



numeri om. D. 
senarii D. ergo om. 
28. quorum videlicet radix substantiam quam 
voluisti signat. 




THE BOOK OF ALGEBRA AND ALMUCABOLA 101 

i.e. one-half of 3. For i| taken twice gives 3. 1 You proceed then in the 
same manner with such multiplications with all roots, whether they are 
integral or fractional. 

Method of dividing 

If, moreover, you wish to divide the root of 9 by the root of 4, you divide 
9 by 4, giving 2j, and the root of this, which is finally i|, completes the 
division. 2 

If you wish to perform the reciprocal division, i.e. divide the root of 4 
by the root of 9, divide 4 by 9, giving f of a unit, and take the root of this, 
i.e. § . This is, of course, the result of the division. 3 

Another method of division 

You may desire to divide the root of 9 by the root of 4, without dividing 
the square by the square. 4 Double or gather up the root of 9 as many times 
as desired and of the resulting number you find the root. This number di- 
vide by 4 or by any other number by which you wished to divide the first 
root ; for in this way the root of it will be found. You proceed then in like 
manner if you wish to divide the root of 9 by 3 or 4 or less, or by \ or less, 
or by anything else ; with all of these the rule is the same. Follow then 
the rule which we have explained and so you will find the result, if God will. 

Similar multiplications explained by things, or roots, and numbers 

Now if you wish to multiply the root of 9 by the root of 4, multiply 9 by 4, 
giving 36. Take the root of this, i.e. 6. This is the product of the root of 
9 multiplied by the root of 4. Likewise if you wish to multiply the root of 
5 by the root of 10, you multiply 5 by 10 giving 50. 5 The root of this is 
the desired product. 

1 1 Vr> — V9 — 3 

2 v 9 - V ? ~ 2- 

3 "^4 _ -J* - 2 

, - V ? - 3- 

V 9 

4 There seems to be something incorrect about Robert's translation. Probably this should be 
as in the Arabic, according to Rosen, op. cit., p. 30 : " If you wish to divide twice the root of 9 by 
the root of 4, or of any other square, you double the root of nine in the manner above shown 
to you in the chapter on Multiplication, and you divide the product by four, or by any number 
whatever. You perform this in the way pointed out. 

In like manner, if you wish to divide three roots of nine, or more, or one-half or any mul- 
tiple or sub-multiple of the root of nine, the rule is always the same: follow it, the result 
will be right." 

Rosen here follows the custom of modern translators of Arabic in leaving out the reference to 
the Deity which is actually given in the Arabic text. 

5 V5 . Vio= V50, both quantities being surds. 



102 LIBER ALGEBRAE ET ALMUCABOLA 

i Quod si radicem tertiae cum radice medietatis multiplicare volueris, tertiam 
cum medietate multiplica, et producitur vna sexta. Radix igitur sextae ipsum 
est quod ex radice tertiae cum radice medietatis multiplicata, nobis excreuit. 
Si autem duas radices numeri nouenarii, cum tribus radicibus numeri quater- 
s narii multiplicare volueris, accipe duas radices numeri nouenarii secundum 
quod iam diximus, vt scias cuius substantiae radicem compleant. Similiter de 
tribus radicibus numeri quaternarii facias, quatenus cuius substantiae sint radix 
reperias. Has igitur substantias inter se multiplica, vnam videlicet earum cum 
altera, atque huius producti .radicem accipe, quoniam hoc est quod ex duabus 
io radicibus numeri nouenarii cum tribus radicibus quaternarii multiplicatis excreuit. 
Quotcunque igitur radices simul colligere, vel quas a quibusdam minuere vo- 
lueris, cum errore abiecto iuxta hoc exemplar multiplicare poteris. 

INCIPIUNT CAPITUM QUAESTIONES 

Dixit Mahomet Algoarizim, hactenus praemisimus numerorum capita, quae sub 
is sex quaes tionibus pro numero capitum in libri principio a nobis proposita sunt 
atque ibi etiam diximus, numerum restaurationis et oppositionis in his sex ca- 
pitibus omnino versari. Sed quoniam ea quasi sub inuolucris te edicta sunt, 
igitur haec ipsa, quo omnium studium exerceatur, et scientia facilius elucescat, 
adducemus ac fusius explicamus. 

20 Caput primum, quaestio prima 

Modus huius quaestionis est, vt dicas : denarium numerum in duo diuide vt 
eius vna pars cum altera multiplicata, numerum ex multiplicatione concreet seu 
producat, qui quater acceptus aequalis fuit numero, ex multiplicatione vnius 
partis semel cum se ipsa genera to. 
25 Similitudo talis est, vt vnam partem numeri denarii rem constituas, et alteram 
10 sine re. Multiplica igitur rem cum 10 sine re, fient 10 res absque substantia. 
Item multiplica 10 res absque substantia cum quatuor, quoniam quater dixisti, 

2. fietque b (6 a D). igitur om. D. ipsa. tegumento sunt dicta, hie illud in quo sciencia et 

4. cum tribus radicibus numeri quaternarii om.D. animi studium faciliori exerceantur aditu elucescet 

5. extrahe. introducimus. C add.: Quid si quaestio capitis 

7. fuit D. primi? te pro tibi C. 

8. inuicem pro inter se. et unam (id est una 20. Titulum om. et sic ubiqae. 

D) earum in alteram ducas, et huius summe radicem 21. In marg. C Textus, et sic ubique in has 

accipias. quaes t. Modus huius capituli. decenum pro 

9- erit. denarium fere ubique. ita in duo et sic saepe. 

10. radicibus om. numeri 4^. diuido. 

n. Quotquot. in simul colligere volueris. 22. diuisio pro pars vel pars diuisionis saepius. 

inuicem sed del. et minuere superscr. D man. 2. 22-24. deducta, numerus ex multiplicatione 

12. omni/>rocum. poteris + laus deo etc. V ; increet et quater acceptus similis sit numero (non 
poteris om. D. D) ex multiplicatione unius diuisionis semel in 

13. Titulum om. seipsam deducte generato. 

14-16. Mahumed Algoarizim, primum quidem 25. In marg. C Minor, et sic ubique. de- 

capitula numeri proposuimus (premisimus D) que cem pro denarii. alteram + eiusdem (eius D) 

sub sex questionibus ad similitudinem 6 capitu- diuisionem. 

lorum in libri principio propositorum constituimus. 26. re 1 + proponas V; + propones D. re 2 

Vbi eciam diximus quam numerus. om. V. fientque 10 radices. 

17. procul dubio versatur. 27. 10 sine re V; rem D pro 10 res absque sub- 

17-19. ea que in questionibus quasi sub quodam stantia. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 103 

If you wish to multiply the root of ^ by the root of \, you multiply \ by \, 
giving \. The root of this one-sixth is that which we obtain by the multi- 
plication of the root of \ by the root of \. 

Again if you wish to multiply two roots of 9 by three roots of 4, 1 take two 
roots of 9 according to the method which we have explained so that you may 
know the square of which this is the root. Treat similarly the three roots 
of 4 in order to find the square of which this is the root. Then multiply these 
squares by each other, i.e. one of them by the other, and take the root of 
this product since this is the result of the multiplication of two roots of 9 
by three roots of 4. 

Therefore using this process you are able, casting aside error, to multiply 
as many roots as you wish to join together or as many as you wish to sub- 
tract from other quantities. 2 

Problems Illustrating the Chapters 3 

Says Mohammed Al-Khowarizmi : Up to this point we have set forth 
the chapters on numbers which we proposed in the beginning of this work, 
under six problems, one for each chapter, and we also mentioned in that place 
that every problem of restoration and opposition necessarily falls within the 
scope of one of these six chapters. But since the explanations were some- 
what involved we present and more fully explain these further problems, 
by which each type is illustrated and the science is more easily elucidated. 4 

First problem, illustrating the first chapter 

Divide ten into two parts in such a way that one part multiplied by 
the other and the product, or result, taken four times, will be equal to the 
product of one part by itself. 5 

The method is to let x represent one part of ten, and the other 10 — x. 
Therefore multiply x by 10 — x, giving iox — x 2 . Also multiply 10 x — x 2 
by 4, as it was to be taken four times, giving 40 x — 4X 2 as four times the 

1 2 Vg by 3 V4 = V^6 by V^6 - V^6 = 36. 

2 Rosen, op. cit., p. 31, translates this paragraph: "You proceed in this manner with all 
positive or negative roots " ; he follows with the geometrical explanation of the two problems 
given in footnote 4, p. 33, and a further elucidation of the third problem of that set. 

3 Scheybl adds to the statement of each problem the marginal word, Textus, and to the expla- 
nation, Minor; also the numbering of the problems appears to be his addition. 

4 Rosen, op. cit., p. 35 : "Of the six problems. 

Before the chapters on computation and the several species thereof, I shall now introduce six 
problems, as instances of the six cases treated of in the beginning of this work. I have shown 
that three among these cases, in order to be solved, do not require that the roots be halved, and 
I have also mentioned that the calculating by completion and reduction must always necessarily 
lead you to one of these cases. I now subjoin these problems, which will serve to bring the 
subject nearer to the understanding, to render its comprehension easier, and to make the argu- 
ments more perspicuous." 

5 4 x(io - x) = x 2 ; 5 x 1 = 40 x ; x = 8. 



104 



LIBER ALGEBRAE ET ALMUCABOLA 



i fientque quatuor aequales multiplicationes partis vnius cum altera, 40 res absque 
4 substantiis. Postea rem cum re, id est alteram partem cum seipsa, multiplica, 
et producetur substantia, rebus 40 absque 4 substantiis aequalis. Restaura 
igitur numerum, hoc est substantiae substantias quatuor adiicias, venientque 

s quinque substantiae 40 res coaequantes. Vnius ergo substantiae radicem octo- 
narius numerus assignat. Ipsa deinde substantia in 64 terminatur, cuius scilicet 
radix vnam partem numeri denarii cum seipsa multiplicatam, demonstrat, et 
residuum numeri 10 in binario terminatur numero. Duo itaque alteram partem 
numeri 10 obtinent. lam igitur haec quaestio ad vnum sex illorum capitum, 
10 illud nimirum in quo diximus, substantiae radices coaequant, te perduxit. 

Caput secundum, quaestio secunda 

Denarium numerum sic in duo diuido, vt si denarius numerus semel cum seipso 
multiplicetur, numerus qui ex multiplicatione producetur aequalis sit duplo 
numeri eius, qui ex multiplicatione vnius partis cum seipsa producitur, septem 

is nouenis partibus eiusdem producti numeri superadditis. 

Huius rei expositio est, vt numerum 10 cum seipso multiplices, fientque 100, 
duas substantias et septem vnius substantiae nonas coaequantia. Haec ad vnam 
conuertas substantiam, id est ad nouem vigesimas quintas partes quae quintam 
et £ quintae continent, atque centenum numerum ad eius quintam et quatuor 

20 quintae quintas, id est ad 36, vnam substantiam coaequantes conuerte ; et erit 
radix substantiae 6, vnam diuisionis partem numeri decern exprimens, vnde 
altera deinde in quaternario numero procul dubio terminatur. Igitur haec 
quaestio ad secundum sex capitum te perduxit, in quo diximus, substantiae 
numeros coaequant. 

2 S Caput tertium, quaestio tertia 

Denarium numerum ita in duo diuido, vt, vna eius parte in alteram diuisa, inde 
exiens particula in quarternario numero finietur. 

Expositio talis est, vt vnam partem, rem constituas ; atque alteram, 10 sine re 

proponas, deinde 10 sine re in rem diuidas, et exeunt 4. lam autem manifestum 

30 est, si id quod ex aliquo diuiso exierit, cum eo in quod ipsum diuiditur, multipli- 



1. similitudines multiplicationis. in alteram 
+ et erunt. 

2. re + multiplica. deducas. 

3. net. 4 + 4 or D. coequales V. 

4. igitur + vel comple C. et super sub- 
stantiam, substantias (res D) adicias, fientque 40 
res 5 substantias. 

6. et ipsa. deinde om. 8 pro scilicet V. 

8. ternario D. Nam et {om. D) duo. 

9. obtinebit C. ad unum capitulorum te 
perduxit in quo diximus, Substantia radices coequat. 

I3 _ i5- ducatur, numerus qui ex multiplicatione 
excreuerit (decreuerit D) similis sit numero qui ex 
multiplicatione unius diuisionis in semetipsam bis 
deducte (+ tollitur D) vii nouenis superadditis. 

16. semetipso V. -que om. 

17. nouenas. coequantur D. Haec + igitur. 



18. 9 partes ex (et D) 25 que ( + numerum 
D). 

19. 4 or s 4 * quintas. continent, et centenum 
numerum ad eius quintam V; om. D. 

20-22. Radix igitur huius {om. D) substantie id 
est 6 unum numeri 10 diuisionem ostendunt. 
Altera uero eius diuisio in 4 rio . 

22-24. Igitur hec questio quo ad unum vi capi- 
tulorum usque perduxit in quo diximus, substantia 
numeri coequat D ; om. V. 

26. et eiuus (?) una D. 4 eueniant V; Vna- 
quoque particula in ternario numero finiatur D 
pro inde . . . finietur. 

29. ergo pro deinde. super rem diuide ut 
fiant 4. Et iam. autem om. 

30. quam pro si. exierit + si V. multipli- 
catur V. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 105 

product of one part by the other. Then multiply x by x, i.e. one part by 
itself, giving x 2 , which equals 40 x — 4 x 2 . Therefore restore * or complete 
the number, i.e. add four squares to one square, and you obtain five squares 
equal to 40 x. Hence 8 is the root of the square which itself is 64. The root 
of this is that part of 10 which is to be multiplied by itself, and the difference 
between this number and ten is 2. So that 2 is the other part of 10. Now 
this problem has led you to one of the six chapters 2 and, indeed, to that one 
in which we treat the type, squares equal to roots. 

Second problem, illustrating the second chapter 

I divide 10 into two parts in such a way that the product of 10 by itself 
is equal to twice the product of one part by itself, adding seven-ninths of the 
same product. 3 

Explanation. You multiply 10 by itself, giving 100, which is equal to ix 2 
and ^ x 2 . You reduce this to one square, i.e. to ^, equal to \ and f of \ 
of itself. And so reduce \ and \ of \ of 100, i.e. 36, to one square. The 
root of the square is 6, representing one part of the division of 10, whence 
then the other part is necessarily 4. Therefore this problem has led you 
to the second of the six chapters in which we treated the type, squares equal 
to numbers. 

Third problem, illustrating the third chapter 

I divide 10 into two parts in such a way that when one part is divided by 
the other, the resulting fraction equals 4. 4 

Explanation. You let x represent one part, and consequently the other 
you propose as 10 — x. Then you divide 10 — x by x, giving 4. Now 
it is evident that if you multiply the quotient by the divisor you 

1 This is strictly the operation corresponding to the term algebra, and the verb used in the 
Arabic text is in fact from the same stem jbr as the word ' algebra ' ; the quantity 40 x above is 
regarded as incomplete by the amount 4 x 2 . 

2 The reference is to the six types of quadratic equations which are discussed extensively in the 
first part of the work, namely, 

ax 2 = bx, ax 2 + bx = n, 

ax 2 = n, ax 2 + n = bx, 

ax = n, ax 2 = bx + n. 

The first six of the problems in this set are chosen to illustrate each of these six types, in order. 

3 2 x 2 + I x 2 = 100 ; -% 5 - x 2 — 100 ; x 2 — 36 and x = 6. 

The Arabic text of this problem is somewhat lengthier, and includes the statement of a 
second problem which does not appear either in our text or in the Libri version. Following 
Rosen's translation, op. cit., p. 36: "I have divided ten into two portions: I have multiplied 
each of the parts by itself, and afterwards ten by itself : the product of ten by itself is equal to 
one of the two parts multiplied by itself, and afterwards by two and seven-ninths ; or equal to 
the other multiplied by itself and afterwards by six and one-fourth." But a solution is given 
only for the first part of the problem. 

4 =4; 10 — x = 4 x; 10 =5*, and x = 2. 



io6 LIBER ALGEBRAE ET ALMUCABOLA 

i caris, quod turn substantia quae diuisa est, adimpleatur. Et in hac quaestione 
4 diuisum obtinet et id per quod diuisum diuiditur, rem proposuimus ; numerus 
igitur 4 cum re multiplicands erit, et producentur 4 res, substantiam quam 
diuisimus, hoc est 10 sine re exequantes. Restaura igitur 10 sine re, et ipsam rem 

s rebus 4 adde, et venient 10 quinque res coaequantia. Haec autem res in binario 
finitur numero. Vnde haec quaestio ad tertium sex capitum te perduxit, vbi etiam 
diximus, radices numeros coaequant. 

Caput quartum, quaestio quarta 

Tertiam rei et vnam drachmam cum quarta rei et vna drachma sic multiplico. 

10 vt huius multiplicationis productum in 20 terminetur. 

Expositio talis est, vt tertiam rei cum quarta rei multiplices, et producetur 
medietas sextae vnius substantiae, et drachma cum quarta rei multiplicata, 
quartam rei generat addendam. Similiter tertia rei cum drachma, tertiam 
rei procreat atque tandem drachma cum drachma, drachmam producit. Haec 

is porro multiplicatio ad medietatem sextae vnius substantiae, et ad tertiam ac 
quartam rei atque ad vnam drachmam vinginti drachmas coaequantia extenditur. 
Vnam igitur drachmam ex 20 subtrahas, et manebunt 19 drachmae medietatem 
sextae vnius substantiae simul tertiam ac quartam rei coaequantes. lam ergo 
substantiam compleas, hoc est, quicquid habueris cum duodecim multiplices, et 

20 producetur substantia et 7 res, 228 ex numeris coaequantes. Radices igitur 
media, id est per medium diuide, et vnam medietatem cum seipsa multiplica, et 
producentur 12 et \. Haec 228 ex numeris adiicias, et veniunt 240 et \; hinc 
radicem accipe, 15 et ^; atque ex ea 3 et dimidium subtrahe, et manebunt 12, 
radicem substantiae adimplentia. lam igitur haec quaestio ad quartum sex 

25 capitum te perduxit, in quo diximus, substantia et radices numeros coaequant. 

Caput quintum, quaestio quinta 

Denarium numerum ita in duo diuido, vt vnaquaque diuisionis parte cum 
seipsa multiplicata, duorum productorum summa ad 58 perueniat. 

1. substantiam cuius diuisiones (diuisionis D) ducta, 4 am (4 te D) rei progenerat. Hec igitur 
fuerint adimplebit pro quod . . . adimpleatur. summa. 

hac diuisione D. 15. et rei 4 am et etiam. 

2. cum quatuor, ipsum exeuntem, id deinde in 16. drachmas om. 

quod diuiditur, rem proponamus C. Multi- 18. sextam D. et 3 am et. coequantiaV; 

plica ergo rem per 4 et fient 4 res V ; Multiplicatam coe + adequantia D. Sic igitur substantiam 

ergo iiii or et fient xi res D. compleas et quidquid habueris. 

4. coequantes. cum re D pro sine re. et 20. 7 radices. numero V ; om. D sed spat. 
super 4 res ipsam adde, fientque. relict. 

5. ergo. 21. in ipsam. fientque. 

6. Ergo pro Vnde. usque ad unum sex. 22. Hec igitur super 228 adiciaset fient. Huius 
ut et V ; nos (sed del.) ubi et D pro vbi etiam. (Hanc D) ergo. 

9. Terciam substantie et unum dragma in 4 an » 23. et ex eis tria et medium diminuas. 
substantie et uno dragmate. 24. Hec ergo V; lam in hoc D. ad unum 6. 

10. summa pro productum ubique. 25. substantie. numerum. 

. j . j . A . 27. sic pro ita. In marg. \(h-\- 3*^2. D - 

12-15. in dragmate ductum, dragma generat I r" ' S v^ 

(generatur D) adiectiuum et 3 a rei in dragmate unaquaque diuisione in semetipsam deducta, tota 

ducta 3 am rei procreat. Sed et 4 a rei in dragmate multiplicationis summa ad 58 tendatur. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 107 

obtain thus the quantity 1 which was divided. And as in this problem the 
quotient is 4 and the divisor is given as x, you multiply 4 by x, giving 4X for 
the quantity which was divided, i.e. equal to 10 — x. Therefore complete 
10 — x by adding x to 4X, giving 10 equal to $x. Whence it follows that 
x is 2. Thus this problem has led you to the third of the six chapters in 
which we treated the type, roots equal to numbers. 

Fourth problem, illustrating the fourth chapter 

Multiply 5 £ and one unit by jxand one unit so as to give as the product 20. 2 
Explanation. You multiply ^# by \x, giving \ of \x 2 , and a unit mul- 
tiplied by \x gives \x to be added. Similarly \ x multiplied by a unit gives 
\ x and then a unit by a unit gives a unit. Then this multiplication 
amounts to § of \x 2 , and \x and \x and one a unit, equal to 20 units. 
You subtract one unit from 20 units, giving 19 units equal to § of ^x 2 
together with %x and \x. Now then you complete the square, 3 i.e. you 
multiply throughout by 12. This gives x 2 and jx equal to 228. Then 
halve the roots, i.e. divide them equally, and multiply one-half by itself, 
giving i2j. You add this to 228, and you will have 240^. From the root 
of this, 15I, subtract 3I, leaving 12 as the root of the square. Now then 
this problem has led you to the fourth of the six chapters in which we 
treated the type, a square and roots equal to numbers. 

Fifth problem, illustrating the fifth chapter 

I divide ten into two parts in such a way that the sum of the products 
obtained by multiplying each part by itself is equal to 58.* 

1 Robert of Chester employs substantia here in the non-technical sense of "substance " or " quan- 
tity." So also census in the Libri version and mat in the Arabic version are used with the same 
significance. This was a common usage of Arabic writers. Abu Kamil followed this practice, 
on occasion, and Leonard of Pisa, who drew extensively from Abu Kamil, copied this peculiarity 
from the Arabs: see Scritti di Leonardo Pisano, Vol. I, p. 422, and my article, The Algebra of 
Abu Kamil Shoja' ben Aslant, Bibliotheca Mathematica, third series, Vol. XII (1912-1013), p. 53. 
In particular, also, census appears in this sense in the Liber augmenti et diminutionis vocatus nu- 
meratio divinationis, ex eo quod sapientes Indi posuerunt, quem Abraham compilavit et secundum 
librum qui Indorum dictus est composuit, published by Libri, Histoire des sciences mathematiques en 
Italie, Vol. I, Paris, 1838, pp. 304-371. This work is probably by Abraham ibn Esra. 

2 {\ x + 1) {\ x -\- 1) = 20; ^ x 2 + \ x + \ x -\- 1 = 20; x 2 + 7 x + 12 = 240 ; x 2 + 7 x = 228 ; 

I of 7 is 35, (3^) 2 = 12I, 228 + i2j = 2405, V 2401 = 155, 155 — 3! = 12, which is the value of x. 

3 The present usage of the expression "to complete the square" is quite different from that of 
our text. Here it means, of course, to make the coefficient of x 2 unity and this also corresponds 
to the operation termed by the Arabs, algebra, as opposed to the operation of almuqabala; 
see the article al-Djabr wa-'l-Mukdbala by Professor H. Suter in The Encyclopedia of Islam, 
Vol. I (Leyden, 19 13), pp. 989-990. 

4 x 2 + (10 - x) 2 = 58; 2 x 2 - 20 x + 100 = 58; 2 x 2 + 42 = 20*; x 2 + 21 = 10 x, which is a 
problem that appears earlier in the text. 



108 LIBER ALGEBRAE ET ALMUCABOLA 

i Expositio talis est, vt 10 sine re cum seipso multiplices, fientque ioo et substantia 
absque 20 rebus. Postea multiplica rem cum suo aequali et producetur substantia. 
Deinde haec duo multiplicationis producta in vnum collige, et veniunt 100 et 2 
substantiae absque 20 rebus, 58 coaequantes. Comple igitur 100 et 2 substantias 

5 absque 20 rebus, et ipsas 20 res 58 ex numeris adiicias, et venient 100 drachmae et 
2 substantiae, 58 drachmas et 20 res coaequantia. Hoc iam ad vnam conuertas 
substantiam, atque oppositione deinde 29 ex 50 proiicias, et manebunt 21 et sub- 
stantia, 10 res coaequantia. Res igitur mediabis, et veniunt 5 ; haec sum seipsis 
multiplicata, producunt 25. Ex his 21 abiicias, et relinquentur 4. Accipe horum 
ioradicem 2, atque hanc a 5, id est a medietate radicis, subtrahe, et manebunt 3 
quae videlicet vnam partem numeri decern adimplent. Igitur haec quaestio ad 
quintum sex capitum te perduxit, in quo diximus, substantia et numeri radices 

coaequant. 

Caput sextum, quaestio sexta 

15 Rei tertiam et eius quartam sic multiplico, vt multiplicationis productum ipsam 
rem, viginti quatuor drachmis superadditis, coaequent. 

Expositio est, vt primum scias, quod quando tertiam rei cum quarta rei multi- 
plicaueris, medietas sextae vnius substantiae, rem et 24 drachmas coaequans, 
oriatur. ' Multiplicatio igitur medietatis sextae substantiae cum duodecim, sub- 

20 stantiam reddet completam. Similiter multiplicatio rei et 24 drachmarum cum 
12, radices 12 ducenta et 88, substantiam coaequantes, adimplebit. Diuide 
igitur radices per medium, et mediam partem cum seipsa multiplica, atque multi- 
plicationis productum numero 288 iunge, et venient 324. Horum nunc radicem 
accipe, id est 18, quibus medietatem radicum etiam adde, et radix in 24 finietur. 

25 Haec igitur quaestio ad sextum sex capitum iam te perduxit, in quo diximus, 
radices et numeri substantiam coaequant. 

Ad hue restat, vt de sedecim aliis tractemus quaestionibus, quae ex sex prae- 
missis oriri videntur, vt quicquid ex numero huic arti addicto opifici propositum 
fuerit, omni errore abiecto facilius elici queat. 

1. in semetipsis. 16. substantiam. coequant. 

2. radicibus. 18. et om. D. 
3-10. has multiplicationis summas in unum 19. orietur. 

collige et habebis 100 et duas substantias absque 20. Similiter + et. 34 V. in (+ in D) 12. 

20 radicibus 58 coequantes. Comple igitur 21. ducentos et 88 et radices xii D. coequans 

100 et duas substantias absque 20 radicibus cum V; om. D. 

re quam diximus, et adde earn super 58, et fient 22. ac pro atque. 

100 et due substantie, 58 et 20 (10 D) res coequancia. 23. summam super 288 adicias (addicias D et sic 

Res ergo mediabis et erunt 5. Hec igitur in saepius) et erit hoc totum324; erit hoc totum C 

seipsis (semetipsis D) multiplica et erunt 25. Ex sed del. igitur pro nunc. 

his ergo 21 (om. D) abicias et remanebunt 4. 24. accipias. quibus videlicet medietatem 

Sume ergo horum radicem (harum radices D) id radicum adicias, id est 6, et sic substantia (substan- 

(om. D) est (om. D) duo qui ab 5 prius positis, id tiam D) in 24 (cxiiii D) terminetur. 

est a medietate radicum, subtrahas. 25. unum sex. 

n. videlicet om. 27. Hee ergo sunt (+vi D) questiones de qui- 

12. unum 6. te + iam. numeri om. D. bus superius me tractaturum promisi. Sed adhuc 

R 1 pro radices. pro Ad hue. primis pro praemissis. 

15. Substantie pro Rei. in pro et. et pro 28. quidquid V; quid D. huius artis intento. 

vt et sic infra (3). In marg. I </> + "^? D - 2g ' abiecte D - eliciatur pro elici queat. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 109 

Explanation. You multiply 10 — x by itself, giving 100 and x 2 less 
20 x. Then multiply x by itself, giving x 2 . Collecting the products of 
these two multiplications you obtain 100 and 2x 2 less 20X equal to 58. 
Complete the 100 plus 21 2 less 20X by adding the 20X to 58. This gives 
100 units + 2X 2 equal to 58 units and 20 x. Now you reduce this to one 
square, and then by opposition 1 you take 29 from 50, leaving 21 + x 2 equal 
to 1 ox. Therefore ybu halve the roots, giving 5. Multiply this by itself, 
giving 25. From this you subtract 21 and 4 is left. Take the root of this, 
2, and subtract it from 5, i.e. from the half of the roots. This gives 3 which 
represents, of course, one part of ten. So this problem has led you to the 
fifth of the six chapters in which we treated the type, a square and num- 
bers equal to roots. 

Sixth problem, illustrating the sixth chapter 

I multiply \x and \x in such a way as to give x itself plus 24 units. 2 
Explanation : first you observe that when you multiply \ x by \ x you 
obtain \ of \x 2 equal to x + 24 units. The multiplication of \ of \x 2 by 12 
gives the complete square. Similarly, the multiplication of x + 24 units 
gives 12X -f- 288 s which equal x 2 . Therefore take one-half of the roots and 
multiply the half by itself. Add the product of this multiplication to 288, 
giving 324. Take now the root of this, i.e. 18, and add it to the half of the 
roots. The root finally is 24. So this problem has led you to the sixth of the 
six chapters in which we treated the type, roots and numbers equal to a 



square. 4 



Sixteen Additional Problems 5 



It now remains for us to treat sixteen other problems which seem to arise 
out of the six which we have set forth, in order that the craftsman versed 
in this art may, more easily and without any error solve any problem 
proposed. 6 

1 This operation corresponds to the Arabic term almuqabala. In this instance the 29 on the 
right balances or cancels an equal amount of the 50 on the left. 

2 \x .; x = x + 24; x 2 = 1 2 * + 288; \ of 12 is 6, 6 2 = 36, 288 + 36 = 324, V324 = 18, 
18 + 6 = 24, which is the value of the unknown. 

3 Evidently Robert of Chester wrote the word for two-hundred out in full, or in Roman nu- 
merals, and the 88 in Hindu-Arabic numerals, for both the Dresden and Scheybl versions use this 
form, separating the two-hundred from the 88. 

4 The above six problems appear in this order in all of the versions. 

The terminology of the fourth and sixth problems is the same in the Arabic text, although 
Rosen, op. cit., pp. 38, 40 translates differently the same expression: "I have multiplied one- 
third of thing and one dirhem by one-fourth of thing . . . " ; "I have multiplied one-third of 
a root by one-fourth of a root ..." The Libri text has census in both problems, and the 
Boncompagni text has muUUudo; the Arabic word is mal with the meaning (footnote 1, p. 107) 
'' unknown quantity." 

6 I have added this title to correspond to the Arabic (Rosen, op. cit., p. 41). 

6 This paragraph is an addition by Robert of Chester. 



no 



LIBER ALGEBRAE ET ALMUCABOLA 



i Quaestio prima 

Denarium numerum sic in duo diuido, vt vna parte cum altera multiplicata, 
productum multiplicationis in 21 terminetur. 

lam ergo vnam partem, rem proponimus quam cum 10 sine re, quae alteram 
s partem habent, multiplicands, et producuntur 10 res absque substantia, drachmas 
21 coaequantes. Comple igitur 10 res cum substantia, et substantiam numero 
21 adde, et venient 10 res, substantiam et 21 drachmas coaequantes. Accipe 
medietatem rerum, hoc est 5, et earn cum seipsa multiplica, et producentur 25. 
Ex his 21 subtrahe, et manebunt 4. Horum radicem, 2 scilicet, accipe, atque 
10 earn tandem ex medietate radicum subtrahe, et manebunt 3, quae vnam partem 
diuisionis demonstrant. 



Quaestio secunda 

Numerum denarium sic in duo diuido, vt vtraque parte cum seipsa multiplicata, 
si productum partis minoris ex producto partis maioris auferatur, quadraginta 

J 5 maneant. 

Exempli expositio talis est, vt 10 sine re cum suo aequali multiplices, et pro- 
ducentur 100 ex numero, vna substantia absque 20 rebus. Multiplices etiam 
rem cum se, et producetur substantia, quam ex 100 et substantia absque rebus 
diminuas, et manebunt 100 absque 20 rebus, quadraginta drachmas coaequantia. 

20 Comple igitur 100 drachmas cum 20 rebus, et eas drachmis 40 adiicias et habebis 
100, quadraginta drachmas et 20 res coaequantes. Igitur 40 ex 100 auferas, et 
manebunt 60 drachmae 20 res aequantes. Res igitur ternario aequantur numero, 
qui vnam partem diuisionis demonstrat. 

Quaestio tertia 

25 Denarium numerum sic in duo diuido, vt vtraque parte cum seipsa multiplicata, 
et multiplicationum productis simul collectis, ac quantitate deinde, quae est inter 
duas partes, illis addita, tota summa ad 54 drachmas excrescat. 

Huius exempli expositio talis est, vt 10 sine re cum suo aequali multiplices, et 



2. vt om. D. deducta, summa. In marg. 



5+ <f> D. 



5. fiuntque. drachmas om. 

6. sine re pro res. super pro numero. 

7. Die ergo 10 res. coequant. Sume 
medietatem radicum. 

8. in semetipsam (semetipsa D). 

9. A quibus 21 demptis V ; om. D. et om. V. 
Accipe eorum (earum D) radicem, id est 2, et 
earn ex. 

10. et om. D. 

13. ut (et D) unaquaque diuisione in semetipsa 
(semetipsam D) deducta, si multiplicatio minoris 
diuisionis ex multiplicatione maioris tollatur 40 
{om. D) remaneant. 

16. Expositio huius talis. consimili. 

17. 100 et substantia absque (+ absque vel 



minus in marg. D man. 1) 20 radicibus. Multi- 
plica igitur rem in re, et net. 

18. que V. absque om. V; et D. 20 
rebus. 

20. 100 cum 20, et eas super 40 adicias. 

21. drachmas om. Hoc ergo centeno opponas 
(apponas D) numero et 40 ex. inferas D. 

22. 60 (4, + cum D) 20 res coequancia. 

23. unam mensurat diuisionem. 

25. unaquaque. semetipsa. multiplica 



D. 



In marg. 3 "4~ *^» D - 



26. et utriusque multiplicatione in unam col- 
lecta (+ aD) quantitate que. 

27. addita + nam et unamquamque earum 
in semetipsa multiplicasti D. 

28. Inde igitur talis datur expositio, ut. con- 
simili. 



THE BOOK OF ALGEBRA AND ALMUCABOLA in 

First Problem 1 

I divide ten into two parts in such a way that the product of one part 
multiplied by the other gives 21. 2 

Now then we let x represent one part, which we multiply by 10 — x, 
representing the other part. The product 10 x - x 2 is equal to 21 units. 
Complete 10 x by x 2 and add this x 2 to 21. This gives iox equal to x 2 + 21 
units. Take one-half of the unknowns, i.e. 5, and multiply this by itself, 
giving 25. From this subtract 21, giving 4. Take the root of this, 2, and 
subtract it from half of the roots, leaving 3, which represents one of the parts. 

Second Problem 

I divide ten into two parts in such a way that each part being multiplied 
by itself, the product of the smaller part taken from the product of the larger 
part leaves 40. 3 

Explanation. You multiply 10 - x by itself, giving 100 + x 2 — 20a;. 
You multiply x by itself, giving x 2 , which you take from 100 + x 2 — 20 x, 
leaving 100 — 20X equal to 40 units. Therefore by adding 20X to the 
40 units complete 100 units by 20X. This gives iooequal to 40 units + 2ox. 
Therefore you take 40 from 100, leaving 60 units equal to 20a;. Three is 
then the value of x and represents one part. 

Third Problem 

I divide ten into two parts in such a way that when to the sum of the 
products of each part by itself is added the difference between the two parts 
the sum total will be 54 units. 4 

1 The sixteen problems which follow are selected by Robert of Chester from twice that num- 
ber in the Arabic text. The Boncompagni version presents nine, including the first, second, 
third, and fifth of this list. 

The Boncompagni version interjects (Joe. cit., pp. 45-46) before these problems: "pro mul- 
titudine data assignatur horum muttatione quaslibet questiones secundum restauracionem pro- 
positas in predictos modos solubiles esse palam est. Cuius utilitas ad documentum libri 
elementorum precipua est, in inveniendis scilicet lineis alogis et medialibus binomiis et residuis 
sive reccisis que per notum numerum assignari non possunt. In practica quoque geometrie et 
universis questionibus ignotorum secundum arismeticam formatis certissima via est." 

2 *(io — .v) = 21, which leads to the type equation given earlier in the work, x 2 + 21 = 10 x, 
and indeed, it incidentally appears a second time in the preceding set of problems. The Arabic 
text precedes the statement of this problem, Rosen, op. cit., p. 41, as follows : " If a person puts 
such a question to you as . . . " ; the Boncompagni version, loc. cit., p. 46, precedes: "Igitur 
sub formas precedencium et alias questiones proponimus. Queritur. ..." 

3 (10 — x) 2 — x 2 = 40, whence 100 — 20 x = 40; 20 x = 60, x = 3, which is the value of the 
root. 

4 x 2 + (10 — x) 2 + 10 — 2 x = 54, whence 2 x 2 — 22 x + no =54; 2 x 2 + no = 54 + 22 x; 
x 2 + 55 = 27 + 11 x; x 2 + 28 = 11 x; § of n is si, (5D 2 = 30?, 3°i ~ 28 = 2\, V 2 i = if, 
5§ -i§ = 4> which is the value of the unknown quantity. 



112 



LIBER ALGEBRAE ET ALMUCABOLA 



i producentur ioo ex numero et vna substantia, absque 20 rebus. Rem etiam cum 
re multiplices, et producetur substantia, quam 100 ex numero et substantiae, 
absque 20 rebus iunge, et venient 100 ex numero et 2 substantiae, absque 20 
rebus. lam quantitatem quae inter vtramque partem est, 10 sine 2 rebus, toti 
s summae adde, atque hoc totum ad no ex numero et 2 substantias, duabus et 
20 rebus abiectis, 54 drachmas coaequantes, exuberando peruenit. Compleas ergo, 
et die: no drachmae et 2 substantiae 54 drachmas et 22 res coaequant. Hoc 
autem ad vnam conuertas substantiam, et die : 55 drachmae et vna substantia 
coaequant 27 drachmas et n res. Igitur2 7 ex 55 subtrahe, et manebunt 28 drachmae 
10 et vna substantia, n res coaequantia. Res igitur per medium diuide, et venient 
5 et medium ; haec cum suo aequali multiplica, et producentur 30 et quarta, ex 
his 28 subtrahe, et radicem ex residuo, vnum scilicet et \ accipe, quae simul ex 
medietate rei auferas, et manebunt 4. Haec igitur vnam partem diuisionis adim- 
plent. 



15 



Quaestio quarta 



Denarium numerum sic in duo diuido, vt vna eius parte cum seipsa multiplicata, 
numerus inde productus alteram octagesies et semel comprehendat. 

Expositio est, vt 10 sine re cum suo aequali multiplices, et producentur 100 ex 
numeris et substantia, absque 20 rebus, 81 res coaequantia. Comple igitur 100 

20 ex numeris et substantiam, et adde 20 res numero 81, et venient 100 ex numeris 
et substantia, 100 res et vnam coaequantia. Mediabis ergo res, et venient 50 
et dimidium. Haec cum suo aequali multiplica, et producentur 2550 et quarta. 
Ex his nunc 100 subtrahe, et manebunt 2450 et quarta. Horum radicem accipe, 
id est 49 et dimidium, atque hanc ex radicum medietate subtrahe, et manebit 

25 vnitas, vnam partem numeri decern ostendens. 

Quaestio quinta 



Duas substantias duabus drachmis differentes, diuido, maiorem scilicet in 



I. ent. vna om. 

1-4. Rem . . . rebus 1 om. V; Rem quoque in 
re multiplicata, et erit substantia. Hec insimul 
iunge et erunt C et due substantie absque xx 
rebus D. 

4. Et iam. vtramque + substantiam D. 
super totam summam addidisti. Dicas ergo 
quantitas que inter utramque est diuisionem 10 
absque duabus rebus designat. Hoc igitur totum 
ad 100 et 10 et [om. D) duas pro 10 sine . . . et 2. 

5. absque duabus rebus in (et D) pro duabus et. 

6. Tunc ergo compleas. 

8. conuerte. vna om. et sic infra. 
10. erunt. 

II. haec + igitur. consimili. erunt. 

12. his + ergo. subtrahas, et radicem (ra- 
dices D) remanentis (remanentes D) id est duorum 
et 4 e accipias ( + Accipe radicem D) id est unum et 
medium que simul. 

13. radicisV; radices D pro rei. igitur +4. 



16. et pro vt D. in se ipsam semel deducta 
(deducto D), 81 vicibus alteri (iubet altera D) com- 
prehendetur. 

18. Expositio + huius rei. duo pro suo D. 
consimili. ex numeris om. saepe. 

19. minus. 

20. et substantiam om. et 2 om. D. ra- 
dices super {om. D). 81, et erunt. 100 om. D. 

21. et substantia om. D. 100 et unam ra- 
dicem. radices. erunt. 

22. Has ergo. suo consimili, sibi aequali = 
cum seipsis: se in marg. C; in suo consimili V D. 
erunt 2000,500 et 50 et 4 a . 

23. ergo pro nunc. 2000 et 450 (400 D). 
Horum + ergo. 

24. et ipsam pro atque hanc. medietate 
+ id est 50 et medio. subtrahas. 

25. unum, unam denarii ostendens diuisionem. 
27. diuido om. V. id est earum minorem 

super maiorem. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 113 

The explanation of this problem is of the following nature : you multiply 
10 — x by its equal, which gives 100 + x 2 - 20 x. Also you multiply 
x by x, giving x 2 , which you add to 100 -f- x 2 — 20 x. This gives 
100 + 2 x 2 - 20 x. To the sum total now add the difference between 
the two parts, 10 - 2 x, and this total amounts to 110+2X 2 -221, 
which equals 54 units. You complete therefore (by adding 22 #), and you 
obtain no units + 2 x 2 equal to 54 units + 22 x. This, moreover, you 
reduce to one square, giving 55 units + x 2 equal to 27 units +nx. So 
subtract 27 from 55, giving 28 units + x 2 equal to nx. Halve the un- 
knowns, giving 5§. Multiply this by itself, giving 305. From this subtract 
28 and the root of the remainder, i|, taken from one-half of the roots, will 
leave 4. Therefore this represents one part. 

Fourth Problem x 

Divide ten into two parts in such a way that the product of one part by 
itself equals 81 times the other part. 2 

Explanation : you multiply 10 — x by itself, giving 100 + x 2 — 20 x, 
which equals 81 x. Complete 100 + x 2 by adding 20 x to Six. This 
gives 100 + x 2 equal to ioix. You halve the unknowns, giving 50^. 
Multiply this by itself, giving 25505. Now subtract 100 from it, leaving 
2450 j. Take the root of this, i.e. 49^, and subtract it from the half of 
the roots. This will give unity, representing one part of ten. 3 

Fifth Problem 

Two squares (i.e. two quantities or numbers) being given with a difference 
of two units, I divide the smaller by the larger in such a way that the 
fraction resulting from the division shall equal one-half. 4 

1 The order of the problems is from this point different from the order of the corresponding 
set of problems as given in the Libri, Arabic, and Boncompagni versions, but the variations are 
due to the omission of problems, as will be noted below, rather than to actual changes in order. 
See chapter V of our Introduction. The Boncompagni text is the least complete, giving only 
ten problems, which follow, in the main, those of the Libri text. 

2 This problem (10 - x) 2 = 81 x, appears on pages 47-48 of Rosen's book, and appears twice 
in the Libri text, pp. 282-283 and pp. 289-290. 

In modern notation, the solution proceeds : 

100 - 20 x + x 2 = 81 x; x 2 + 100 = 101 x; 5 of 101 is 505, (505) 2 = 25505; 2550J - 100 
= 2440J ; V 24405 = 49! ; 50^ - 49^ = 1, which is the value of one root. The other root would 
be 505 + 495, or 100. 

3 The second root, 100, is not given, since it leads to 100 and negative 90 as the two 
parts into which 10 is divided. This was rejected by the Arabic writer as an impossible 
solution, nor, indeed, was such a solution regarded as possible for centuries after the time of 
the Arab. 

4 Rosen, pp. 50-51, and again pp. 62-63 5 Libri, p. 283 and p. 295. By error Scheybl makes 
this one-half of the larger square. 



ii 4 LIBER ALGEBRAE ET ALMUCABOLA 

i minorem, sic, vt vna diuisionis particula, quae est exiens, medietatem substantiae 

maioris compleat. 

Rem ergo et duas drachmas cum medietate quam diuisionis particulam diximus, 

multiplico, et veniet rei medietas et drachma, rem coaequans. Propter medie- 
s tatem ergo rei vtrinque medietatem rei abiicio, et manebit drachma medietatem rei 

coaequans. Hanc igitur duplico, et venient duae drachmae, quae ipsam rem, 

constituunt, vnde quatuor deinde alteram. 

Quaestio sexta 

Substantiam et eius radicem sic multiplico, vt multiplicationis productum tres 
10 similitudines substantiae, adimpleat. 

Expositio est, quoniam quando radicem cum tertia substantiae multiplicaueris. 
tota producitur substantia. Tria igitur huius substantiae radicem adimplent. 
et 9 ipsam substantiam. 

Quaestio septima 

is Tres radices substantiae cum quatuor eius radicibus ita multiplico, vt tota 

multiplicationis summa ipsi substantiae et quadraginta quatuor drachmis coae- 

quetur. 

Expositio talis est, vt cum quatuor radicibus substantiae tres eius radices 

multiplices, et producentur 12 substantiae, substantiam vnam et 44 drachmas 
20 coaequantes. Minue igitur substantiam ex 12 manebunt 11 substantiae, 44 

drachmas coaequantes. Substantia ergo vna 4 aequat. 

Quaestio octaua 

Quatuor substantiae radices cum quinque eiusdem substantiae radicibus sic 
multiplico, vt tota multiplicationis summa ipsius substantiae duplo et triginta 
25 sex drachmis coaequetur. 

Quatuor radices substantiae cum suis quinque radicibus multiplico, produco 
autem sic 20 substantias, duas substantias et 36 drachmas coaequantes. Ex 20 
igitur substantiis duas substantias aufero, et manebunt 18 substantiae 36 drachmas 
coaequantes. Igitur 36 in 18 diuido, et exeunt 2, quae ipsam substantiam adimplent. 

I. sic diuido, ut unam diuisionis particulam 16. ipsa D. 

compleat medietas. 18. Huius est expositio, ut 4 eius radices in suis 

4. fiet. coequantes D. Cum medietate; tribus radicibus. 

Ccorr.inV C. 20. 12 + et. substantiae + et V. 

5. vtrinque om. rei 2 om. V. proicio. 21. coequans V. Substantiam V; Sub- 
drachma om. V. stantie D. coequant. 

6. Hoc. dico pro venient. quae om. ; 23. eius. substantiae om. 

in marg. C. rem + hoc est vnam substantiam 24. vt + et. ipsiV; ipse D. duplo om. 

vel numerum vnum C. 25. xxxvi radicibus D. 

7. et alteram 4. 26. Expositio est, vt quatuor C. Quatuor + 

9. in pro et. ergo. substantiae om. multiplica V. et 

10. substantiae + hoc tres substantias propo- erunt 20 substantie. 

sitas C. 28. xi pro 18 et xliiii pro 36 D. 

II. Expositio + huius. substantiam ante 29. coaequantes + ex xx ergo substantiis duas 
substantiae D. substantias aufero et remanebunt xviii substantie 

12. totam D. oriatur. igitur om. D. 36 super (sunt D) 18. erunt. sub- 

13. ipsam om. stantiam om. D. 
15. et pro vt C. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 115 

I multiply x + 2 units by \, representing the quotient. 1 This gives \ x 
and a unit equal to x. On account of the one-half x I take one-half x from 
both sides, leaving a unit equal to \ x. I double this, giving two units, 
which equal the unknown. Whence four is the other. 

Sixth Problem 

I multiply a square by its root in such a way that the product equals three 
similar squares. 2 

Explanation. From what is given it follows that when you multiply 
the root by one-third of the square, the total gives the square. 3 Therefore 
3 represents the root of this square, and 9 the square. 

Seventh Problem 

I multiply three roots of a square by four of its roots in such away that the 
sum total of the multiplication will equal the same square and 44 units. 4 

Explanation. Multiplying four roots of a square by three of its roots, 
we have twelve squares, which are equal to one square and 44 units. 5 Take 
therefore the one square from the 12 squares, leaving n squares equal to 
44. Hence one square equals 4. 

Eighth Problem 

I multiply four roots of a square by five roots of the same square in such 
a way that the sum total of the multiplication will equal double the square 
and 36 units. 6 

Explanation. I multiply four roots of a square by five of its roots and 
I have as a product 20 squares, which are equal to 2 squares and 36 units. 
Therefore I take the two squares from the 20 squares, leaving 18 squares 
equal to 36 units. I divide 36 by 18, and obtain 2, which represents the 



square. 7 



i, whence x = \ (x +2); x =\x +1; \x = 1, x =2. 



x + 2 

2 Rosen, pp. 54-55; Libri, p. 291. 

Al-Khowarizmi carefully avoids the term for the cube of the unknown, with which he was 
certainly familiar, for Diophantus and other Greek writers employed the term. His continuator, 
Abu Kamil, discussed not only cubics which, like this one, are reducible immediately (to equa- 
tions of lower degree), but also equations in quadratic form. However, systematic discussion 
of the general cubic was first attempted by Omar Khayyam. 

3 x 2 ■ x = 3 x 2 . 

4 Rosen, p. 55; Libri, p. 291. 

5 3 x • 4 x = x 2 +44. 

6 Rosen, p. 55 ; Libri, p. 284. 

7 4 x ■ 5 x = 2 x 2 + 36 ; 18 x 2 = 36 ; x 2 =2. 



n6 LIBER ALGEBRAE ET ALMUCABOLA 

Quaestio nona 

Radicem substantiae cum eius quatuor radicibus ita multiplico, vt tota multi- 
plicationis summa tribus substantiis et quinquaginta drachmis coaequetur. 

Expositio est, vt radicem cum quatuor radicibus multiplicem, et 4 substantias 
s ita productas tribus substantiis et 50 drachmis coaequem. Ex 4 igitur substantiis 
tres substantias aufero, et manebit substantia, 50 drachmas coaequans. Igitur 
radix substantiae 50 est radix numeri 50. 

Quaestio decima 

Ex substantia eius partem tertiam et tres drachmas aufero, et quod residuum 
iofuerit cum suo consimili multiplico, et oritur ex multiplicatione ipsa substantia. 
Expositio est, vt quando tertiam et tres drachmas subtraxero, duae tertiae 
absque tribus drachmis maneant. Sunt autem res. Vnde duas tertias rei absque 
3 drachmis cum suo aequali multiplicamus, et producuntur quatuor nonae sub- 
stantiae et 9 drachmae, absque quatuor radicibus, radicem coaequantia. Adde 
15 igitur absque 4 radicibus vni radici, et venient quatuor nouenae substantiae et 9 
drachmae quinque radices coaequantes. Oportet ergo vt quatuor nouenas sub- 
stantiae compleas, vt ipsa substantia perficiatur, et hoc quidem vt singula cum 2 
et quarta multiplices, et venient n res et \, vnam substantiam et 20 drachmas 
cum quarta coaequantes. Operare igitur cum eis quemadmodum in medietatione 

20 radicis tibi diximus. 

Quaestio vndecima 

Tertiam substantiae cum eius quarta sic multiplico, vt tota multiplicationis 
summa ipsi substantiae coaequetur. 

Expositio huius est, vt tertiam rei cum quarta multiplicem, et producetur 
25 medietas sextae vnius substantiae, rem coaequans. Radix igitur substantiae in 
12 terminatur, et substantia 144 in se continet. 

3. tribus + similitudinibus. substantie. 13. multiplico (multiplicamus D) + Due ergo 
coequatur V; adequatur D. 3 e in duas 3 as ducte 4 substantiae nouenas com- 

4. Huius est expositio. et erunt 4 sub- ponunt et cum ( + et cum D) tribus in duabus rei 
stantie, 3 substantias. 3 ii8 multiplicate, duas res componunt similiter ( + 

5. coequantes; C add. in marg. aequales eodem duas D) diminutiuas et sine 3 bus in 3 bus 9 drag- 
preferam. 4 + ex D. igitur om. mata constituunt adiectiua. Erunt ergo, nouene. 

6. tollo. coaequans + et ipsa et radix 1' 14. 8 pro 9. 

in iiii or radicibus 1' D. Radix ergo 50 in 4 15. sine 4 radicibus super radicem, et fient. 

radicibus 50 ducta (sunt D) ducenta que tribus nouenas. 

similitudinibus substantie et 50 dragmatibus equan- 16. coequans. 

tur constituit (manet equalis D). 17. vt ' + et. compleatur et {om. D) hoc 

7. C add. : Sequitur examen. est ut illas in duabus (duobus D) et (aut D) 4 ta . 

Radix V50. Substantia 50. 18. multiplices ( -f vel, si placet, singula in 4/9 

V50 vt vna radix, cum V800, quatuor diuidas C). Multiplica ergo 9 in duo et 4 a et 

radicibus, multiplicata, producit 200. Atque tan- erunt xx (n res V) et 4* ( + et multiplica v radices 

turn sunt etiam 50 ter, hoc est tres substantiae, et in duo et 4 to et erunt 11 res et 4 te D). Habebis 

quinquaginta. ergo substantiam et 20 dragmata et quartam, 11 

9. partem om. tollo. res et 4 am rei coequancia. Oppone ergo. 

10. et om. ex + qua videlicet. 20. radicum. 

11. Huius est {om. D) expositio; Huius C 24. tertiam +etD. deducam V ; deductam D . 
sed del. 25. sextae + hoc est vna duodecima C. rei add. 

12. remanebunt, et ipse erunt radix. Unde super substantiae el numerum super rem C. 
( + et V). 



THE BOOK OF ALGEBRA AND ALMUCABOLA 117 

Ninth Problem 

Multiply the root of a square by four of its roots in such a way that the 
sum total of the multiplication shall equal three squares and 50 units. 1 

Explanation. I multiply the root by 4 roots and I obtain four squares 
which are equal to three squares and 50 units. I take the 3 squares from the 
4 squares, leaving a square equal to 50 units. 2 Therefore the root of this 
square is the root of 50. 3 The root of 50 multiplied by 4 roots of 50 gives 
200, which is equal to three of the squares and 50 units. 

Tenth Problem 

I take from a square one-third of it and three units, and multiply the 
remainder by itself ; the product is the square. 4 

Explanation. When I have subtracted one-third and three units, two- 
thirds less three units remain. Now let the square be represented by x; 
then we multiply § x - 3 units by itself and have | x 2 + 9 units - 4 roots 
equal to one root. Add the one root to the four roots, giving | x 2 + 9 units 
equal to 5 roots. It is now necessary to complete the four-ninths of a 
square, so as to make it a whole square. You multiply each side by 2j, b 
giving iijx equal to x 2 + 20^ units. You perform the operations with 
these, then, in the manner which we have explained to you in the sections 
on the halving of the root. 

Eleventh Problem 

I multiply one-third of a square by one-fourth of it in such a way that 
the sum total of the multiplication will give the square. 6 

Explanation. I multiply 3 x by | (x) and I obtain \ of \ x 2 equal to x. 
The root of the square, then, is 12, and the square is 144. 7 

1 Rosen, p. 56; Libri, pp. 291-292. 

2 x • 4 x = 3 x 2 + 50. 

3 Scheybl adds : "The root of fifty multiplied by four roots of 50 gives 200, which is equal to 
three of the squares and 50 units." Robert of Chester's text is more closely followed by the 
Vienna and Dresden manuscripts. 

The symbol for square root V used by Scheybl was introduced by Adam Riese in his Coss 
written in 1524, but not printed until recently (Berlet, Leipzig, 1892) ; for the word Coss see 
page 38. 

4 Rosen, pp. 56-57; Libri, pp. 284-285. In modern notation : (x -\x - $) 2 = %■ Between 
this and the preceding problem the Arabic text includes a problem, leading to the equation : 

x 2 +20 = 1 2 x. 

6 Scheybl adds that you may, if you prefer, divide both sides by f . 

6 Rosen, p. 58 ; Libri, p. 292. \x • \x = x. 

7 In the Arabic text this problem follows : (| x + i)(^ x + 2) = x + 13. 



u8 LIBER ALGEBRAE ET ALMUCABOLA 

i Quaes tio duodecimo, 

Drachmam et medium ita in duo diuido, vt maior pars duplex minori habeatur. 
Expositio huius est, vt maiorem partem ad minorem vnum et rem constituam ; 
vnde etiam dicam drachmam et dimidium super drachmam et rem diuisa, et duae res 
s drachmam constituunt. Duas igitur res cum drachma et re multiplico, produ- 
cuntur autem duae substantiae et duae res, drachmam et dimidium coaequantes. 
Eas igitur ad vnam conuerto substantiam, hoc est, vt ex omni re suam auferam 
medietatem. Dico ergo, substantia et res tres quartas drachmae coaequant. 
Operare nunc cum his quemadmodum tibi iam diximus. 

io Quaestio decima tertia 

Substantiam cum eius duabus tertiis multiplico, et hunt quinque. 

Expositio talis est, vt rem cum duabus tertiis rei multiplicem, et producuntur 

duae tertiae substantiae quinque coaequantia. Comple igitur f substantiae cum 

similitudine earum medietati, et veniet substantia. Similiter comple 5 cum sua 

is medietate, et habebis 7^ quae substantiam coaequant. Ipsam igitur substantiae 

radicem quae cum suis duabus tertiis multiplicata quinarium numerum producit. 

Quaestio decima quarto 

Inter puellas drachmam sic diuido, vt vnicuique earum aequalis particula rei 
contingat ; quibus etiam si vnam insuper puellam adhibuero, illis omnibus par- 

20 ticula primae particulae minus vna sexta aequalis contingit. 

Expositio huius est, vt ipsas puellas cum minutia particulae qua differunt 
multiplicem, postea quod ex multiplicatione excreuerit cum numero puellarum 
postremarum multiplicem atque tandem productum hoc in id quo puellae priores 
a posterioribus differunt, diuidam, et complebitur substantia. Puellas igitur 

25 priores, vnam rem scilicet, cum sexta, quae est inter eas, multiplico, et producitur 
£ rei. Deinde multiplico earn cum numero puellarum postremarum, quae sunt 
res et vnum, et producitur sexta substantiae et sexta rei in drachmam diuisa, 
drachmae aequalis. Substantiam igitur compleo, id est, substantiam cum senario 
numero multiplico, et sic substantiam et radicem habebo. Drachmam etiam cum 

30 sex drachmis multiplico, et venient substantia et radix, sex drachmas coaequantes. 

2. per 2 V ; in duo inequaliter D. 18. unum dragma. uniuscuiusque. 

3. constituas V. 19. insuper om. His pro illis D . 

4. et pro etiam. diuisi V ; diuisum D. 20. minus sexta tocius. contingat. 

5. et re om. V. et {om. V) fiuntque. 21. ut primas puellas V; om. D. cum om. 

6. autem om. V. que D. 

7. conuerte. et hoc. 22. Ac postea. numerum V. 

8. Dicam. tres om. 23. posteriorum D. multiplicabo (multitudo 

9. Oppone ergo. hiis D. D). Postea quod ex hac multiplicatione collectum 

11. sic multiplico, ut fiant. fuerit super illud quod ( + quod V) inter puellas 

12. talis om. primas est {om. D) et posteriores diuido. 

13. unius substantie 5. 24. diuidunt C. et + tunc. Tunc etiam 

14. erit. Et similiter. unum pro v D puellas primas que sunt res in sexta. 
del. 25. multiplica. 

15. substantia pro medietate D. quae om. 26. radicis. 
substantias D. coequantem. Eius ergo radix 27. 5 pro vnum. radicis. 

est res que quando in. 28. coequalis. ipsam pro substantiam. 2 

16. fuerit ad quinarium excrescet numerum. 29. ergo in 7 pro etiam cum sex. 
C add. : radix numeri quae est VtI componet. 30. erunt. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 119 

Twelfth Problem 

I divide a unit and one-half in such a way that the larger part shall be 
double the smaller. 1 

Explanation. I let the greater part be to the lesser as one is to x ; whence 

1 say divide 1^ units by one unit + x, giving 2 x. Therefore I multiply 

2 x by one unit + x, giving 2 x 2 + 2 x, which is equal to i\ units. I reduce 
this, therefore, to one square, that is, of each thing I take the half. I say 
then that x 2 + x is equal to f of a unit. You now proceed in the manner 
which we have explained. 

Thirteenth Problem 2 

I multiply a square by two-thirds of itself and have five as a product. 3 
Explanation. I multiply x by two-thirds x, giving § x 2 , which equals 

five. Complete § x 2 by adding to it one-half of itself, and one x 2 is obtained. 

Likewise add to five one-half of itself, and you have 7^, which equals x 2 . 

The root of this, then, is the number which when multiplied by two-thirds 

of itself gives five. 

Fourteenth Problem 

I divide a unit among girls in such a way that each one receives the same 
fractional part of the thing. Now if I add one girl to the number, each 
receives for her part one-sixth (of a unit) less than before. 4 

Explanation. I multiply the number of girls at the first by the fractional 
part representing the difference. Then I multiply this product by the 
second number of girls, and finally I divide this product by the difference 
between the first and second number of girls. This completes the given 
quantity. 5 Hence I multiply in this instance one x, representing the first 
number of girls, by the difference between the two amounts, ^, and ^ x is 
obtained. Then I multiply this product by the second number of girls, 
which is 1 -f x, and ^ x 2 + | x is obtained ; this being divided by a unit 
equals a unit. I complete the square, that is, I multiply the square by six, 
and I have x 2 + x. Likewise I multiply the unit by six units, giving x 2 + x 

1 Strictly, i\ — x = 2 x, whence x = \. The English translation of this problem is adapted by 
Rosen (p. 59) to conform to the solution in the explanation, and this follows closely our explana- 



tory text. The equation which is given by Rosen is : ? — = 2 x, and to this our text leads. 

1 am indebted to Professor W. H. Worrell for the following precise translation of the Arabic 
text of the passage : "If it is said to divide a unit and a half between a man and a part of a man, 
then the man has received the like of the fraction." The Libri text, pp. 285-286, varies from 
this only in stating that the man receives double that which the fractional part (of a man) receives. 

2 The following problem precedes in the Arabic text : (x — j x — \ x - 4)* = x +12. 

3 x • f x =5, whence x = ^15/2. 

4 Rosen, pp. 63-64; Libri, p. 286. Our text is faulty. The problem is =$• 

6 Latin substantia, 'square,' obviously an error. 



120 LIBER ALGEBRAE ET ALMUCABOLA 

i Radices igitur per medium diuido, et earum medium cum suo aequali multiplico, 
et quod producitur ad 6 adiicio, atque huius aggregati radicem accipio, vnde tan- 
dem medietatem radicum subtraho. Nam hoc quod residuum fuerit, numerum 
puellarum priorum designabit, et ipsae sunt duae. 

5 Quaeslio decima quinta 

Si ex substantia quatuor radices subtraxero, ac postea tertiam residui accepero, 
et haec ipsa tertia quatuor radicibus aequalis fuerit, tunc substantia in ducenta 
quinquaginta sex terminatur. 

Expositio huius est vt scias, quod cum tertia residui, posterioris scilicet, aequalis 
10 sit quatuor radicibus, residuum prius duodecim radicibus aequale erit. Adde 
igitur illas super quatuor, et venient 16 radices, quae sunt radix substantiae. 

Quaestio decima sexta 

Ex substantia tres radices subtraho et postea quod residuum est cum suo aequali 
multiplico, sitque tota multiplicationis summa aequalis substantiae. 
15 Manifestum est, quod residuum sit radix, quam quaternarius adimplebit 
numerus ; substantiam verb numerus 16 component. 

Hae igitur sunt sedecim quaestiones quae ex prioribus nasci videntur, vt dixi- 
mus. Quicquid igitur iuxta artem restaurationis et oppositionis multiplicare 
volueris, facile per ea quae tradita sunt expedies. 

20 CAPUT RERUM VENALIUM 

Res autem venales et omnia, quae ad ipsas attinent, duobus modis et quatuor 
numeris disponuntur. Horum igitur numerorum primus iuxta Arabes, Almuzahar, 
qui et primus propositus nominatur. Secundus vero, Alszian, id est secundus per 
primum dinotus, appellatur. Tertius Almuhen id est ignotus. Quartus Alche- 
25 mon, id est per primum et secundum dinotus. Hi porro quatuor numeri sic dis- 
ponuntur, vt eorum primus, qui est Almuzahar vltimo, qui est Alchemon, oppo- 
natur. Horum autem quatuor numerorum tres semper noti ac certi ponuntur, 
quatuor vero numerus ignotus ponitur, et is ipse est cum quo quantum inquiritur. 

2. atque earn multiplicationem super 6 adicio, 18. QuotquotV; Quidquid D. 

et huius summe. ex qua pro vnde tandem. 19. per earum aliquam illud multiplicatum 

3. abstraho. Nam + et. reperies. 

4. et sunt due V; om. D. 20. Titulum om. 

6. ex om. D. et postea; et C in ras. ter- 21. Item res omnis venales D. ad om 
tiam om. D. ipsis. 

7. si pro et haec. et pro in D. 22. vero pro igitur. Almuzarar sine Almu- 

9. quoniam pro quod cum. posterioris sarar ubique V ; Almuzaar siue Almusaar D. 
scilicet om. 23. id est pro qui et. nuncupatur D. Alter 

10. et quam residuum simile sit (est D) 12 Alszarar ubique V ; Alzazar siue Alszazar D. 
radicibus. 24. Almuthemen ubique V; Almute siue 

n. quatuor + radices. Almuthemon D. Althemen ubique V; Altemon 

13. ac pro et. est + et D. siue Althemon D. 

15. Unde manifestum est, quod illud residuum 25. Sed et hii 4. 

similiter sit radix et quod substantia sit 4 et fiunt 27. eciam. 

(fuerit D) 16 dragmata. 28. numerus om. ponitur + et incertus. 

17. Hec. ex 6 primis. vt + iam. ipse est ille. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 121 

equal to 6 units. I take one-half of the roots and I multiply the half by 
itself. I add the product to 6, and of this sum I take the root. The re- 
mainder obtained after subtracting one-half of the roots will designate the 
first number of girls, and this is two. 

Fifteenth Problem 

If from a square I subtract four of its roots and then take one-third of the 
remainder, rinding this equal to four of the roots, the square will be 256. 1 

Explanation. Since one-third of the remainder is equal to four roots, 
you know that the remainder itself will equal 12 roots. Therefore add this 
to the four, giving 16 roots. This (16) is the root of the square. 

Sixteenth Problem 

From a square I subtract three of its roots and multiply the remainder 
by itself ; the sum total of this multiplication equals the square. 2 

Explanation. It is evident that the remainder is equal to the root, 
which amounts to four. The square is 16. 

These now are the sixteen problems which are seen to arise from the 
former ones, as we have explained. Hence by means of those things which 
have been set forth you will easily carry through any multiplication that 
you may wish to attempt in accordance with the art of restoration and 
opposition. 

CHAPTER ON MERCANTILE TRANSACTIONS 3 

Mercantile transactions and all things pertaining thereto involve two 
ideas and four numbers. 4 Of these numbers the first is called by the Arabs 
Almuzahar and is the first one proposed. The second is called Alszian, and 
recognized as second by means of the first. The third, Almuhen, is unknown. 
The fourth, Alchemon, is obtained by means of the first and second. Further, 
these four numbers are so related that the first of them, the measure, is 
inversely proportional to the last, which is cost. Moreover, three of these 
numbers are always given or known and the fourth is unknown, and this 

1 Rosen, p. 66; Libri, p. 296. \ (x 2 — 4 x) = 4 x. 

In the Arabic text these two problems precede : x 2 . 3 x = 5 x 2 and (x 2 — \x 2 ) . 3 x = x 2 . 

2 Rosen, p. 67 ; Libri, p. 296. (x 2 — 3 x) 2 = x 2 , whence x 2 — 3 x = x. 

The problem, x + V x 2 - x = 2, precedes. This is one of two problems given in the German 
excerpt of 1461 from the algebra of Al-Khowarizmi (Gerhardt, M onatsbericht d. Konigl. Akad. der 
Wissenschaften zu Berlin, 1870, pp. 142-143). 

3 The famous 'rule of three' is the subject of discussion in this chapter. 

4 The two ideas appear to be the notions of quantity and cost ; the four numbers represent 
unit of measure and price per unit, quantity desired and cost of the same. These four technical 
terms are al-musa 'ir, al-sir, al-thaman, and al-ntulhammin, and further al-maqul; see p. 44. 



122 



LIBER ALGEBRAE ET ALMUCABOLA 



i Talis igitur ad hanc artem regula datur, vt in omni huius inquisitione tres numeri, 
qui noti ac certi positi sunt, considerantur, quoniam eorum duo semper inter se 
oppositi inueniuntur. Horum igitur duorum vnus cum altero multiplicands, 
atque multiplications productum in notum tertium ac certum positum, qui 

5 ignoto opponitur, diuidendum erit. Nam quod ex diuisione exierit, erit numerus 
de quo dubitatur, et ipse ei numero opponitur in quern facta est diuisio. Sed ne 
hanc artem aliquis error incurrere arbitretur, tale damus exemplum. 

De primo modo 

Decern pro sex, quot pro quatuor ? 

10 Vide nunc, quo modo, pro eo vt diximus, praefati numeri disponuntur. Nam 
quando 10 dixisti, numerum Almuzahar pronunciasti ; et quando pro 6 dixisti, 
Alszian protulisti ; et quando quot dixisti, numerum Almuhen siue Magol, id est 
ignotum, pronunciasti ; et quando pro 4 dixisti, numerum Alchemon edidisti. 
Vides igitur quod eorum tres, id est 10, 6 et 4, noti et certi sint, de quarto vero 

15 adhuc ignoto, dubitetur. Si igitur ad regulam prius datam respexeris, primum 
cum vltimo,id est 10 cum 4,multiplicabis,sunt etenim oppositi numeri, noti quoque 
ac certi, et quod ex multiplicatione excreuerit, id est 40, in alterum numerum 
notum ac certum, qui est Alszarar, id est in 6 oportet diuidere, et exeunt 6 et | 
vnius, numerum ignotum designantes. Et hie numerus numero senario, qui 

20 Arabice Alszarar nominatur, est oppositus. 



De secundo modo 

Secundus modus huius artis est, vt dicas, decern pro 8, pro quot quatuor ? 

Decern igitur sunt Almuzahar, qui videlicet numero Almuhen ignoto, cum quo 
quantum acquiritur, est oppositus, et 8 designat numerum Alszarar, qui numero 
25 Alchemon, qui sunt 4 opponitur. Vnum igitur duorum numerorum cognitorum 
atque oppositorum cum altero multiplica, id est 4 cum 8, et producentur 32. Haec 
ergo 32 in tertium cognitum numerum 10, qui est Almuzahar, diuide, et exeunt 
3^, qui numerum Alchemon designant, quique ei numero in quern diuiditur est 
oppositus. 



1. quoque pro igitur. hac D. 

2. certe V. quoque pro quoniam D. ad- 
inuicem pro inter se. 

3. inuenientur. unus est ; unius C. 

4. et eorum pro atque. per. atque V. 
certe D. 

5. diuidenda V ; diuidendus D. 

6. dubitabatur. per. 

7. errorem. 

8. Titulum om. et sic infra. 

9. Secundum ergo primum modum, sic dicas, 
10 pro 6, quot pro 4 ? 

10. secundum quod diximus. 
n. quoniam pro quando. 

12. Alszaran C. Magul V; Magulum D. 

13. nunciasti. Almuhemen C. 

14. qualiter pro quod. 10 + et D. ac 
certi ponuntur et quomodo de quarto, adhuc in- 



cognito, dubitatur. 

16. omnino pro etenim D. numeri om. 
quoque om. D. 

17. per alterum (alium D). numerum om. D. 

18. per. erunt ( 4- 7 siue D). 

19. numeri {om. D) almagul (maghulis D) id 
est incognitum. Et huius ( + -modi D). 



23. Almuszarar C. videlicet om. D. 



24. inquiritur V. 

25. Primum D. 



quo 



duorum om. D. 

26. in alterum (alium D). fient. Hunc 
ergo numerum per alterum (alium D) cognitum ; 
Hunc C sed del. 

27. Almuzahar + id est 32 per 10 diuide. 
Almuszarar C. erunt V; exeund D. 

28. qui om. numeri. designantes, qui 
videlicet eo numero per quern. est om. D. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 123 

is indicated by the question as to the quantity. The rule of this kind of 
problem is to consider the three quantities which are given or known, of 
which two are always found to be inversely proportional to each other. 
These two are to be multiplied one by the other and the product of the mul- 
tiplication is to be divided by the third number, likewise known, which is 
inversely proportional to the unknown. Now the quotient of this division 
will be the number which is sought, and it is inversely proportional to the 
number by which you divide. But lest some error be made in this type 
of problem we give an example of it. 1 

A Problem of the First Type 

Ten for six, how many for four ? 

Observe now in what manner the given numbers are related, according to 
what we have said. For when you say " ten," you give the measure, and when 
you say "for six," you state the price. When you ask, "how much ?" you 
give the unknown, called Almuhen or Magul, and saying "for four," you 
mention the cost. You note further that three of these, that is, 10, 6, and 
4, are known and definite numbers, and the question is concerning the 
fourth or unknown number. If now you take account of the rule given, you 
multiply the first by the last, that is, 10 by 4, for they are the known and 
definite numbers which are inversely proportional to each other. It is 
necessary to divide the product, that is, 40, by the other known and definite 
number, that is, the measure, which is 6. This gives 6f , designating the 
unknown number. This number is inversely proportional to the number 
six, which in Arabic is called Alszarar. 

A Problem of the Second Type 

An example of the second type of such problems is given by the question, 
" ten for eight, what is the cost of four ? " 

Ten now is the measure which is inversely proportional to the unknown 
cost, and eight designates the price which is inversely proportional to the 
quantity, 4. Therefore multiply one of the two known and inversely pro- 
portional numbers by the other, that is, 4 by 8, and you will have 32. Divide 
32 by the third known number, 10, which is the measure. This gives 3^, 
designating the cost which is inversely proportional to the number by which 

1 As we have noted in the introduction, page 44, this paragraph is not carefully translated by 
Robert of Chester, and he added the last sentence. The corresponding passage in the Libri ver- 
sion {op. cit., pp. 268-269) is entitled, Capitulum conventionum negotiator um, and begins as fol- 
lows : Scias quod conventiones negotiationis hominum, que sunt de emptione et venditione et 
cambitione et conductione et ceteris rebus, sunt secundum duos modos, cum quattuor numeris 
quibus interrogator loquitur. 

Leonard of Pisa {Liber abbaci, p. 2 and p. 83) follows somewhat the terminology of this ver- 
sion in his discussion. The title (p. 2) is given : De emptione et venditione rerum nenalium et 
similium. The section opens (p. 83), as follows: 'In omnibus itaque negotiationibus quattuor 
numeri proportionales semper reperiuntur, ex quibus tres sunt noti, reliquus uero est ignotus.' 



124 



LIBER ALGEBRAE ET ALMUCABOLA 



i His igitur duobus modis omnia quae venalia dicuntur, absque omni errore pos- 
sunt tractari, si deus voluerit. 

Quaesiio sen interrogatio vltima 

Homo in vinea 30 diebus pro decern denariis conducitur, ex quibus operatus est 
5 sex diebus, quantum ergo precii totius debet accipere ? 

Expositio huius est, quoniam manifestum est quod sex dies quintam partem 
totius temporis adimplent, et quod hoc quod ex precio ei contingent, sit secundum 
quod ipse ex toto tempore, scilicet ex 30 diebus, sit operatus. Quod autem dixi- 
mus, sic exponitur. Quoniam quando mensem id est 30 dies dixisti Almuzahar 
10 protulisti ; et quando 10 dixisti, Alszarar ; quando verb 6 dies, Almuhen pronun- 
ciasti, quando deinde dixisti quantum precii ei contingent, Alchemon nunciasti. 
Multiplica ergo Alszarar, qui sunt 10, cum Almuhen qui ei opponitur, id est cum 6, 
et quod ex multiplication excreuerit, 60 scilicet, in 30 Almuzahar diuide, et exeunt 
2 denarii, et ipsi erunt Alchemon, id est pars quae homini contingent. 

15 Hoc igitur modo quicquid huius tibi propositum fuerit, ex rebus venalibus siue 
ponderibus, seu ex omnibus quae ad haec attinent, agendum erit. 

Laus deo praeter quern non est alius. 

Finis libri restaurationis et oppositionis numeri quern Robertus Cestrensis de 
Arabico in latinum in ciuitate Secobiensi transtulit, [Era] anno millesimo centesimo 
20 octogesimo tertio. 



I. omni om. D. 4. dragmata. 

5. precium. 

6. quod iam pro quoniam. quoniam pro 
quod. partem om. 

7. mensis pro totius temporis. quod l om. D. 
eius pro ei V. 

8. ex mense. scilicet ex 30 diebus om. 

9. Almuszarat C. 

10. dixisti 10 + dixisti 
dies. 

II. Et quando dixisti. 
12. id est 10 V; 10 D. 



et quando 6 dixisti 



6 + et fient 60. 



13. 60 scilicet om. super 30, id est super 
Almusarar (Almuzaar D) et erunt duo dragmata et 
ipsa. 

14. erunt om. D. contingit D. 

15. Hoc ergo modo quotquot tibi huius positum 
V; Huius ergo modi quicquid nisi positum D. 
venalibus + ac aliis. 

16. his V ; hiis D pro ad haec. est. 
17-20. om. D. 

17. Deo gracias V. 

18. Explicit V. 

19. Sectobiensi V. Era M C lxxxiii V. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 125 

you divided. According to these two methods it is possible to treat all 
commercial problems, without error if God will. 

The Last Problem or Question 

A man is hired to work in a vineyard 30 days for 10 pence. He works 
six days. How much of the agreed price should he receive ? 

Explanation. It is evident that six days are one-fifth of the whole time ; 
and it is also evident that the man should receive pay having the same 
relation to the agreed price that the time he works bears to the whole time, 
30 days. What we have proposed, is explained as follows. The month, 
i.e. 30 days, represents the measure, and ten represents the price. Six 
days represents the quantity, and in asking what part of the agreed price is 
due to the worker you ask the cost. Therefore multiply the price 10 by 
the quantity 6, which is inversely proportional to it. Divide the product 
60 by the measure 30, giving 2 pence. This will be the cost, and will 
represent the amount due to the worker. 

This, then, is the method by which all proposed problems concerning 
commercial transactions or weights and measures and all related problems 
are to be solved. 

Praise be to God, beside whom there is no other. 

Here ends the book of restoration and opposition of number which in 
the year n 83 (Spanish Era) Robert of Chester in the city of Segovia trans- 
lated into Latin from the Arabic. 



126 LIBER ALGEBRAE ET ALMUCABOLA 



i REGULE 6 CAPITULIS ALGABRE CORRESPONDENTES 

Prima. Quando numeris assimilantur "2^_ committatur $> per 7^_ et productum 
ostendit quesitum. 

2 a . Quando <f> assimilantur ^- committatur <j> per ^f" et radix propositi (producti) 
s ostendit quesitum. 

3 a . Quando ^ assimilantur % committatur [per <P] per ^* et productum ostendit 
quesitum. 

4 a . Quando <f> assimilantur 7£_ 7 <^> ^ 7 "^ debent per £- committi ; radix 
mediari, medium in se duci, productum numero addi. Radix tocius aggregati 
io minus medietate 7^_ ostendit quod queritur. 

5 a regula. Quando *^. assimilantur <f> 7 J~, ^ 7 4* debent per % committi ; "%{_ 
mediari, medium in se duci, a producto (f> subtrahi, ~%l residui a medietate 7^ 
tolli et hiis residuum ostendit quesitum. Quod si 73^ residui a medietati 7^ 
subtrahi non potest, addere licet eandem. 

15 6 a . Quando %- assimilantur cj> 7 *^L- Hec debent per J- committi ; 75^ mediari, 
medium in se duci, productum <p addi. Radix aggregati plus medietate 72^ os- 
tendit quod queritur. 

1-17. Regule . . . queritur add V. 



THE BOOK OF ALGEBRA AND ALMUCABOLA 127 

RULES CORRESPONDING TO THE SIX CHAPTERS OF 

ALGEBRA 

First. When roots are equal to a number, divide the number by (the 
number of) the roots, and the quotient represents the desired quantity. 1 

Second. When squares are equal to a number, divide the number by 
(the number of) the squares, and the root of that which you obtain repre- 
sents the desired quantity. 2 

Third. When roots are equal to squares, divide (the number of) the roots 
by (the number of) the squares, and the quotient represents the desired 
quantity. 3 

Fourth. When a number is equal to the sum of squares and roots, divide 
by (the number of) the squares. Take one-half of (the number of) the 
roots after the division and multiply it by itself. To this product add the 
number. The root of this sum less one-half of the number of roots, repre- 
sents that which is sought. 4 

Fifth. When roots are equal to a number and squares, divide the roots 
and the number by (the number of) the squares. Take one-half of (the 
number of) the roots after the division, and multiply it by itself. From this 
product subtract the number ; the root of the remainder subtracted from 
one-half of (the number of) the roots is the desired quantity. 5 But if it is 
not possible 6 to subtract the root of the remainder from one-half of (the 
number of) the roots, it is permissible to add the same. 

Sixth. When squares are equal to a number and roots, on this side divide 
by (the number of) the squares. Take one-half of (the number of) the roots 
after the division, and multiply it by itself. To this product add the 
number ; the root of this sum plus one-half of (the number of) the roots 
represents that which is sought. 7 

1 ax = n ; x = a/n. 

2 ax 2 = n; x 2 = a/n ; x = *a/n. 

3 ax 2 = bx ; x = b/a. 

4 ax 2 + bx =n; x = ^(b/ia) 2 + n/a -b/2 a. 

5 ax 2 +n = bx; x = b/2 a ± ^(b/2 a) 2 — n/a. 

6 It is always possible to subtract the root of the remainder here from one-half of the num- 
ber of the roots, even from the standpoint of the mathematician of the fourteenth century who 
made this table of rules, for the remainder would always be positive, since the square root of 
(b/2 a) 2 — n/a is always less than b/2 a. 

1 ax 2 =bx + n ; x = b/2 a + ^{b/2 a) 2 + n/a. 

The six rules with similar algebraic symbolism are found also in Codex Dresdensis, C. 8o m ; 
see Wappler, Programm, loc. cit., p. n. 



x ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

Combinationem triplicem in censibus reperimus, aut enim census et radices 
aequantur numeris, aut census et numeri aequantur radicibus, aut vero radices 
et numeri censibus aequantur. 

5 Exemplum combinationis primae 

Census et octo radices eius 33 denariis aequipollent. Quaeritur ergo, quantus 
est census ? 

Respondetur : 9. 
Regula est, vt productum ex medietate radicum cum seipsa multiplicata, 
10 scilicet 16, ad numerum, 33, addamus et resultant 49, cuius radix est 7. Atque 
ab hac radice medietatem radicum, 4 scilicet, subtrahamus, et manebunt 3, quae 
sunt radix census. Est igitur census 9. 

Item si dicatur, duo census et octo radices sunt aequales quadraginta duobus 
denariis. Reduc quaestionem ad vnum censum sic. Si 2 census et 8 radices sunt 
1542 denarii, igitur vnus census et 4 radices sunt 21. Operando igitur per illud ex 
duplicatione et peruenies in fine ad intentum. 

Item si dicas, medietas census et 4 radices sunt aequales 16^ denariis. 
Reduc quaestionem ad integrum censum, et dimidia totum, et patebit in fine 
intentum. 

20 Exemplum combinationis secundae 

Census et 1 5 denarii valent octo radices, quantus igitur est census ? 

Respondetur: 9, vel 25. 
Regula. Due medietatem radicum in se, et fiunt 16. Ex quibus subtrahe 15 
denarios et manet vnum, cuius radicem, scilicet vnum, subtrahe a medietate radi- 
25 cum, et manent 3, quae sunt radix census. Vel illud vnum adde ad medietatem 
radicum, et veniunt 5 quae sunt radix census. Patet igitur, quaestionem secun- 
dum vtranque partem esse veram, ideo oportet vt eas determines distinctiue. 

Corollarium 

Notandum si quadratum medietatis radicum minus fuerit quam est vltimus 
3°numerus propositus, vt si dicatur, census et 15 numeri sunt aequales 6 radicibus, 
tunc quaestio impossibilis erit. Si vero fuerit vltimo proposito numero aequalis 
vt si dicatur, census et 9 denarii valent 6 radices, tunc medietas radicum est radix 
census. Si autem quaestio venerit cum pluribus censibus, aut paucioribus vno 
censu, reduc ad vnum censum, et operare vt supra dictum est. 

35 Exemplum combinationis tertiae 

Quatuor radices et 12 denarii censui aequipollent. 

Reduc, hoc est multiplica, medietatem radicum in se, et fiunt 4. Quibus adde 

128 



ADDITIONS TO THE ALGEBRA 129 

SOME ADDITIONS IN EXPLANATION OF THE ALGEBRA 

We find a triple combination of squares, 1 namely squares and roots equal 
to numbers, squares and numbers equal to roots, and finally roots and 
numbers equal to squares. 

An Illustration of the First Type 

A square and 8 of its roots equal 33 units. The question is, what is the 
square ? 2 Answer : 9. 

The rule is to add to the number the product of half of the root multiplied 
by itself, that is 16 to 33, and 49 is obtained, of which the root is 7. Now 
from this root we subtract one-half of the number of the roots, namely 4, 
which leaves three. This is the root of the square, and the square is g. 3 

Likewise if you are given that two squares and eight roots are equal 
to 42 units. Reduce in the following manner to one square. If two squares 
and eight roots are equal to 42 units, then one square and four roots are 
equal to 21. You operate therefore by duplication and arrive in the end 
at the desired result. 4 

Likewise if you are given one-half a square and four roots equal to 16^ 
units. Reduce the problem to a whole square, take one-half of the whole, 
and in the end you arrive at the desired result. 5 

An Illustration of the Second Type 

A square and fifteen units equal eight roots. What then is the square ? 

Answer : 9 or 25. 

Rule. Multiply one-half of the roots by itself, and 16 is obtained. From 
this subtract the 15 units and one is obtained, of which you subtract the 
root, namely one, from one-half of the roots. This gives 3 as the root of 
the square. Or add that one to one-half of the roots and 5 is obtained as 
the root of the square. It is clear then that the problem is solved by each 
value, and hence it is necessary to determine separately both solutions. 6 

Corollary 

It should be noted that if the square of half of the roots is less than the 
proposed number, as, for example, a square and the number 15 equal to 
6 roots, then the problem is not solvable. 7 If the square of one-half of the 
roots is equal to the given number, as, for example, a square and 9 units 
equal to 6 roots, then the half of the roots is the root of the square. More- 
over, if a problem is proposed involving more than one square or less than 
a whole square, reduce to one square, and operate as indicated above. 

1 Scheybl changes from substantia to census; see p. 69, n. 1. 

2 Textus in the margin precedes the statement of each problem, and Minor the solution. 



13° 



ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 



1 12 denarios et veniunt 16, cuius radicem 4 adde ad medietatem radicum, et 
resultant 6, quae sunt radix census. Omne etiam, quod aut maius est vno censu 
aut minus, reducas ad vnum censum, atque deinde operare vt dictum est. 



Proponit causas trium combinationum 

Causa combinationis primae'est haec : 

Sit quadratum a b quod censum significet, sitque huius quadrati latus ipsius 
census radix. Et quoniam ductus vnius lateris in numerum radicum, quantitas 



B 



radicum sumptorum existit. Applicetur igitur ad vnumquodque latus quadrati 
quarta pars numeri radicum, scilicet duae radices lateri vni, et duae alii, et ita 
10 deinceps. Resultat autem sic quadratum aliud, praeter quatuor eius angularia 
quadrata, quorum singulorum vnumquodque latus duo, hoc est quartam partem 
numeri radicum, continebit. Ex ductu igitur quartae partis numeri radicum in se 
quater, resultabunt ilia quatuor parua angularia quadrata, atque haec eadem 



A 
















B 



A 
















B 



d 



resultant ex multiplicatione medietatis numeri radicum in se semel. Si igitur 
15 productum ex medietate numeri radicum in se multiplicata, addatur ad censum et 
ad radices, quadratum cuius vnius lateris quantitas aequalis est lineae c d de- 
scribetur. Hoc autem latus excedit latus census in medietate numeri radicum, 
quia per quartam partem in vno extremo, et per quartam partem in altera huius 
census latus auctum est. Ideo subtracta medietate numeri radicum a tota linea 
20 c d manebit quantitas radicis census quae quaerebatur. 

Ilia probatio procedit ex propositione quarta secundi Euclidis. 



ADDITIONS TO THE ALGEBRA 131 

An Illustration of the Third Type 

Four roots and twelve units are equal to a square. 

Find, that is to say multiply, half of the roots by itself, and four 

is obtained. To this add the 12 units, and 16 is obtained. Add the root p- 

' 130 

of this, 4, to one-half of the roots and 6 appears as the root of the square. 
Also whatever is given either greater or less than one square, reduce to one 
square and operate as indicated above. 1 

The Explanation of the Three Types 

The explanation of the first type is as follows : let the square AB repre- 
sent x 2 . Then the side of this square will represent the root of x 2 . or x. 
When one side of this square is multiplied by the number of the roots, the 
quantity of the assumed roots is represented. Hence let there be applied to 
each side of the square the fourth part of the number of the roots, namely 
two roots to one side, two to another, and so on. Thus another square 
appears, lacking only four small corner squares of which each has every side 
equal to two, that is the fourth part of the number of the roots. There- 
fore by multiplying the fourth part of the number of the roots by itself four 
times, these four small corner squares are obtained, and this same result is 
obtained by the multiplication of half of the roots by itself once. If then the 
product of half the roots multiplied by itself be added to the square and 
roots, the square is formed of which one side is equal in value to the line cd. 
Moreover, this side exceeds the side of the unknown square by the half of 
the roots, since the side of the unknown square has been extended by the 
fourth part at one extremity and by the fourth part at the other. Hence 
by subtracting the half of the number of the roots from the whole line cd 
the value appears of the root of the square which is sought. 

The proof of this follows from the fourth proposition of the second book 
of Euclid. 2 

3 x 2 + 8 x = 33 ; x 2 + 8 x +16 = 49 ; x +4 = 7 ; x = 3 ; x 2 = 9. 

* 2 x 2 + 8 x =42; x 2 + 4 x = 21 ; x + 4 x +4 =25; x +2 =5; x =3. 

6 \ x 2 + 4 x = 165 ; whence x 2 + 8 x = 33, as above. 

6 x 2 + 15 = 8 x; x 2 — 8 x +16 = 1 ; x — 4 = =*= 1 ; x = 3 or 5. Both roots are positive, 
and so two solutions are given. 

1 x 2 + n = bx; x = b/2 ± V{b/2) 2 - n. 

The roots are imaginary if {b/2) 2 is less than n, as in the illustration which Scheybl gives; 
similarly, the two roots are equal to each other and each equal to one-half the coefficient of the 
roots if the square of this quantity is the same as the given constant, that is {b/2) 2 = n. 

1 4 x + 12 = x 2 ; x 2 — 4X+4=i6;x— 2 =4;* =6. 

2 Euclid II, 4, following Heath, The Thirteen Books of Euclid's Elements: "If a straight line 
be cut at random, the square on the whole is equal to the squares on the segments and twice 
the rectangle contained by the segments." In modern notation, {a + b) 2 = a 2 + b 2 + 2 ab. 



132 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

i Aliter sic 

Sit census quadratum a b, cuius vni laterum applicetur medietas numeri radicum, 
et alteri eius lateri applicetur medietas numeri radicum altera, sit autem primum 
additum a c, secundum vero a d. Ad perficiendum igitur quadratum c d, deficit 

s quadratum quod vocetur b e, cuius latus medietati numeri radicum est aequale. 
Patet igitur causa operationis ; nam ex ductu medietatis numeri radicum in se, 
resultat quadratum b e, et residuum est census cum adiectis radicibus. Vnde cum 
illud magnum quadratum notum sit, et eius radix nota, per subtractionem radicis 
quadrati b ea. radice quadrati a e, necessario radix census manebit. 

10 Sequuntur figurae geometricae. 

d a\ 1 \d 



a 







- 


6 



Exemplum combinationis secundae 

Sit census quadratum abed, cuius lateri b c applicabo parallelogrammum cb ef, 
et ponam ipsum 15. Totum igitur parallelogrammum da e f est census et 15 
denarii, et continet octo radices census. Diuidam ergo lineam d f aequaliter in 

15 punctum g et erigam super latus g d quadratum g k m d, et protraham c b vsque ad / 
et ponatur litera h in locum vbi g k secat a e lineam. Et quoniam d b est quadra- 
tum, cum ideo ex structura et propositione sexta secundi libri Euclidis, b k quad- 
ratum sit, duo parallelogramma gb etb m inter se aequalia erunt ; mox deinde, per 
communem quandam noticiam, g a et c m aequalia. Atque tandem cum g a ex 

20 propositione 36 libri primi Eucli[dis], g e parallelogrammo aequale sit, c m eodem 
parallelogrammo g e aequale erit. Subtractis igitur quindecim, hoc est gnomone 
I d h k quadrato g m quod est sedecim, manet vnitas quadratum b k. Et quia 
notum, latus igitur vel radix, quae est linea I b, erit nota. Sed et tota / c nota est, 
manet igitur, post subtractionem I b &b I c linea, et b c linea nota. Et in hoc casu, 

25 quo punctus g cadit in parallelogrammum applicatum censui, sequuntur figurae 
geometricae. 

m I 




a 



d 



k 



ADDITIONS TO THE ALGEBRA 133 

Another Demonstration 

Let the square ab represent x 2 . To the side of it apply one-half the given 
number of roots and to another side of it apply the other half of the number 
of the roots. Let the first addition be represented by ac and the second by 
ad. To complete the square cd the square called be is lacking and the side 
of this is equal to one-half the number of the roots. The reason for our 
procedure (in halving the roots) is now apparent ; for by multiplying the 
half of the number of roots by itself we obtain the square be, and the re- 
mainder is x 2 together with the added roots. Now since the larger square 
is known, and the root of it is known, then by subtracting the root of the 
square be from the root of the square ae, necessarily the root of our unknown 
square remains. 

The geometrical figures follow. 1 

An Illustration of the Second Type 

Let the square abed represent x 2 , and to its side be apply the parallelogram 
cbef, which we take to be equivalent to 15 units. Then the whole rectangle 
daef is equivalent to x 2 and 15 units, thus containing 8 roots of x 2 (Sx). 
Divide now the line df into two equal parts by the point g and erect a square 
gkmd upon the side gd. Extend cb up to a point I (the point of intersection 
with the upper side of the square upon gd) . Mark the point of intersection of 
gk with ae by the letter h. Since db is a square, it follows by our construction 
and by proposition 6 of the second book of Euclid 2 that bk is a square and 
further that the two parallelograms gb and bm are equal to each other. 3 
Further, then, by a well-known axiom 4 ga and cm are equal. Also finally, 
since ga is equal to the parallelogram ge by the thirty-sixth proposition of 
the first book of Euclid, 4 it follows that cm is equal to the same parallelogram 
ge. Hence subtracting 15, that is the gnomon Idh, from the square gm, 
which is 16, one is left as the area of the square bk. Since the area is known, 
the side or root, which is the line lb, is also known. But the whole line Ic is 
known, whence it follows by subtraction of lb from Ic that the line be is 
known. In this instance it is to be noted that the point g falls within the 
parallelogram which is applied to the square. 

The geometrical figures follow. 

1 Compare these and subsequent figures with the corresponding figures on pages 77-89. 

2 Euclid II, 6, following Heath, The Thirteen Books of Euclid's Elements, p. 1. "If a straight 
line be bisected and a straight line be added to it in a straight line, the rectangle contained by the 
whole with the added straight line and the added straight line together with the square on the half 
is equal to the square on the straight line made up of the half and the added straight line." In 
modern algebraic notation, (a + x) x + (a/2) 2 = (a/2 + x) 2 . 

3 By Euclid I, 43, following Heath, loc. cit., "In any parallelogram the complements of the 
parallelograms about the diameter are equal to one another." 

4 Euclid I, 36, following Heath, loc. cit., "Parallelograms which are on equal bases and in the 
same parallels are equal to one another." 



134 



ADDITA QUAE DAM PRO DECLARATIONE ALGEBRAE 



i Sed esto iam, quod punctus g cadat in latus census vel quadrati. 

Sit census vt prius a b c d, et parallelogrammum, numero quindecim, lateri b c 
applicatum c b e f. Tunc super medietatem lineae d f, erigo quadratum g k m f 
et accipio lineam c o aequalem lineae cf et protraho lineam nh o p aequedistantem 
s lineae df. Et quoniam lineae d c et c b inter se aequales sunt, lineae quoque c o 
et c f aequales ; et parallelogrammum igitur d o propter aequalitatem linearum, 
parallelogrammo c e aequale'erit. Sed quia aequalia etiam inter se sunt ex propo- 
sitione 43 primi libri Euclidis, duo supplementa g et m, cum per subtractionem 
aequalium ab aequalibus, ex communi quadam noticia, aequalia relinquantur, 

10 parallelogrammum d h parallelogrammo / e cum quadrato c p aequale erit. Sed 
quia d h aequale etiam est, ex propositione 36 primi, parallelogrammo g cum 
quadrato c p ; haec igitur duo, g parallelogrammum et c p quadratum, prioribus 
duobus I e parallelogrammo et c p quadrato ex communi quadam noticia, aequalia. 
Atque mox, per ablationem quadrati c p communis, g ipsi I e parallelogrammo 

15 aequale. Sunt vero c et I m lineae inter se aequales. Ergo et linea m e lineae 
g c aequalis erit, atque ita aequalis etiam lateri quadrati h I. Cum ergo quadra- 
tum g m sit notum, eo quod eius latus sit medietas numeri radicum, si subtrahantur 
ab eo 15, quae gnomonem valent, quadrato h I circumiacentem et cui etiam paral- 
lelogrammum d aequale est, manebit quadratum h I notum. Vnde et latus eius 

20 notum. Sed quia illud est aequale lineae g c vel m e, latere igitur quadrati hi did 
medietatem numeri radicum m f, latus quadrati g m addito, constituitur linea e f 
nota. Atque haec est radix census. Patet ergo propositum. 
Sequuntur figurae geometricae. 




a 



n 



d 



k 


I 








h 








e 
m 

V 



9 



Exemplum combinationis tertiae 



25 Quatuor radices et duodecim denarii censui aequipollent. 

Causa huius combinationis est haec. Sit census quadratum abed ignotum 
continens quatuor radices et 12 denarios. Ex isto igitur quadrato resecabo 
parallelogrammum beef continens quatuor radices ; manebit ergo parallelo- 
grammum a e continens precise duodecim denarios. Deinde diuidam lineam e c, 

30 numerum radicum, aequaliter in puncto g, et super latus e g erigam quadratum 
g e h k. Super latus etiam d g erigam quadratum g d ml, et secet linea m I lineam 
ef in puncto n. Et quoniam lineae a d et d c inter se aequales sunt, linea quoque 
m d et d g aequales, cum per subabiectionem aequalium ab aequalibus, linea a m 
lineae g c ex communi quadam noticia, aequalis sit. Eadem a m linea propter aequa- 

35 litatem, ex altera quadam noticia, lineae e g, atque mox etiam lineae h k, aequalis 
erit. Item, quia lineae h e et e g inter se aequales sunt, lineae quoque n e et d g 



ADDITIONS TO THE ALGEBRA 135 

Now let the point g fall within the side of the square. 

Let the square as before be represented by abed and the parallelogram cbef, 
equivalent to the given number 15, be applied to the side be. Then upon 
half of the line df erect the square gkmf. Construct the line co equal to 
the line cf and draw the line nhop everywhere equally distant from the line 
df. Since the lines dc and cb are equal to each other (being sides of a square) , 
and further, the lines co and cf equal (by construction), it follows that the 
parallelogram do is equal to the parallelogram ce on account of the equality 
of the sides. But further, since the two supplementary rectangles go and 
om are equal to each other by the forty-third proposition of the first book 
of Euclid, by subtracting equals from equals according to the well-known 
axiom, the remainders are equal, giving the parallelogram dh equal to the 
parallelogram le together with the square cp. But also since dh is equal to 
the parallelogram go plus the square cp by the thirty-sixth proposition of 
the first book (of Euclid), it follows that these two, the parallelogram go 
and the square cp, are equal to the preceding two, the parallelogram le 
and the square cp by another well-known axiom. 1 Whence by subtracting 
the common part, the square cp, go is equal to the parallelogram le ; and in- 
deed the lines co and Im are also equal to each other. Further, the line 
me will be equal to gc, and so also equal to the side of the square hi. 
Since the square gm has a known area, by the fact that its side is one-half 
of the given number of roots, if from it there be subtracted 15 which is repre- 
sented by the gnomon gfm, circumscribed about the square hi and equal, 
as we have just shown, to the parallelogram do, the square hi remains known 
in area ; whence also the side of it is known. But since this side of the 
square hi is equal to the line gc or me, when added to the half of the number 
of the roots mf, a side of the square gm, the line ef is then known. And 
this is the root of the unknown square. The proposed question is solved. 

The geometrical figures follow. 

An Illustration of the Third Type 

Four roots and twelve units are equal to x 2 . 

The explanation of this type is as follows. Let x 2 be represented by 
the unknown square abed which contains 4 roots and 12 units. From this 
square cut off the parallelogram beef containing four roots. The paral- 
lelogram ae consequently will contain precisely 12 units. Then divide the 
line ec, representing the number of the roots, into two equal parts by the 
point g and upon the side eg erect the square gehk. Also upon the side 
dg erect a square gdml, and let the line ml cut the line ef in the point n. 
Since the lines ad and dc are equal to each other, and further the lines md 

1 Following Heath, loc. cit., "Things which are equal to the same thing are also equal to one 
another." 



136 



ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 



1 aequales, cum per subtractionem aequalium ab aequalibus, linea n h lineae d e 
aequalis sit. Eadem n h linea, propter aequalitatem, lineae m n aequalis erit. 
Igitur superficies h I et m f aequales. Duae igitur superficies d n et h I simul 
sumptae, ex communi ilia noticia, si aequalibus aequalia adiiciantur, vni superficiei 

s d f quae 12 continet, aequales erunt. Vnde si numerus 12 addatur ad quadratum 
e k, cuius latus vel radix est linea e g, medietas numeri radicum, resultabit quadra- 
tum d I, cuius latus vel radix est linea d g. Si igitur illi radici addatur medietas 
numeri radicum, quae est linea g c, linea d c, quae est latus census, proueniet. 
Patet ergo propositum. 



10 Sequuntur figurae geometricae. 
/ 




a 



m 



f 



d 





n 


I 
k 


h 







Vel sub alia forma, sic : 

/ 




9 





f 


m 

h 


n 






k 





d 



9 



a 



Sequuntur multiplicatio cum odditis et diminutis 

Sicut idem est multiplicare compositum cum composito, et multiplicare vtram- 
que partem compositi cum vtraque parte compositi, vt sic fiat quadruplex multi- 
is plicatio, scilicet articuli cum articulo et digiti cum articulo, deinde articuli cum 
digito, et quarto digiti cum digito. Comsimiliter potest fieri quadruplex multipli- 
catio, vbi articulus et digitus cum articulo praeter digitum multiplicare debet, et 
similiter vbi articulus praeter digitum cum articulo et digito, vel contra, multiplicari 
debet. Vnde indicimus quadruplicem multiplicationem, atque in omni multi- 
20 plicatione tali praedicta haec regula notanda est. 

Si digiti tarn in multiplicando quam in multiplicante additi cum articulis aut 



ADDITIONS TO THE ALGEBRA 137 

and dg are equal, then by subtracting equals from equals the line am will 
be equal to the line gc by this well-known axiom. On account of this 
equality the same line am will, by another axiom, 1 be equal to the line eg 
and hence also to hk. Further, since the lines he and eg are equal to each 
other and also the lines ne and dg are equal, then by subtracting equals from I3 ' 6 
equals the line nh is equal to the line de. On account of this equality the 
same line nh will be equal to the line mn. Hence the areas hi and mf are 
equal. Therefore the two areas dn and hi taken together, by the well- 
known axiom "if equals be added to equals," will be equal to the single 
area df, which contains 12 units. Whence if the number 12 be added to the 
square ek, whose side or root is the line eg, the half of the number of the 
roots, the square dl is obtained whose side or root is the line dg. If, then, the 
half of the number of the roots, which is represented by the line ge, be 
added to that side (dg) the line dc will be obtained, which is the side of the 
unknown square. The proposed problem is solved. 
The geometrical figures follow. 

[Or in another form, thus] 

Multiplication with Positives and Negatives 2 

Since the multiplication of a composite 3 number by a composite number 
is the same as the multiplication of each part of the one composite by each 
part of the other, so it follows that the multiplication is fourfold, namely 
article by article, digit by article, then article by digit, and fourthly digit 
by digit. Similarly, you may have a fourfold multiplication when an article 
and a digit is to be multiplied by an article less 4 a digit, or the reverse. 5 
It seems desirable to indicate the fourfold nature of this multiplication (by 
some examples), and in every multiplication of this kind this rule is to be 
followed. 

If the digits in the multiplicand as well as in the multiplier are added to 

1 The halves of equals are equal, and the axiom of the preceding note. 

2 Attention is called to a similar use of this expression in the text (p. 32). Johannes de Muris 
in the third book of the Quadripartitum numerorum, doubtless familiar to Scheybl, entitles a 
similar chapter, De multiplication et diuisione additorum et diminutorum (Cod. Pal. Vind. 4770, 
fol. 230 6 ) ; see also Karpinski, The "Quadripartitum numeronim" of John of Meurs, loc. cit., 
p. no. 

3 In this discussion Scheybl uses the more common terms such as "composite numbers," 
"articles," and "digits," instead of "nodes" and "units" as in the corresponding section of 
Robert of Chester's text (pp. 88-96). 

4 praeter is used to suggest the idea of negative, illustrating in fact the Arabic conception of 
negative, namely that the "article" or ten (in this instance) is incomplete by the digit which is 
subtracted from it. 

5 Probably including all four types, (x + a)(x + b), (x + a) (x - b) , (x - a) (x + b) , and 
(x — a)(x — b). That multiplication is commutative was doubtless felt, even though not 
expressed. 



138 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

1 diminuti fuerint in vtroque, tunc multiplicatio quaelibet debet addi. Si autem 
vnus fuerit additus et alter diminutus, tunc ista multiplicatio subtrahi debet a 
productis. 

Sequitur huius rei vel regulae exemplum 

5 Idem est multiplicare 8 cum 17, et multiplicare 10 praeter 2 cum 20 praeter 3, 
et hoc fiet sic. Multiplica 10 cum 20, et resultabant 200. Deinde multiplica 
praeter 2 cum 20, et resultabunt 40, subtrahenda a. 200 et manent 160. Tertio 
multiplica 10 cum praeter 3, et resultabunt 30, subtrahenda a 160 et manent 130. 
Quarto multiplica praeter 2 cum praeter 3 et resultabunt 6 addenda, et veniunt 

10 136. Igitur octo decies septies sunt 136. 



Sequitur huius rei calculus. 



10 praeter 2 
20 praeter 3 



200 praeter 40 
is praeter 30 et 6 

136 et caet. 

Aucta minuta simul minues ; sed caetera iunges. 
Aequantur numero radices censibus ambo, 
In medio minues, alibi quod colliges addes. 

20 Alia tria carmina, quae trium aequationum exempla proponunt 

Census et 8 res 30 valent simul et tres. 
Census cum seno, res quinque valere notato. 
Aequiualent censum res 4 et duodenus. 

Sequuntur nunc in declaratione exempla multiplications alia 

25 Si cum 10 praeter rem debeas multiplicare 10. 

Multiplica 10 cum 10, et hunt 100. Deinde multiplica 10 cum re, et hunt 10 
res diminuendae. Die ergo quod resultent 100 denar. praeter 10 res. 
Item secundo : si cum 10 et re multiplicares deberes 10. 
Multiplica 10 cum 10 et re, et resultabunt 100 denarii et 10 res. 
30 Item tertio. Si cum 10 et re multiplicare deberes 10 et rem. 

Multiplica 10 cum 10, et hunt 100. Deinde multiplica rem cum 10, et hunt 10 
res addendae. Tertio multiplica 10 cum re, et hunt 10 res addendae. Quarto 
multiplica rem cum re, et fit census addendus. Erit ergo totum 100 dena., 20 
res et census. 
35 Item quarto. Si cum 10 praeter rem multiplicare debes 10 praeter rem. 

Multiplica 10 cum 10, et hunt 100. Deinde multiplica rem diminutam cum 10, 
et hunt 10 res diminuendae. Manent autem 100 praeter 10 res. Tertio multi- 
plica 10 cum re diminuta, et hunt 10 res diminuendae. Manent autem 100 praeter 
20 res. Quarto multiplica rem diminutam cum re diminuta, et fit census addendus. 
40 Totum igitur erunt 100 denarii, census praeter 20 res. 



ADDITIONS TO THE ALGEBRA 139 

the articles, or both negative, then the fourth product is positive. If, how- 
ever, one term in one binomial is positive and the other corresponding term 
is negative then this product should be subtracted from the other products. 

There Follows an Illustration of this Rule 

To multiply 8 by 17 is the same as to multiply 10 less 2 by 20 less 3, and 
it will be done in this way. Multiply 10 by 20, giving 200 ; then multiply 
negative 2 by 20, giving 40, which subtracted from 200 leaves 160; in the 
third place multiply 10 by negative 3, giving 30, which subtracted from 160 
leaves 130; in the fourth place multiply negative 2 by negative 3, giving 
positive 6, which being added gives 136. Therefore 8 times 17 gives 136. 

The calculation follows: 10-2 

20 ~ 3 
200 — 40 

-30+6 
136, etc. 

Positives by negatives you subtract, but other products you add. When 
roots are equal to both number and squares, the half (is multiplied by the 
half) and you subtract, otherwise you add that which you obtain. 1 
Three other lines of verse which illustrate the three types of equations. 

A square and 8 roots equal 33. 

A square together with 6 equals 5 roots. 

Four roots and 12 are equal to a square. 

In further explanation : other examples of multiplication. 

Suppose you are to multiply 10 by 10 less x. 

Multiply 10 by 10, giving 100. Then multiply 10 by (negative) x, giving 
negative 10 x. Hence the product is 100 units less 10 x. 

Second. Suppose you are to multiply 10 by 10 plus x. 

Multiply 10 by 10 and by x, giving 100 units and iox. 

Third. Suppose you are to multiply 10 plus x by 10 plus x. 

Multiply 10 by 10, giving 100. Then multiply x by 10, giving positive 
10 x. Thirdly, multiply 10 by x, giving positive 10 x. Fourthly, multiply 
x by x, giving positive x 2 . Hence the product is 100 units, 20 x and x 2 . 

Fourth. Suppose you are to multiply 10 less x by 10 less x. 

Multiply 10 by 10, giving 100. Then multiply negative x by 10, giving 
negative 10 x. The total, so far, is 100 less 10 x. Thirdly, multiply 10 
by negative x, giving negative 10 re, or, in all, 100 less 20 x. Fourthly, 
multiply negative x by negative x, giving x 2 positive. The final product will 
be 100 units, and x 2 less 20 x. 

1 These verses are somewhat obscure. The meaning probably is that in the type .r 2 + n = bx 
the number is subtracted from the square of half the coefficient of x, whereas in the other two 
types of complete quadratics you subtract the number from this square. 



140 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

i Item quinto. Si cum 10 praeter rem debes multiplicare 10 cum re. 

Multiplica 10 cum 10, et fiunt ioo. Deinde multiplica rem cum 10, et fiunt 
10 res addendae. Tertio multiplica 10 cum praeter rem, et fiunt 10 res dimin- 
uendae. Quarto multiplica rem adiectam cum re diminuta, et sit census diminu- 
s tus. Est ergo totum ioo dena. praeter censum. 

Item sexto. Si cum 10 praeter rem deberes multiplicare rem. 
Multiplica rem cum 10, et fiunt 10 res; deinde multiplica etiam rem cum re 
diminuta, et fit census diminutus. Est igitur totum, 10 res praeter censum. 
Item septimo. Si cum 10 et re deberes multiplicare rem praeter 10. 
10 Multiplica rem cum 10, et fiunt 10 res. Deinde multiplica praeter 10 cum 10, 
et fiunt ioo diminuenda. Tertio multiplica rem cum re, et fit census addendus. 
Quarto multiplica praeter 10 cum re, et fiunt 10 res diminuendae. Est igitur to- 
tum, census praeter ioo denar. 

Item octauo. Si cum 10 et medietate rei multiplicare deberes medietatem 
is denarii praeter quinque res. 

Multiplica medietatem denarii cum 10, et fiunt 5. Deinde multiplica praeter 
5 res cum 10, et fiunt 50 res diminuendas. Tertio multiplica medietatem denarii 
cum medietate rei, et fit quarta rei addenda. Quarto multiplica praeter 5 res cum 
medietate rei, quod est multiplicare duas res et dimidiam cum vna re, et fiunt duo 
20 census et 5 diminuendi. Est igitur totum, 5 denarii praeter 2 census et \ et praeter 
49 res et f rei. 

Item nono. Si cum vno denario praeter sextam denarii, multiplicare deberes 
denarium praeter sextam. 

Multiplica denarium cum denario, deinde praeter sextam denarii cum denario, 

25 et fit denarius praeter sextam denarii. Tertio multiplica denarium cum praeter 

sextam denarii, et fit vna sexta minuenda ; manent ergo quatuor sextae. Quarto 

due vel multiplica praeter sextam cum praeter sextam denarii, et fit vna trigesima 

sexta addenda. Est igitur totum viginti quinque trigesimae sextae vnius denarii. 

Item decimo. Si cum 10 et re multiplicare deberes 10 et rem praeter 10. Idem 

30 est ac si cum 10 et re multiplicare deberes rem. 

Et tunc fit totum, census et 10 res. 

De radicum duplatione, triplatione et quadruplatione 

Notandum quod cum census radicem, siue notam siue surdam duplare volueris, 
multiplica duo cum duobus, et cum producto multiplica censum. Et erit huius 
35 producti radix, dupla radix census propositi quae quaerebatur. Consimiliter si 
eius triplum habere volueris, multiplica ternarium cum ternario. Et si eius 
quadruplum, multiplica quatuor cum quatuor, et ita deinceps, et cum producto 
multiplica censum propositum, et erit producti radix, census propositi radix 
duplata, triplata vel quadruplata et caet. 

40 Idem in fradionibus obseruandum 

Vt si medietatem radicis habere consideras, multiplica medietatem cum me- 
dietate, et cum producto deinde censum, et tunc radix producti ostendit quaesitum. 
Similiter si tertiam partem habere volueris, multiplica tertiam partem cum tertia 
parte. Et ita deinceps. 



ADDITIONS TO THE ALGEBRA 141 

Fifth. Given to multiply 10 plus x by 10 less x. 

Multiply 10 by 10, giving 100. Then multiply x by 10, giving positive 
10 £. Third, multiply 10 by negative x, giving negative 10 x. Fourth, 
multiply the positive x by the negative x, giving negative x 2 . The sum 
total is 100 units less x 2 . 

Sixth. Given to multiply x by 10 less x. 

Multiply x by 10, giving iox; then multiply x by negative x, giving 
negative x 2 . The total is therefore iox less x 2 . 

Seventh. Given to multiply x less 10 by 10 plus x. 

Multiply x by 10, giving 10 x. Then multiply negative 10 by 10, giving 
negative 100. Third, multiply x by x, giving positive x 2 . Fourth, multiply 
negative 10 by x, giving negative 10 x. The total is therefore x 2 less 100 
units. 

Eighth. Given to multiply one-half a unit less 5 x by 10 plus \ x. 

Multiply the half unit by 10, giving 5. Then multiply negative 5 x 
by 10, giving negative 50 x. Third, multiply the half unit by \ x, giving 
\ x positive. Fourth, multiply negative 5 x by \ x, which is the same as 
multiplying 2\x by x, giving 2\x 2 negative. The total therefore is 5 
units less 2\ x 2 , and less 49! x. 

Ninth. Given to multiply a unit less |bya unit less I of a unit. 

Multiply a unit by a unit, then negative £ of a unit by a unit, giving a 
unit less ^ of a unit Third, multiply a unit by negative ^ of a unit, giving 
negative § of a unit. These give f of a unit. Fourth, multiply negative ^ 
by negative ^ of a unit, giving positive 35. The total is therefore twenty- 
five thirty-sixths of a unit. 

Tenth. Given to multiply 10 plus x less 10 by 10 plus x. This is the 
same as to multiply x by 10 plus x. Hence the product is x 2 plus 10 x. 

Concerning the doubling, tripling, and quadrupling oj radicals 

It should be noted that when you wish to double the root of a square, 
either a definite root or a surd, you multiply 2 by 2 and multiply the given 
square by this product. The root of this product will be the double which 
you seek of the root of the proposed square. Similarly if you wish its triple, 
you multiply 3 by 3. And if you wish the quadruple of it, you multiply 
4 by 4, and so on; and finally you multiply the proposed square by the 
product ; the root of the product thus obtained will be the double, triple, 
or quadruple, etc., of the proposed square. 

A similar note on fractions 

If you have it in mind to obtain half of the root, multiply one-half by 
one-half, then the given square by the product. The root of this final 
product gives the desired result. Similarly if you wish to have a third part, 
multiply a third part by a third part, and so on. 



142 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

! De radicum diuisione 

Nota, si radicem nouenarii in radicem numeri 4 diuidere volueris, diuide 9 in 4 

et erit exiens 2 et quarta, atque exeuntis huius radix, quae est vnum et semis, erit 

numerus exiens diuisionis radicis in radicem. Quod si duas radices nouenarii in 

s radicem numeri 4 diuidere volueris, quaeras primo duplum radicis nouenarii 

secundum quod docuimus, et illud diuide in radicem numeri quatuor. 

De radicum multiplicatione 

Si radicem nouenarii cum radice numeri 4 multiplicare volueris, multiplica 
9 cum 4, et producti radix est quod quaeris. Ita cum aliis. 
10 Quod si radicem tertiae cum radice medietatis multiplicare volueris, multiplica 
tertiam cum medietate, et producitur sexta, cuius radix est quod quaeris. 

Quod si duas radices nouenarii cum tribus radicibus quaternarii multiplicare 

volueris, inquiras primo secundum quod supra docuimus censum cuius radix est 

duplum radicis numeri 9. Consimiliter inquiras censum cuius radix est triplum 

is radicis numeri 4, et multiplica vnum censum cum altera et radix producti erit 

quaesitum. 

Sequuntur nunc quatuor aenigmata 

Primum 

Radix ducentorum subtractis 10, addita ad duplum subtracti, scilicet ad 20, 

20 subtracta ducentorum radice aequaliter erit 10. 

Dicit aenigma, si a radice numeri 200 subtrahantur 10, id deinde quod relinqui- 
tur ad 20 addatur, ab hoc collecto postea radix numeri 200 auferatur, quod sub- 
tractum tandem, hoc est 10, aequaliter maneant. Hoc autem sic probabitur. Sit 
linea a b radix ducentorum, a qua resecabo lineam a c, quae sit 10. Deinde lineae 

25 a b adiungam b d, lineam quae sit 20, a qua resecabo lineam b e, aequalem lineae 
a b, et a linea b e resecabo lineam b f aequalem lineae a c. Erit igitur linea c b 
aequalis lineae / e. Est autem linea c b radix ducentorum exceptis 10. Ac linea 
e d, 20 excepta radice ducentorum, et c b linea est aequalis/ e lineae, igitur linea fd 
radix ducentorum erit exceptis 10, addita ad 20 excepta radice ducentorum. 

30 Quod autem haec linea / d sit praecise 10, probabo. Linea b d est 20 et cum b f 
linea aequalis sit lineae a c, quae 10 posita est, oportet igitur quod et / d linea 
propter aequalitatem 10 sit, quod erat probandum. 

Sequitur figura. 

Radix ducentorum Radix ducentorum g '-5 § 

, , 1 a 






\b f \e ° g d 

- -i i 



I 10 J «|, 10 g| 

•OX' 'OX I 

— ■ ho ____._• 20""3S* 'o 

1- o ^ o 

© © 



ADDITIONS TO THE ALGEBRA 143 

On the division of radicals 

Notice, if you wish to divide the root of 9 by the root of 4, divide 9 by 
4, giving 2 j. Of the result take the root, which is 1^, and the resulting 
number will be the quotient of the root divided by the root. But if you 
wish to divide two roots of 9 by the root of the number 4, you seek first the 
double of the root of 9 according to that which we have explained, and 
divide the product by the number 4. 

On the multiplication of radicals 

If you wish to multiply the root of the number 9 by the root of the num- 
ber 4, multiply 9 by 4 and the root of the product is that which you seek. 
Other multiplications are similar. 

Thus if you wish to multiply the root of \ by the root of \, you multiply 
\ by 2, giving e, and the root of this is that which you seek. 

In the same way if you wish to multiply two roots of 9 by three roots of 
4, you first try to find, as we have explained above, the square whose root 
is twice the root of the number 9. Similarly you try to find the square 
whose root is three times the root of the number 4, and you multiply the one 
square by the other. The root of the product will be that which you seek. 

Now follow four problems l : First problem 

The root of 200 less 10, added to the double of that which is subtracted, 
in other words to 20, will be equal to 10 when the root of 200 is subtracted. 2 

The problem says that if from the root of the number 200, 10 is subtracted, 
and if then 20 is added to that which remains, and if from this sum the root 
of the number 200 is taken away, we finally have left that which was sub- 
tracted, namely 10. This will be proved in the following manner : 

Let a b represent the root of 200, and from it cut off the line a c, repre- 
senting 10. Then to the line a b join b d, a line which is 20 (units in length). 
From this cut off b e, equal to the line a b, and from b e cut off the line b f equal 
to the line a c ; the line c b will be equal to the line fe. Moreover the line 
c b is the root of 200 less 10, and the line ed is 20 less the root of 200. The 
line c b is equal to the line/e, whence the line f d equals the root of 200 less 
10 plus 20 less the root of 200. Moreover I shall prove that this line fd 
is exactly 10. The line b d is 20, and as 6/is equal to the line a c, which was 
taken as 10, it follows then that the line fd, on account of this equality, is 
10, which was to be proved. 

1 This and the following three problems are not given by Robert of Chester, although the 
Libri and Arabic versions give the first, second, and fourth problems, and the Boncompagni text 
the first and second. Scheybl was probably familiar with the other Latin text, the one published 
by Libri. The geometrical figure in the Libri text is an L-shaped figure, and the same reversed 
in the Arabic text. In each of these the vertical line represents V 2 oo. 

2 The geometrical demonstration of this simple algebraic addition is relatively compb'cated ; 
in algebraic symbolism, V200— 10+20— V2oo= 10, and similarly below. 



144 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

i Secundum aenigma 

Radix ex 200, subtractis 10, diminuta a duplo subtracti, scilicet a 20 excepta 
radice ex 200 est triplum subtracti, scilicet 30, praeter duas radices ducentorum. 
Dicit aenigma, si a radice numeri 200 subtrahantur 10, id deinde quod relinquitur 

5 a 20 subtrahatur, atque ex hoc residuo postea radix numeri 200 auferatur, quod 
tandem triplum subtracti, hoc est 30, et duae radices numeri 200 maneant. Hoc 
autem sic probabitur. Sit linea a b radix ducentorum, et sit b c sibi aequalis, b d 
vero 20, atque postea a e 10. Secabo autem de linea b a lineam b f aequalem a e. 
Erit igitur tota linea df 30. Porro ex linea c d secabo lineam c h aequalem lineae 

10 e b. Et quoniam linea e b est radix ducentorum exceptis 10, cum linea c d sit 20 
excepta radice ducentorum, facta subtractione e b lineae a linea c d, manet linea 
h d quae erit 30 exceptis duabus radicibus ducentorum. Quod autem haec eadem 
linea h d praecise tantum sit, probabo. Linea / d est 30 et cum linea b c radix sit 
ducentorum, linea deinde a e lineae / b, linea etiam e b lineae c h aequalis ; et 

is aggregatum ergo ex lineis / b et c h radix ducentorum erit. Manet autem sub- 
tractione facta linea h d. Probatum ergo quod probandum erat. 

Sequitur figura. 



Tertium aenigma 

Duae radices alicuius numeri sunt vna sui quadrupli. 
20 Hoc satis patet ex eo quod quadratum est quadruplum ad aliud quadratum, 
cuius costa est dupla ad costam alterius. 

Quartum aenigma 

Centum et census exceptis 20 radicibus adiuncti ad 50 et ad 10 radices exceptis 
duobus censibus, sunt 150 exceptis censu et 10 radicibus. 
25 Probatio. Vbi enim subtrahuntur 20 radices, et adduntur 10 radices, idem est 
ac si solum 10 radices subtraherentur, et vbi additur census et subtrahuntur duo 
census, idem est ac si solum modo vnus census subtraheretur, ex quo patet pro- 
positum. 

Sequuntur nunc harum regularum exercitii maioris causa quaestiones decern et octo 

30 Prima 

Diuisi 10 in duas partes et multiplicaui vnam partem cum altera, et postea 
vnam cum seipsa, et produxit haec multiplicatio partis cum seipsa tantum quan- 
tum multiplicatio vnius partis cum altera quater : quae igitur fuerunt partes ? 



ADDITIONS TO THE ALGEBRA 145 

Second problem 

The root of 200 less 10, taken from twice that which is subtracted, namely 
from 20, less the root of 200, is the triple of that which was first subtracted, 
namely 30, less two roots of 200. 

The problem states that when you subtract 10 from the root of the num- 
ber 200, and this in turn from 20, and from the remainder you take the root 
of the number 200, then the triple, 30, of the quantity originally subtracted, 
10, less * two roots of the number 200 remains. This will be proved in the 
following manner : 

Let the line a b represent the root of 200, and let b c be equal to it. Let 
b d be 20, and further a e be 10. Cut off from the line b a the line bf equal to 
a e. Then the whole line df is 30. Further, from the line c d cut off the line 
c h equal to the line e b. Since the line e b is the root of 200, less 10, and the 
line c d is 20 less the root of 200, making the subtraction of the line e b from 
the line c d, you obtain the line h d which is then 30 less two roots of 200. 

Moreover that this line h d is exactly of that length, I will prove. The 
line fd is 30, and since the line b c is the root of 200, the line a e equals / b, 
and also the line e b equals c h. Hence by adding the two lines / b and c h 
we shall obtain the root of 200. The subtraction being made there remains 
the line hd. That therefore which was to be proved, has been proved. 

Third problem 
Two roots of any number make one of the quadruple. 
This is sufficiently evident from the fact that any square is the quadruple 
of another square if the side of the first is double the side of the other. 

Fourth problem 
One hundred and x 2 less 20 x, added to 50 plus 10 x less 2 x 2 , gives 150 
less x 2 and less also iox. 

Proof. Where 20 roots are subtracted and 10 roots are added, the result 

is the same as if only 10 roots were subtracted. Also where a square is 

added and two squares are subtracted, the result is the same as if only one 

square were subtracted. From this follows that which was proposed. 

There follow eighteen questions for greater practice in these rules 2 

First question 
I divided 10 into two parts and multiplied one part by the other, then 
I multiplied one part by itself. This product of one part by itself gave as 
much as four times the product of one part by the other. What were the 
two parts? The answer by the rule is 8 and 2. 

1 Not "and," as in the text. 

2 The eighteen problems are all, with minor changes, from Al-Khowarizmi's algebra, but prob- 
lems 10, n, and 14, which follow, Robert did not include in his text: 

(lo) 7^ + SL f i - 2 - (II) l ■ ^- + s* - *°> ™ ^^ =si 

10 — x x 2 10— x 10 — 2 x 



146 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

1 Respondetur ex regula : 8 et 2. 

Regula. Vnam sectionem ponas rem et alteram 10 praeter rem. Multiplica 
deinde rem cum 10 praeter rem, et fiunt 10 res praeter censum. Postea quadrupla 
hoc totum et producuntur 40 res praeter 4 census, et illud aequatur productum ex 

5 multiplicationis rei cum re, quod est census. Ex hypothesi ergo quaestionis 
census aequalis est 40 rebus praeter 4 census. Vnus igitur census aequatur 8 
rebus. Sed ille est 64, radix vero numerus 8. Vna igitur sectio est 8, et per con- 
sequens reliqua 2. Ista autem quaestio reducitur ad caput vbi census radicibus 
aequatur. 



10 



Quaestio secunda 



Diuisi 10 in duas partes, et multiplicaui, 10 cum seipso, fuitque resultans ex 
tali multiplicatione aequale vni duarum partium multiplicatae cum seipsa bis 
et septem nonis multiplicationis vnius : quae igitur fuerunt partes ? 

Respondetur ex regula : 6 et 4. 

x s Regula. Ponatur vna duarum partium res, quae multiplicetur cum seipsa, et 
fit census. Is census dupletur et addantur septem nonae census et resultant 2 
et %, hoc est viginti quinque nonae, census. Igitur si census esset nouem partes, 
istae essent quintae pars et quatuor quintae vnius quintae totius, quod esset 
viginti quinque nonae. Sed ex hypothesi totum multiplicatum vel productum 

20 est 100. Accipiatur igitur quinta pars et £ quintae partis ex numero 100, erunt 
autem haec 36. Atque tantus est census, cuius radix scilicet 6 vna diuisionis pars 
in proposita quaestione. Reducitur aut haec quaestio ad caput in quo census 
numero aequatur. 

Quaestio tertia 

2 S Diuisi 10 in duas partes, et diuisi vnam partem in aliam et exiuerunt 4. Quae- 
ritur, quae sint partes. 

Respondetur ex regula : 8 et 2. 

Regula. Sit vna pars res, et altera 10 praeter rem. Si igitur ex diuisione 10 

praeter rem in rem exierint 4. Igitur ex multiplicatione 4 cum re resultabunt 10 

30 praeter rem. Quatuor igitur res sunt 10 praeter rem, atque ita quinque res 

decern praecise. Ergo vna res 2. Reducitur autem haec quaestio ad illud caput 

vbi res numero aequatur. 

Quaestio quarta 

Multiplicaui tertiam partem census et denarium vnum cum quarta parte 
35 census et cum denario vno, et prouenerunt 63 : quantus igitur est census ? 

Respondetur ex regula : 24. 
Regula. Multiplica tertiam partem cum quarta, et producetur pars duo- 
decima census; deinde multiplica vnum denarium cum \ et producetur \ rei; 
multiplica etiam tertiam cum denario et producetur \ rei, atque tandem denarium 
40 cum denario, et producetur denarius. Est autem totum multiplicationis produc- 
tum yV census, quarta rei et tertia rei atque denarius, aequantes 63 denarios. 
Demas igitur primo denarium vtrique, casum deinde hunc, vt quidem semper 



ADDITIONS TO THE ALGEBRA 147 

Rule. You would let one part equal x and the other 10 — x ; then 
multiply x by 10 — x, giving iox — x 2 . Afterwards take the quadruple 
of the total, which gives 40 x — 4 x 2 . This equals the product of x multi- 
plied by x, which is x 2 . Now under the supposition of our question, x 2 is 
equal to 40 x — 4 x 2 . Hence x 2 equals 8 x, whence x 2 is 64, and the root of 
it is the number 8. One of the parts of 10 is then 8, and consequently 
the remaining part is 2. This problem, then, is reduced to the chapter 
wherein a square is equal to roots. 

Second question 

I divided 10 into two parts, and multiplied 10 by itself. The result 
of this multiplication was equal to the product of one of these parts multi- 
plied by itself and by two and seven-ninths. What are the parts ? The 
answer in accordance with the rule is 6 and 4. 

Rule. Let x represent one of the two parts, which multiplied by itself 
becomes x 2 . Double this square and add to it seven-ninths of one square, 
giving 2-| x 2 or 2 g 5 x 2 . Hence if the whole square consists of nine parts (ninths) 
that will be the fifth part and four-fifths of one-fifth of the whole, which is 
twenty-five ninths. But under our supposition the product is equal to 
100. Therefore take one-fifth, and | of one-fifth, of the number 100, 
namely 36, which is the value of x 2 . The root of this, namely 6, is one part 
of 10 in the proposed problem. This question then is referred back to the 
chapter in which a square is equal to a number. 

Third question 

I divided 10 into two parts in such a way that the one part divided by 
the other equaled 4. The question is, what are the parts? The answer in 
accordance with the rule is 8 and 2. 

Rule. Let x represent one part, and 10 — x the other. Now if by the 
division of 10 — x by x there results 4, it follows that by the multiplication 
of 4 by x you will obtain 10 — x ; 4X, then, equals 10 — x, and hence 10 pre- 
cisely equals 5 x. The value of x is 2. This question is referred back to 
that chapter in which a root is equal to a number. 

Fourth question 

I multiplied one-third x 2 plus one unit by one-fourth x 2 plus one unit 
and the result was 63. What is the value of x 2 ? The answer in accordance 
with the rule is 24. 

Rule. Multiply one-third (x) by one-fourth (x), and one- twelfth x 2 will be 
the result ; then multiply one unit by \ (x), and \ x will be the result ; again 
multiply \ (x) by one unit, giving ^ x, and finally a unit by a unit, giving a 
unit. The sum total of this multiplication is yg- x 2 + £ x + % x + 1 which 
equals 63 units. Take one unit from both sides, and then, according to the 
rule that it is always necessary to reduce to an integral square, the whole 



148 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

i necesse, ad censum integrum reducas, totum scilicet, vel singula, cum 12 multi- 
plicando, et venient 1 census et 7 res aequales 744 denariis. Per primam igitur 
regulam dimidia radices, et medietatem cum seipsa multiplica et producentur 
12^; quibus additis ad 744 resultabunt 756^. Hinc elice radicem quadratam, 

s veniunt 27^, atque ab his medietatem radicum 3 scilicet et \ subtrahe et relin- 
quuntur 24, quantitas census. Haec autem quaestio ad caput in quo census et 
res numero aequales sunt, reducitur. 

Quaestio quinta 

Diuisi 10 in duo, et multiplicaui vtrumque cum seipso, et aggregatum quadra- 
10 torum fuit 58 : quae igitur sunt partes ? 

Respondetur ex regula : 7 et 3. 

Regula. Multiplica 10 praeter rem cum seipsa, et producentur 100 den. et 

census, exceptis 20 rebus ; deinde multiplica rem cum seipsa, et producetur census. 

Habes igitur 100 den. et 2 census, exceptis 20 rebus, quae omnia aequantur 58. 

is Restaurando igitur 20 res diminutas, patet quod 100 denarii et 2 census valent 

58 den. et 20 res, atque reducendo ad vnum censum, veniunt 50 den. et 1 census 

aequales 29 den. et 10 rebus. Postea vero subtrahendo 29 a 50 relinquuntur 21 

den. et vnus census aequales 10 rebus. Operare igitur per caput secundum, et patet 

quod altera pars 7 et altera 3 sint. Haec autem quaestio reducitur ad caput in 

20 quo census et numeri rebus aequantur. 

Questio sexta 

Multiplicaui tertiam census cum quarta eiusdem, et prouenerunt census et 24 
denarii : quantus igitur est census ? 

Respondetur ex regula : 24. 

25 Regula. Quia satis nosti, quod ex multiplicatione tertiae partis rei cum quarta 
parte rei, duodecima proueniat pars census, quae in hoc casu vni rei et 24 denariis 
aequalis est. Integra igitur censum, multiplicando totum cum 12, et perficias 
quod census vnus 12 rebus et 288 denariis aequetur. Age igitur, multiplicando 
medietatem radicum cum seipsa, et addendo productum seu quadratum ad 288, 

30 et prouenient 324, cuius radix est 18, quae addita ad medietatem radicum, veniunt 
24, census propositae quaestionis. Reducitur autem haec quaestio ad caput in 
quo radices et numeri substantias coaequant. 

Quaestio septima 

Diuisi 10 in duo, vt multiplicatio vnius cum altero producat 24: quae igitur 
35 sint partes, quaeritur. 

Respondetur ex regula : 6 et 4. 

Regula. Scias vnam partem esse rem, et alteram 10 praeter rem. Multiplica 

igitur vnam cum altera, et producuntur 10 res praeter censum, quae valent 24 

denarios. Restaurando igitur die, quod 10 res valeant 24 den. et vnum censum. 

40 Age nunc per caput quo census et numeri rebus aequantur, et patebit quod vna 

pars diuisionis 6 et altera 4 sint. 



ADDITIONS TO THE ALGEBRA 149 

or each part is now multiplied by 12; you will obtain x 2 + 7 x equal 
to 744 units. By the first rule you take one-half of the roots and multiply 
the half by itself, obtaining i2j, which being added to 744 will make a 
total of 7565. Take the square root of this and you obtain 2 1\. Now from 
27^ subtract the half of the roots, namely 3 and \, and 24 remains as the 
value of the square. This question is referred back to the chapter in which 
squares and roots are equal to a number. 

Fifth question 

I divided 10 into two parts and I multiplied each part by itself, and the 
sum of the squares was 58. What are the two parts ? The answer in accord- 
ance with the rule is 7 and 3. 

Rule. Multiply 10 — x by itself, and you obtain 100 units + x 2 — 20 x ; 
then multiply x by itself, and x 2 is obtained. You have then 100 units + 
2 x 2 — 20 x, a total equal to 58. By restoration then of the negative 20 x, it 
follows that 100 units + 2 x 2 equal 58 units + 20 x, and reducing this to 
one square, 50 units + x 2 are found equal to 29 units + 10 x. Accordingly, 
by subtracting 29 from 50, 21 units + x 2 remain equal to iox. Proceed 
therefore according to the second chapter, and it becomes clear that the 
parts are 7 and 3. This problem is referred back to the chapter in which 
squares and numbers are equal to roots. 

Sixth question 

I multiplied ^ x by \ of it and obtained x + 2 units. What is the value 
of x ? The answer in accordance with the rule is 24. 

Rule. Since you well know that the product of ^ x by \ x, is T x 2 x 2 , it fol- 
lows that Y2 %2 > m this instance, equals x +■ 24 units. Make the square 
whole by multiplying all by 12, and you find that x 2 + 12 x is equal to 288 
units. Treat this (equation), then, by multiplying the half of the roots by 
itself and adding the product, or square, to 288. You obtain 324, whose 
root is 18 ; this added to the half of the roots gives 24, the square which was 
sought in the proposed question. This problem is referred back to the 
chapter in which roots and numbers are equal to squares. 

Seventh question 

I divided 10 into two parts in such a way that the product of one by the 
other is 24. The question is, what are the parts ? The answer in accord- 
ance with the rule is 6 and 4. 

Rule. You know that you can let the one part equal x, and the other 
10 — x. Then multiply the one by the other and you obtain 10 x — x 2 , 
which equals 24 units. Now by restoration say that iox amounts to 24 
units + x 2 . Treat this then according to the chapter in which squares 
and numbers are equal to roots, and it will be clear that one part is 6 and 
the other 4. 



150 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

i Quaestio odaua 

Diuisi 10 in duo, atque vtroque multiplicata cum seipso, subtraxi minus de ma- 
iori et manserunt 40. Quaeritur de duobus. 

Respondetur ex regula : 7 et 3. 
s Regula. Multiplica rem cum re, et proueniet census. Deinde multiplica 
etiam 10 praeter rem cum 10 praeter rem, et prouenient 100 dena. et census, 
exceptis 20 rebus. Subtrahe igitur censum a 100 et censu exceptis 20 rebus, et 
manent 100 exceptis 20 rebus quae aequantur 40 dena. Restaurando igitur die, 
quod 100 denarii aequentur 40 denariis et 20 rebus. Subtrahendo deinde 40 a 100 
iopatet quod 60 denarii aequantur 20 rebus; tres igitur denarii rei vni; estque 
ternarius vna diuisionis pars, quare 7 altera. 

Quaestio nona 

Diuisi 10 in duo et vtramque multiplicaui cum seipso, adiunxi deinde producta 
simul et insuper addidi differentiam horum duorum antequam multiplicarentur 
15 cum seipsis, et prouenerunt 54. Quaeritur. 

Respondetur ex regula : 6 et 4. 
Regula. Multiplica rem cum re, deinde etiam 10 praeter rem cum 10 praeter 
rem, et adde simul producta, et veniunt 100 denarii et 2 census, exceptis 20 rebus. 
Cum igitur superfluum inter partes, vel partium differentia, sit 10 praeter 2 res, 
2oadiecto hoc superfiuo, erit totum no dena. et 2 census exceptis 22 rebus, quae 
omnia aequantur 54 denariis. Die igitur, res integrando, quod no den. et 2 
census aequentur 54 denariis et 22 rebus. Reducendo deinde ad vnum censum, 
die, quod 55 denarii et 1 census aequentur 27 denariis et 11 rebus; subtrahendo 
vero 27 a 55, die quod 28 denarii et vnus census aequentur 11 rebus. Age igitur 
25 per caput quo census et numeri rebus aequantur, et patebit quod vna pars diui- 
sionis 6 et altera 4 erunt. 

Quaestio decima 

Diuisi 10 in duo et vtrunque diuisi in alterum, et aggregatum ex diuisionibus, 
id est exeuntium, fuit 2 et vna sexta. Quaeritur. 

30 Respondetur ex regula : 4 et 6. 

Regula. Aggregatum ex multiplicatione vtriusque cum seipso, aequum est 
multiplicationi vnius cum altero et producti cum aggregato ex diuisionibus 
vtriusque in alterum quod in hoc casu est 2 et vna sexta. Ideo multiplica 10 
praeter rem cum seipso, et re cum re, et prouenient 100 et 2 census, exceptis 20 

35 rebus, et hoc totum aequatur multiplicationi rei cum 10 praeter rem, et producti 
cum 2 et vna sexta. Sed multiplicatio rei cum 10 praeter rem producit 10 res 
praeter censum, quibus cum 2 et \ multiplicatis 21 res et f rei praeter 2^ census 
resultabunt, quae aequantur 100 dena. et 2 censibus exceptis 20 rebus. Re- 
staurando igitur census et res, veniunt 41 res et § rei aequales 100 denariis et 4 

40 censibus cum sexta parte census. Reduc igitur totum ad vnum censum sic. 
Consideretur censum esse 6, et erunt 4 census et sexta, 25. Atque huius vnus 
census, 6 scilicet, est vna quinta et quinta quintae. Totius igitur quod habes 



ADDITIONS TO THE ALGEBRA 151 

Eighth question 

I divided 10 into two parts, and each being multiplied by itself, I sub- 
tracted the smaller from the larger, and 40 remained. The question is 
as to the parts? The answer in accordance with the rule is 7 and 3. 

Rule. Multiply x by x, and you obtain x 2 . Then multiply also 10 — x 
by 10 — x and you obtain 100 units + x 2 — 20 x. Subtract, therefore, x 2 
from 100 + x 2 — 20 x, and you have 100 — 20 x, which equals 40 units. 
By restoration say, then, that 100 units are equal to 40 units and 20 x. 
Then by subtracting 40 from 100 it is plain that 60 units are equal to 20 x, 
and hence 3 units to one x. Three is one part ; hence 7 is the other. 

Ninth question 

I divided 10 into two parts, and multiplied each part by itself; then I 
added these products together, and the difference between these two, be- 
fore each was multiplied by itself, and the result was 54. The question 
is stated. The answer in accordance with the rule is 6 and 4. 

Rule. Multiply x by x, and also 10 — x by 10 — x ; add the products and 
you obtain 100 units + 2 x 2 — 20 x. Since the excess of one part over the 
other, or the difference of the parts, is 10 — 2 x, when this excess is added 
we shall have no units + 2 x 2 — 22 x as the sum total, which equals 54 
units. Say therefore, by adding the 22 x, that no units + 2 x 2 equals 54 
units + 22 x. Then by reduction to one square, you say that 55 units + x 2 
equal 27 units and in. By subtracting 27 from 55, you say that x 2 + 28 
units equals in. Treat this by the chapter in which squares and num- 
bers equal roots, and it will be plain that one part is 6, and the other 4. 

Tenth question 

I divided ten into two parts, and I divided each of these by the other ; 
the sum of the two quotients, that is to say, the result, is two and one-sixth. 
The question is stated. The answer in accordance with the rule is 4 and 6. 

Rule. The sum of the products of each (of the parts) multiplied by 
itself is equal to the product of the one by the other when this product is 
multiplied by the sum of the quotients of each of the divisions, which in 
this case is 2^. Hence you multiply 10 — x by itself, and x by x, obtain- 
ing 100 + 2 x 2 — 20 x. This total is equal to the product of x by 10 — x, 
multiplied by 2^. But the product of x by 10 - x gives iox- x 2 , which 
being multiplied by 2% there results 2\ x + \x — 2\x 2 equal to 100 units 
+ 2 x 2 — 20 x. By restoration, then, of the squares and the roots, 41 x + 
§ x are obtained equal to 100 units + 4 x 2 + \ x 2 . Reduce the whole then 
to one square in the following manner : If a square is supposed to be 6, 
then 4^ squares would be 25 ; of this, one square, namely 6, is \ and \ of \. 
Take therefore of everything which you have \ and \ of £, and it will 
be plain that 10 x is equal to x 2 + 24 units. Proceed then by that chap- 



152 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

i accipe quintam partem et quintam quintae, et patebit, quod 10 res vni censui et 
24 denariis aequentur. Age igitur per caput, quo census et numeri rebus aequan- 
tur, multiplicando medietatem radicum cum seipsa, et producuntur 25 ; a quibus 
subtrahe 24 et manet vnitas, cuius radix est vnitas. Hanc radicem subtrahe a 

s medietate radicum, et manent 4, quae sunt vna diuisionis pars. Et nota, quod 
cum illud quod resultat ex diuisione primae partis in secundam alicuius totius, 
multiplicetur cum illo quod resultat ex diuisione secundae partis in primam, illud 
quod prouenit semper idem sit. 

Quaestio vndecima 

10 Diuisi 10 in duas partes et multiplicaui vnam illarum cum 5 et productum diuisi 
in reliquam partem, et exeuntis medietatem addidi ad productum ex multipli- 
catione primae partis cum 5, et totum aggregatum fuit 50. Quaeritur. 

Respondetur ex regula : 8 et 2. 
Regula. Ex 10 accipias rem ; hanc multiplicabis cum 5, et fient 5 res. De- 

15 beres diuidere 5 res in 10 praeter rem, et addere medietatem exeuntis ad 5 res. 
Sed hoc idem est, ac si diuideres medietatem 5 rerum in 10 praeter rem, et adderes 
exeuntem totum ad 5 [res]. Vtraque enim operatio producitur 50. Si ergo diuidas 2 
res et semissem in 10 praeter rem, exeunt 50 praeter 5 res, eo quod addito producto 
ad 5 res, prouenirent 50. Cum igitur constet quod multiplicato illo quod prouenit 

20 ex diuisione cum diuisore redeat census tuus, qui est 2 res et semis. Multiplica 
igitur 10 praeter rem cum 50 exceptis 5 rebus, prouenient 500 et 5 census exceptis 
100 rebus, quae omnia aequantur duabus rebus et semissi. Reduc igitur totum ad 
vnum censum accipiendo quintam partem totius, et patebit, quod 100 et census, 
exceptis 20 rebus aequentur medietati rei. Restaurando igitur, die quod 100 et 

25 census aequentur 20 rebus et medietati rei. Age igitur per caput quo census et 
numeri rebus aequentur, multiplicando medietatem rerum cum se, et prouenient 
105 et ^5-; a quibus subtractis 100 manent 5yg-, cuius radix est 2\; quibus sub- 
tracts a medietate radicum, et manent 8 vna diuisionis pars. 

Quaestio duodecimo, 

30 Diuisi 10 in duos partes, et multiplicatio vnius partis cum seipsa produxit 
numerum continentem alteram partes octogesies semel. Quaeritur de partibus. 

Respondetur ex regula : 9 et 1 . 
Regula. Multiplica 10 praeter rem cum se, et fient 100 et census, praeter 20 
res quae aequantur 81 rebus. Restaurando igitur die, quod 100 et census aequen- 
35 tur 101 rebus. Age nunc per caput quo census et numeri rebus, aequantur, et 
veniet tandem vnitas, vna diuisionis pars. 

Quaestio decitna tertia 

Duo sunt census, quorum maior excedit minorem in duobus, diuisi autem 
maiorem in minorem, et exibat medietas maioris. Quaeritur. 
40 Respondetur ex regula : 2 census minor et 4 maior. 

Regula. Pone rem pro censu, et die, quia res minor diuidens maiorem producit 



ADDITIONS TO THE ALGEBRA 153 

ter in which a square and numbers are equal to roots. Multiplying one- 
half of the roots by itself you have 25 ; from this subtract 24 and there re- 
mains one, of which the root is one. Subtract this root from the half of the 
number of the roots, and four remains as the value of one part. Now note 
that when the quotient obtained by dividing the first part by the second 
part of any whole is multiplied by the quotient of the second part by the 
first, that which is obtained is always the same. 

Eleventh question 

I divided ten into two parts and I multiplied one of these by five, and the 
product I divided by the other part ; one-half of this result I added to the 
product of the first part multiplied by 5, and the sum total was fifty. The 
question is stated. The answer in accordance with the rule is 8 and 2. 

Rule. You may take x as one part of 5 ; this you will multiply by 10, 
giving 5 x. You should divide 5 x by 10 — x, and add £ of the quotient 
to 5 x. But this is the same as if you should divide \ of 5 x by 10 — x 
and add the total result to 5 x ; either operation gives 50. If therefore you 
divide 2\ x by 10 — x, 50 — 5 x is obtained, since when 5 x was added to the 
quotient, the sum was given as 50. Moreover it should be evident that the 
product of the result of any division multiplied by the divisor gives your 
quantity (the dividend), which is 2\x. Therefore multiply 10— x by 
50 — 5 x, obtaining 500 + 5 x 2 — 100 x, all of which is equal to 2\ x. Re- 
duce the whole then to one square by taking the fifth part of the whole, and 
it will be clear that 100 + x 2 — 20 x equals | x. By restoration then say that 
100 + x 2 equals 20 x and \ x. Operate then by the chapter in which 
squares and numbers are equal to roots. Multiplying one-half of the roots 
by itself, 105 and j S is obtained ; from this you subtract 100, leaving 5yg , 
of which the root is 2j, and this being subtracted from one-half of the roots 
8 remains as the value of one part. 

Twelfth question 

I divided 10 into two parts, and the product of one of these parts by itself 
contained the other part 81 times. The question is as to the parts. The 
answer in accordance with the rule is 9 and 1. 

Rule. Multiply 10 — x by itself, giving 100 + x 2 — 20 x, which is 
equal to 81 x. Then by restoration, say that 100 + x 2 equals to 101 x. 
Operate now by the chapter in which squares and numbers are equal to 
roots, and unity will finally appear as the value of one part. 

Thirteenth question 

There are two quantities of which the greater exceeds the less by two. 
I divided the greater by the less and the quotient was one-half the greater 
quantity. The question is stated. The answer in accordance with the 
rule is 2, for the smaller quantity, and 4 for the larger. 



154 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

i medietatem rei maioris, ideo econtra res minor multiplicata cum medietate rei 
maioris producit rem maiorem, et duo multiplicata cum medietate rei maioris, 
producit rem maiorem. Binarius igitur est res minor, et quaternarius maior. 



Quaestio decima quarta 

s Diuisi 10 in duas partes, et multiplicaui vnam partem cum altera, et productum 
diuisi in differentiam inter partes, et resultarunt 5^. Quaeritur, quae sint partes. 

Respondetur ex regula : 3 et 7. 
Regula. Multiplica rem cum 10 praeter rem, et fient 10 res excepto censu; 
deinde diuide 10 res excepto censu in 10 exceptis 2 rebus, quae sunt differentia inter 

10 partes, et exeunt 5^. Si igitur econtra multiplicaueris 5^ cum 10, exceptis 2 
rebus, prouenient 10 res excepto censu. Multiplica igitur 5^ cum 10, exceptis 2 
rebus, et producuntur 52 den. et semis praeter 10 res et semissem. Atque haec 
omnia aequantur 10 rebus, excepto censu. Die igitur, restaurando res et denarios, 
quod 20^ res aequentur 52^ denariis et vni censui. Age igitur per caput quo census 

is et numeri rebus aequantur, multiplicando medietatem radicum in se, et prouenient 
io 5tV e t quae sequuntur et caet. 



Quaestio decima quinta 

Quatuor radices census multiplicatae cum quinque radicibus eiusdem census, 
producunt duplum census et 36. Quaeritur de censu. 
20 Respondetur ex regula : 2. 

Regula. Multiplica 4 res cum 5 rebus, et fiunt 20 census qui aequantur 2 censi- 
bus et 36 denariis. Diuide ergo 36 in 18 et exeunt 2. Atque tantus est census, 
quod examinari poterit. 

Quaestio decima sexta 

25 Subtraxi a censu eius vnam tertiam et tres denarios, multiplicaui deinde re- 
siduum cum seipso, restituit haec multiplicatio ipsum censum. Quantus igitur 
census sit, quaeritur. 

Respondetur ex regula : 9. 
Regula. Subtracta tertia et tribus a tribus tertiis rei, manent f rei praeter 3 dena. 

30 quae sunt radix census. Multiplica igitur f rei praeter 3 den. cum se, et producen- 
tur -§■ census et 9 den. praeter 4 res, et illud aequatur radici. Ergo -f census et 9 
denarii valent 5 res. Reducas f ad vnum censum, eundem, denarios etiam et res 
cum duobus et quarto multiplicando, et inuenies, quod census et 20^ denarii 
aequantur 1 1 rebus et \. Age igitur per caput quo census et numeri rebus aequan- 

35 tur. Accipiendo medietatem radicum quae est 5f et multiplicando earn cum 
seipsa, et fiunt 3i|-j, de quibus subtrahe 20^ et manent nff, cuius radix est 3f, 
quam ad medietatem radicum adde, quia non per subtractionem non deunies ad 
intentum, et veniunt 9, census qui quaerebatur. 



ADDITIONS TO THE ALGEBRA 155 

Rule. Let x represent the one quantity, and say, since the lesser quan- 
tity divided by the greater gives one-half of the greater, that consequently 
the lesser multiplied by one-half of the greater gives the greater quantity. 
But two times one-half of the greater quantity gives also the greater quan- 
tity. Therefore 2 is the value of the lesser quantity, and 4 is the greater. 

Fourteenth question 

I divided 10 into two parts, and I multiplied one by the other and divided 
the product by the difference between the two, obtaining 5! as the result. 
The question is, what are the parts? The answer is 3 and 7. 

Rule. Multiply x by 10 — x, giving iox — x 2 ; then divide iox — x 2 
by 10 — 2 x, which is the difference between the parts, and 5^ is obtained. 
Now, on the other hand, if you multiply 5! by 10 — 2 x, you will obtain 
10 x — x 2 . Hence multiply 55 by 10 — 2 x, which gives 52! units — io| x, 
all of which is equal to 10 x — x 2 . Observe, then, that by restoring to the 
10 x and to the units (the quantities, x 2 and io| x respectively, which are 
subtracted from them) 20^ x is equal to 52^- units + x 2 . Operate there- 
fore by the chapter in which squares and numbers are equal to roots, multi- 
plying the half of the roots by itself, and there will result 105-^g, etc. 

Fifteenth question 

Four roots of a square multiplied by five roots of the same square give 
double the square and 36. The question is as to the square. The answer 
in accordance with the rule is 2. 

Rule. Multiply 4 x by 5 x, giving 20 x 2 , which equals 2 x 2 + 36 units. 
Hence divide 36 by 18, giving 2 as the result. And this amount is the square, 
which may be tested. 

Sixteenth question 

I subtracted from a quantity one-third of it and three units, then I 
multiplied the remainder by itself, restoring the quantity itself by this 
multiplication. The question is, how great is the quantity ? The answer 
in accordance with the rule is 9. 

Rule. Subtracting 3 £ + 3 (units) from | x, there remain § x — 3 units, 
which is the root of the quantity (x) . Therefore multiply § x — 3 units by 
itself, giving ^ x 2 + 9 units — 4 x, and that is equal to the root (x) . Hence 
f x 2 + 9 units equals 5 x. You reduce the f to a whole square by multiply- 
ing it, and also the units and the 5 x, by 2j, and you will find that x 2 + 20j 
units is equal to n| x. Operate then by the chapter in which squares and 
numbers are equal to roots. Taking the half of the roots, 5I, and multiply- 
ing it by itself, you have 31^ : from this subtract 2o|, there remains 1 iff, of 
which the root is 3! . Add this to the half of the roots, since by subtraction 
you will not arrive at the desired result, and 9 appears as the quantity 
which you seek. 



156 ADDITA QUAEDAM PRO DECLARATIONE ALGEBRAE 

1 Quaestio decima septima 

Diuisi drachmam et semissem inter homines et partem hominis, et contigit 
homini duplum eius quod parti. Quanta igitur fuerit pars, quaeritur. 

Respondetur ex regula : §. 
s Regula. Idem est homo et pars, ac si diceres, vnum et res. Diuidatur ergo 
drachma et semis in vnum et rem, et venient 2 res. Multiplica deinde 2 res cum 
drachma et re et fient 2 census et 2 res, quae aequantur drachmae et semissi. 
Reducendo igitur ad vnum censum, die quod census et res, aequentur f drachmae. 
Age igitur per caput, quo census et res numero coaequantur, multiplicando me- 
10 dietatem rei in seipsa, et fit quarta, quae addita ad f facit vnum, cuius radix est 
vnum, a qua subtrahe medietatem rei et manet medietas, pars quae quaeritur. 

Quaestio decima octaua 

Diuisi drachmam inter homines et prouenit simul res, deinde addidi eis hominem 
et postea diuisi drachmam inter eos, et cuilibet contigit minus quod prius sexta 
is parte drachmae. Quot igitur fuerint homines, quaeritur. 

Respondetur ex regula : 2. 
Regula. Huius quaestio consideratio est vt multiplices homines primos cum 
diminuto. inter diuisiones ; deinde multiplices aggregatum cum hominibus primis 
et cum homine addito, et proueniet census tuus. Scias autem, quod hoc non sit 
20 vniuersaliter verum vt credo, sexta census et sexta radicis, quae aequantur drach- 
mae. Die ergo, res integrando, quod census et res aequentur 6 drachmis. Age 
igitur per caput quo census et res numeris coaequentur. Multiplicando medie- 
tatem radicum cum seipsa, et fit quarta, quam adde ad 6 drachmas et veniunt 6^. 
Inde radix quadrata erunt 2 et semis a qua subtracta medietate radicum, manent 
25 2, qui est numerus hominum. 

Sequitur vltimo de rebus venalibus 

Sunt autem conuentiones negociationum quae fiunt in venditione, emptione, 
permutatione, et caeteris rebus, secundum duos modos. 

Primus est modus, vt si dicatur, decern res venditae sunt 6 drachmis, quot 
30 igitur veniunt 4 drachmis ? 

Secundus modus est, vt si dicatur, decern res venditae sunt 6 drachmis, 
quantum igitur est precium 4 rerum ? 

In primo casu, 10 res est numerus appreciati secundum positionem, et 6 drach- 
mae est praecium secundum positionem ; quaestio quot, est numerus ignotus 
35 appreciati secundum quaerentem, et 4 res precium secundum quaerentem. In 
secundo casu, precium et appreciatum secundum positionem, sunt vt prius, et 
quaestio et precium secundum quaerentem et 4 res est appreciatum secundum 
quaerentem. Vnde precium secundum positionem dicitur opponi appreciato 
secundum quaerentem et appreciatum secundum positionem dicitur opponi 
4oprecio secundum quaerentem. Multiplica igitur inter se, et productum diuide 
in tertium modum et exhibit quartus ignotus per regulas quatuor proportionalium 
quantitatum. 

Finis annotationum pro declaratione regularum Algebrae. 



ADDITIONS TO THE ALGEBRA 157 

Seventeenth question 

I divided a drachma and one-half between a man and a part of a man, and 
to the man there fell the double of that which fell to the part (of a man) . 
The question is, how large was the part? The answer is £. 

Rule. A man and part of a man is the same as 1 +x. Hence i| is divided 
by 1 + x, giving 2 x. Then multiply 2 x by 1 + x, giving 2X 2 +2X, which 
is equal to if . Therefore by reduction to one square, say that x 2 + x is 
equal to f of a unit. Operate now by the chapter in which squares and roots 
equal number, multiplying one-half of the roots by itself, obtaining \ ; this 
added to § makes 1 , of which the root is 1 ; from this subtract the half of the 
number of the roots, giving \, the value of the part that is sought. 

Eighteenth question 

I divided a drachma among some men, and each one obtained an unknown 
amount (x) ; I then added one man to the group and again I divided a 
drachma among them ; to each man there now fell 5 drachma less than before. 
The question is, how many men were there? The answer is 2. 

Rule. In considering this problem you multiply the first number of men 
by the decrease ; then you multiply the product by the number of men + 1 , 
and the quantity will be obtained. However you should note that this is not 
a general rule ; you have ^ x 2 + ^ x, which is equal to 1 . Hence by com- 
pleting the quantity you obtain x 2 + x equal to 6 units. Operate there- 
fore by the chapter in which squares and roots equal numbers. Multiply- 
ing one-half of the roots by itself, \ is obtained. Add this to the 6 units, 
giving 6j ; then the square root will be 2\. From this one-half of the 
roots is subtracted, leaving 2, which is the number of men. 

The last section, on commercial transactions 

There are certain customs of business which hold in buying, selling, ex- 
change, and the like, according to two methods. The first method is illus- 
trated : 10 things are sold for 6 drachmas, how many are sold for 4? The 
second : 10 things are sold for 6 drachmas, what is the price of 4? 

In the first case 10 things is the number priced, according to that which 
is given, and 6 drachmas is the price, as given ; the question, how many, 
represents the unknown number of things, according to the question, and 
4 drachmas the price, according to the question. In the second case, the 
price and the quantity, as given, are the same as before ; the question 
represents the price, according to the problem, and 4 is the corresponding 
quantity. Whence the price given is said to be in opposition to the price 
sought, and similarly the quantity given to that sought. Multiply therefore 
among themselves, and divide the product by the third kind, and the fourth 
unknown will appear by the rules of four proportionals. 

End of the annotations to explain the rules of algebra. 



LATIN GLOSSARY 



ABBREVIATIONS 



adj., 


adjective. 


adv., 


adverb. 


comp., 


comparative. 


conj., 


conjunction. 


demonstr., demonstrative 


f., 


feminine noun. 


indef., 


indefinite. 


indecl., 


indeclinable. 


m. 


masculine noun. 


n., 


note. 



neut., 


neuter noun. 


num., 


numeral. 


prep., 


preposition. 


pron., 


pronoun. 


R, 


characterizing a usage of the Regule. 


s, 


characterizing a usage peculiar to Scheybl 




as opposed to Robert of Chester. 


subst., 


substantive. 


super., 


superlative. 


v. , 


verb. 



The first appearance in the original text of each Latin word listed is recorded by reference to page 
and line. In a few instances other references are added to indicate differences in meaning. In cases of 
variation in spelling the catchword presents the form probably used by Robert of Chester ; the current form 
is generally added in parentheses. The limits of the volume preclude a complete study of the Latinity. 



abicio, v., disregard (76, 2) ; with ex, sub- 
tract from (108, 9). 

ablatio, f., S, subtraction (134, 14). 

absque, prep., minus (96, 4, n) ; as adjective, 
S, negative (116, 15). 

abstraho, v., with ex, subtract (120, 3, n). 

accipio, v., take (70, 32) ; extract root 
(72, 1). 

addendus (addo), adj., to be added, positive 

(9°> 13)- 
addicio (-tio), f., addition (74, 22, n). 
additum, neut., S, addition (128, 1). 
addo, v., add (no, 27); with ad, S, add 

(128, 10) ; with super, add to (74, 2, n). 
adequo (adaequo), v., equal (106, 18, n). 
adhibeo, v., add to (118, 19). 
adicio (adiicio, S), v., add to (72, 1) ; with 

super, add to (72, 1, n). 
adiectio, f., addition (74, 22). 
adiectiuus, adj., added, positive (90, 13, n.). 
adiectus, adj., added, positive (96, 23, n). 
adimpleo, v., fill up, complete (74, 21). 
aditus, m., approach (88, 25). 
adiungo, v., add (70, 30). 
adnullo, v., reduce to nothing (74, 29, n). 
ae-, see also e-. 

aenigma, neut., S, problem (142, 17). 
aequalitas, f., S, the equal (80, 12). 
aequatio, f., S, equation, equality (66, 2), 
aequipolleo, v., S, equal (128, 6). 
aequus, adj., S, equal (150, 31.) 



aggregatum, m., S, sum (72, 20). 

ago, v., operate (70, 11). 

algaurizim, Arabic, Al-Khowarizmi (66, 

9, n). 
algebra, f., S, algebra (66, 1). 
aliqui, aliqua, aliquod, indef. pron. adj., 

some, any (74, 21). 
aliquot, indef. indecl. num., some, a num- 
ber (90, 9). 
almucabola, f., S. A transliteration of part 

of the Arabic title al-jebr tv'al-muqabala, 

for algebra (66, 1). 
almuthemen, Arabic, quantity desired (120, 

24, n). 
almuzarar, Arabic, unit of measure, quantity 

(120, 22, n). 
alszarar, Arabic, price per unit (120, 23, n). 
althemen, Arabic, amount, payment (120, 

24, n). 
amplector, v., embrace, surround, include 

(90, 23, n). 
angularis, adj., S, corner (130, 10). 
angulus, m., an angle, corner (78, 11). 
annotatio, f., S, annotation (156, 43). 
appareo, v., S, appear, be evident (84, 8). 
applico, v., apply, place (78, 4). 
appono, v., place by, set in apposition (96, 

23, n). 
appreciatum, neut., S, that priced (156, 36). 
arbitror, consider, judge (122, 7). 
area, f., area, figure (78, 4). 



iS9 



i6o 



LATIN GLOSSARY 



arithmeticus, adj., S, arithmetical (66, 7). 
ars, f., type (70, 28). 

articulus, m., S, multiple of ten (136, 15). 
assigno, v., mark out, indicate (104, 6). 
assimilo, v., R, make like, be equal (68, 12, 

n and 126, 2). 
assumo, v., take, extract square root (72, 

1, n). 
attineo, v., pertain (120, 21). 
auctus, v., S, positive (138, 17). 
aufero, v., with ex, take away (84, 22 ; no, 

21). 
augeo, v., S, increase (130, 19). 

binarium, neut., two (98, 1). 

binarius, adj., (consisting of) two (82, 16). 

bis, adv., twice (98, 16). 

breuis, adj., S, short (82, 23). 

cado, v., S,/all (upon) (132, 25). 

calculus, m., S, calculation (90, 19, n). 

capitulum,„neut., chapter (74, 21, n). 

caput, n., S, chapter (74, 21). 

casus, m., S, event, instance (74, 14). 

causa, f., S, //*£ reason, the explanation 
(130, 4). 

census, m., S, the second power of the un- 
known (128, 2) ; quantity (146, 34). 

centenarius, adj., one hundred (66, 17). 

centenus, adj., one hundred (66, 17, n). 

certus, adj., definite (120, 27). 

circumdo, v., encompass, surround (78, 
12, n). 

circumduccio (-tio), {., perimeter (78, 13). 

circumiaceo, v., S, surround (134, 18). 

coequacio (coaequatio), f., equality (70, 2). 

coequalis (coae-), adj., equal (88, 1, n). 

coequo (coae-), v., equal (68, 7). 

cognitus, adj., known (122, 25). 

collectio, f., -ww (70, 30). 

colligo, v., combine, sum up (68, 6). 

combinatio, f., S, combination (128,2). 

committo, v., R, compare, divide (126, 2). 

compendiose, adv., briefly (88, 23). 

complector, v., S, include, embrace (90, 24). 

compleo, v., fill up, co7nplete (72, 28) ; com- 
plete by transferring negative term (108, 

4). 

completus, adj., complete (108, 20). 

compono, v., compose, make (66, 12). 

compositus, adj., S, composite, used of num- 
bers, including both tens and units, such 
as 16 or 24 (136, 13). 



comprehendo, v., include (112, 17). 
comprobo, v., prove, confirm (80, 1 1, n). 
concipio, v., think (of), coticeive (74, 13). 
concretus, adj., formed, complete (100, 

16, n). 
conduco, v., employ, hire (124, 4). 
coniungo, v., connect, join, add (68, 2). 
considero, v., consider, reflect (66, 10). 
consimilis, adj., equal (80, 6, n). 
constituo, v., constitute, make (74, 18). 
contineo, v., contain (84, 16). 
contingo, v., reach, attain (96, 4, n). 
contra, adv., S, opposite (78, 25, n). 
conuentio, f., S, custom (156, 27). 
conuersio, f., reduction of an equation to 

simpler form (72, 8). 
conuerto, v., arrive (66, 21, n) ; reduce (68, 

23)- 
costa, f., S, side (144, 21). 
cum, prep., S, by (68, 3) ; with (68, 23). 

decenarius, adj., ten (66, 14, n). 
decenus, adj., ten (66, 14, n). 
declaratio, f., S, exposition (128, 1). 
deduco, v., with in, ?nultiply (86, 15, n). 
deficio, v., S,fail, be lacking (132, 4). 
deleo, v., S, efface, cancel (80, 10). 
demo, v., with a, subtract (no, 9, n). 
demonstratio, f., S, demonstration (66, 2). 
demonstro, v., designate, represent (68, 

17)- 
denarius, adj., ten (100, 27). 
denarius, m., S, unit of money, penny (124, 

4) ; unit, S (128, 6). 
descripcio (-tio), f., description, explanation 

(88, 22, n). 
designo, v., represent, designate (72, 14). 
deuinco, v., S, surpass, exceed (84, 1). 
differentia, f., S, difference (150, 19). 
differo, v., differ (112, 27). 
difficultas, f., difficulty (88, 24). 
digitus, m., S, digit, unit (136, 15). 
dimensio, f., S, measuremejit, side (84, 4). 
dimidium, neut., S, half (76, 12). 
diminucio (-tio), f., diminution, subtraction 

(74, 23). 
diminuo, v., with ex, subtract (72, 22), see 

(72, 2, n) ; with ab, subtract (72, 2, n). 
diminutiuus, adj., negative, subtracted (90, 

13, n). 
diminutus, part, and adj., reduced (68, 4); 

lessened, negative (96, 23, n). 
dinosco, v., distinguish, represent (72, 4). 



LATIN GLOSSARY 



161 



dinotus, adj., pointed out, distinguished 
(i 20, 24). 

disciplina, discipline, art, study (88, 25). 

dispono, v., arrange (66, 13). 

distinctus, adj., distinct, separate (70, 22). 

distinguo, v., distinguish, separate, discrim- 
inate (70, 20). 

diuido, v., with in, S (100, 5), with per (100 
5, n), and with super (100, 8, n), divide 
by ; diuido in duo media, bisect or take 
half of (86, 13) ; diuido in duo, separate 
into two parts (102, 21) ; with inter, 
distribute among (118, 18); diuido per 
medium, take the half of (70, 31). 

diuisio, f., division (84, 4, n) ; S, distribution 
(156, 18). 

diuisor, m., S, divisor (152, 20). 

do, v., give (74, 11). 

doceo, v., teach, show (88, 22, n). 

dragma (drachma, S), f., unit (70, 28) ; unit 
of money (124, 4, n). 

dubito, v., consider, doubt, be in doubt 
(122, 6). 

duco, v., with in, multiply (80, 8, n) ; S, draw 
(82, 21). 

duplatio, f., S, doubling (98, 15). 

duplex, adj., twofold, dojible (118, 2). 

duplicacio (-tio), f., doubling, multiplication 
(66, 16). 

duplico, v., double (66, 15) ; multiply (90, 
7, n). 

duplo, v., S, double (140, 33). 

duplum, neut., S, double (142, 19). 

e contrario, S, on the contrary, in the reverse 

way (100, 8). 
e conuerso, in the reverse way (100, 8, n). 
edo, v., give out, publish (66, 9). 
elicio, v., draw forth, solve (108, 29, n) ; 

take square root. S (148, 4). 
elucesco, v., shine forth (102, 18). 
equalis (ae-), adj., equal (78, 4, n). 
equidistans (ae-), adj., equidistant (82, 8, n). 
equiparo (ae-), v., equal (68, 20). 
equo (ae-), v., equal (70, 18). 
ergo, adv., therefore, then, consequently (68, 

6, n). 
erigo, v., S, erect a perpendicular, raise up 

(132, 15)- 
error, m., error (102, 12). 
essentialiter, adv., essentially, naturally 

(66, 13, n). 
et, conj., and (66, 7) ; plus (90, 16). 



euacuo, v., cancel, vacate (80, 10). 
euenio, v., come out, result (82, 25). 
exaequo, v., S, equal (106, 4). 
examen, neut., S, testing (116, 7, n). 
excedo, v., exceed, go beyorid (66, 14). 
excresco, v., grow, iticrease (66, 19). 
exemplar, neut., model, pattern (102, 12). 
exemplum, neut., example, type (122, 7). 
exeo, v., come out (104, 30). 
exerceo, v., exercise, practise (102, 18). 
exercitium, neut., S, exercise, problem (144, 

29). 
exhibeo, v., represent, show (74, 13). 
exigo, v., require, demand (98, 9). 
expedio, v., carry through (120, 19). 
explano, v., explain (86, 1, n). 
expono, v., explain, set forth (88, 22). 
expositio, f., explanation, exposition (104, 

16). 
exprimo, v., represent, show the form of 

(72, 23). 
extendo, v., extend, reach (72, 12). 
extraho, v., with ex, subtract (82, 27). 
extremitas, f., end of a line (78, 23). 
extremum, neut., S, extremity, end (130, 18). 

facile, adv., easily (120, 19). 
facilis, adv., easy (88, 25). 
facio, v., make (66, 16). 
figura, f., figure (88, 23, n). 
finio, v., terminate, come to (104, 27). 
forma, f., S,form (136, 11). 
formula, f., rule, formula (80, 11). 
fractio, f, S, fraction (92, 7). 
fractus, adj., S, fractional (100, 3). 

genero, v ., generate, produce (90, 23). 

genus, neut., class, species, kind (70, 22). 

geometrice, adv., geometrically (76, 20, n). 

geometricus, adj., S, geometrical (66, 7). 

gnomon, m., S, gnomon, or the form of a 
carpenter's square, consisting of three 
rectangles lying around any given rec- 
tangle and forming with it a larger, similar 
rectangle (132, 21). 

habeo, v., have, constitute (70, 15). 
hypothesis, f., S, hypothesis (146, 19). 

igitur, conj., therefore (68, 2). 

ignoro, v., not to know, be ignorant of (76, 

23)- 
ignotus, adj., unknown (78, 11). 



l62 



LATIN GLOSSARY 



imperfectio, f., incompleteness (78, 13). 
imperfectus, adj., incomplete, itnperfecl (80, 

5)- 
in, prep., in (70, 31) ; with verbs of division 

or multiplication, by (68, 3, n). 
inaequalis, adj., S, unequal (78, 10). 
incertus, adj., doubtful, unknown (120, 28, 

n). 
incognitus, adj., unknown (122, 14, n). 
incurro, v., incur, run into (122, 7). 
indigeo, v., need, require (66, 11). 
infinitus, adj., infinite (66, 21). 
infra, adv. and prep., below, less than (68, 

4)- 
inquisicio (-tio), f., problem (122, 1). 
inscribo, v., inscribe (82, 12). 
insuper, adv., in addition, besides (118, 19). 
integer, adj., integral, whole (98, 10). 
integro, v., S, make whole (150, 21). 
intelligo, v., understand, comprehend (88, 

24). 
intentum, neut., S, result proposed (128, 16). 
inter, prep., between (84, 17). 
interrogacio (-tio), f., S, question (70, 31). 
inuenio, v., discover, find (66, 10). 
inuestigacio (-tio), f., investigation, finding 

out (66, 21). 
inuestigo, v., track out, investigate, seek 

after, look into (74, 11). 
inuicem, adv., in turn, alternately (68, 7). 
iungo, v., join, add (1 12, 3). 
iuxta, prep., according to, after the manner 

of, in case of (68, 29) . 

lanx, f., scale of a balance (96, 23, n). 

latitudo, f., breadth (78, 5). 

latus, neut., side (76, 23). 

liber, m., book (66, 8). 

linea, f., line (82, 23). 

liqueo, v., appear, be evident (76, 17). 

locus, m., S, place (76, 24). 

longitudo, f., length (78, 5). 

magnus, comp. maior, maius, adj., great 

(76, 3, n and 78, 13). 
magul, Arabic, unknown (122, 12, n). 
maneo, v., S, remain, abide (72, 2). 
manifestus, adj., evident (72, 14). 
medietacio (-tio), f., halving, half (1 16, 19). 
medietas, f., half (70, 8). 
medio, v., halve (74, 26) . 
medium, neut., middle, half (70, 31). 
medius, adj., middle, mean, half (70, 18). 



mensuro, v., measure (68, 27). 
millenarius, adj., thousand (66, 19). 
millenus, adj., thousand (66, 19. n). 
minor, minus, comp. of parvus, adj., less 

(68, 22). 
minucia (-tia), f., a small particle ; a small 

part (118, 21). 
minuendus (minuo), to be subtracted, nega- 
tive (90, 15). 
minuo, v., with ex, subtract from (74, 14, n). 
minutus, adj., S, negative (138. 17). 
modus, m., manner, fashion (66, 15). 
multiplicacio (-tio), f., multiplication (90, 

10). 
multiplico, v., multiply (66, 17) ; with in, 

7mdtiply by (68, 3, n) ; with cum, S. 

mttltiply by (68, 3). 
multitudo, f., S, multitude, a great number 

(68,29). 

nascor, v., arise, spring forth (82, 16). 
natura, f., nature (98, 9). 
naturaliter, adv., naturally (96, 29). 
necessario, adv., necessarily (86, 8). 
necesse, adj., necessary (78, 25). 
negligo, v., S, neglect, nullify (96, 21). 
nihil, nil, neut., nothing (66, 12). 
nodus, m., node, a tnidtiple of ten (90, 8). 
nomino, v., name, call (122, 20). 
nosco, v., know, recognize (72, 4, n). 
noticia (-tia), f., S, axiom (132, 19). 
notus, adj., known (120, 27); rational, S 

(140, 33)- 
nullus, adj., S, not any, mill, void (74, 29). 
numerus, m., nutnber (66, 9) ; numerus di- 

minutus, fraction (98, 10). 
nuncio, v., announce, report (122, 13, n). 
nuncupo, v., name, call (120, 23, n). 

obtineo, v., maintain, prove, have, obtain 

(78, 6, n). 
omnino, adv., wholly (102, 17). 
omnis, adj., all, every (66, 11). 
operor, v., work, operate (100, 20). 
opifex, m., worker, student (108, 28). 
oppono, v., set in opposition to, oppose (96, 

4, n). 
oppositio, f., opposition, balancing (66, 8). 
oppositus, adj., opposed (122, 16). 
or do, m., arrangement, denomination (66. 

20). 
orior, v., arise, appear, spring up (70, 21). 
ostendo, v., represent, show, reveal (68, 21). 



LATIN GLOSSARY 



163 



parallelogrammum, neut., S, rectangle (82, 8). 
pario, v., produce, obtain (78, 16, n). 
pars, impart (68, 16). 
particula, f., s?nall part, fractional part 

(100, 7). 
pateo, v., S, be clear, follow (128, 26). 
paucior, comp. of paucus, adj., fewer, less 

(72,6, a). 
paucitas, f., fewness, scarcity, paucity (68, 

29). 
perduco, v., bring to, lead to (104, 10). 
perfectio, f., perfection, completion (80, 22). 
perfectus, part, and adj., complete (72, 28). 
perficio, v., complete, make (78, 17). 
permaneo, v., remain (96, 4, n). 
perpendicularis, adj., perpendicular (82, 21). 
perspectus, adj., evident, clear (88, 24). 
pertineo, v., concern, relate to (76, 19). 
peruenio, v., arrive at, come up to (66, 1 7) . 
pluralitas, f, many, plurality (68, 29, n). 
plures, adj., the plural of plus, more (70, 2). 
plus, S, comp. of multus, adj., plus, more 

than (76,3). 
pondus, neut., weight (124, 16). 
pono, v., place, assume, assert, propose (82, 

9) ; with super, place upon (84, 1). 
portio, f., S, portion, segment (82, 22). 
possum, v., be able (102, 12). 
postea, adv., then, afterwards (66, 22). 
posterior, comp. of posterus, adj., S, latter, 

following (76, 16). 
postremus, super, of posterus, adj., last (118, 

23)- 
praeter, adv. and prep., S, minus (136, 17) ; 

negative (138, 7). 

precium (pretium), neut., price (124, 5). 

prefatus (prae-), adj., before-mentioned 
(122, 10). 

premitto (prae-), v., place before (70, 21). 

pretaxo (prae-), v., mention, assign, enu- 
merate (74, 14, n). 

primum, primo, adv., first (74, 13). 

primus, adj., frst (76, 21). 

principium, n., beginning, commencement 
(76, 15)- 

prius, adv., before, previously (72. 28). 

probatio, f., trial, test, proof (76, 23). 

probo, v., try, check, test, prove (76, 21, n). 

procreo, v., create, produce (74, 12, n). 

produco, v., S, produce, give (70, 32). 

productum, neut., S, product (80, 3). 

profero, v., bring forward, produce, give 
(122, 12). 



progenero, v., generate, produce (106, 12, n). 
proicio, v., cast out, take away (96, 4, n). 
pronuncio, v., mention, relate (70, 31). 
propono, v., propose, set forth (68, 26). 
proporcio (-tio), f., connection, proportion, 

ratio (68, 2). 
proposicio (-tio), f., proposition (76, 21). 
propositus, part., proposed (102, 14, n). 
protendo, v., extend (70, 30). 
protraho, v., S, draw, extend (132, 15). 
prouenio, v., S, come forth, appear (100, 16). 
prout, adv., just as, as (76, 17). 
punctum, neut., point (82, 20). 

quadratum, neut., S, square (76, 23). 

quadratus, adj., square (82, 14, n). 

quadrilaterus, adj., S, four-sided, quadri- 
lateral (82, 14). 

quantitas, f., amount, quantity (78, n). 

quantum, adv., how much (76, 19). 

quantus, adj., how great, as great (76, 8, n). 

quaternarius (sometimes quarternarius, S), 
adj., four (68, 20). 

quero (quae-), v., inquire, ask (70, 3). 

questio (quae-), f., question, problem (72, 15). 

quociens (quotiens), adv., how many times 

(90, 7)- 
quotquot, adj., however many (72, 6). 

radix, f., root (72, 1) ; unknown (68, 1). 

rectangulum, neut., S, rectangle (82, 10). 

rectangulus, adj., S, rectangular (82, 8). 

rectus, adj., S, straight (as noun 82, 23) ; 
right angled (82, 15). 

reddo, v., give back, make, render (80, 5). 

redeo, v., S, return, arise (152, 20). 

reduco, v., S, bring back (146, 8) ; multiply 
(128,37). 

refero, v., S, represent (82, 10). 

regula, f., ride (72, 18). 

relinquo, v., S, leave (84, 5). 

remaneo, v., remain, be left (72, 2, n). 

reperio, v., find, discover (66, 13). 

repeto, v., repeat (90, 10). 

res, f., thing (68, 3) ; unknown first power 
(82,5). 

reseco, v., cut off (86, 9). 

residuus, neut., remaining (76, 16). 

respicio, v., look back, refer (122, 15). 

restauracio (-tio), f., restoration, transfer- 
ence of negative terms to the other side 
of the equation (66, 8). 

restauro, v., restore, transfer (104, 3). 



164 



LATIN GLOSSARY 



resto, v., remain (80, 16, n). 

rumbus (for rhombus), m., square (76, 23, n). 

scientia, f., knowledge, science (66, 10). 
scio, v., know, understand (74, 26). 
seco, v., S., cut (132, 16). 
secundum, prep., according to, following 

(66, 20). 
secundus, adj., second, following (120, 25). 
semel, adv., once, a single time (82, 15). 
semis, m., S, one-half (142, 3). 
semper, adv., always (68, 6). 
significo, v., S, signify, represent (78, 17). 
signo, v., mark, represent, signify (78, 

17, n). 
similis, adj., similar, equal (82, 9, n). 
similiter, adv., in like manner, similarly 

(68, 23). 
similitudo, f., similitude, likeness (70, 29) ; 

ad similitudinem, likewise (68, 18). 
simul, adv., at the same time, together, also 

(74,24). 
sine, prep., minus, negative (90, 22). 
singularis, adj., S, one by one, each (78, 7). 
solucio (-tio), f., solution (74, 13). 
solus, adj., alone, pure (68, 1). 
structura, f., S, construction (132, 17). 
studium, neut., study, zeal (102, 18). 
sub, prep., under (102, 14). 
subabiectio, f., S, subtraction (134, 33). 
subiectus, adj., accompanying, adjacent (88, 

22, n). 
substantia, f. , second power of the unknown 

(68, 1) ; quantity (106, 9, n). 
subtractio, f., S, subtraction (84, 5). 
subtraho, v., with ex, subtract from (78, 23) ; 

with ab, subtract from (90, 9). 
sufficienter, adv., sufficiently (76, 19). 
summa, f., amount, sum (72, 20, n). 
sumo, v., assume, take (100, 24). 
super, adv. and prep., above (72, 1, n) ; in 

additions, to. 
superaddo, v., add (104, 15). 
superficies, f., S, area (136, 3). 
superfluum, neut., S, excess (150, 19). 



superius, comp. adv., above (76, 17). 

supersum, v., remain (92, 19). 

supplementa, neut., S, supplementary rec- 
tangles cut off from a larger rectangle by 
two lines parallel to the sides and inter- 
secting on the diagonal (134, 8). 

supra, prep, and adv., above (68, 4). 

surdus, adj., S, surd, irrational (140, 33). 

tantum, adv., with quantum, as much as 

(92, 13)- 
tempus, neut., time (124, 7). 
tendo, v., extend, represent (84. 7). 
termino, v., limit, end (82. 13). 
ternarium, neut., three (98, 4). 
ternarius, adj., (consisting of) three (84, 23). 
tociens (totiens), adv., so often, so many 

times as (90, 7, n). 
tollo, v., take square root (98, 8) ; with ex, 

subtract, take away (no, 13, n). 
totus, adj., whole, all (66, 11). 
tracto, v., treat, use, handle (70, 1). 
trado, v., impart, set forth (120, 19). 
transfero, v., translate (124, 19). 
tribuo, v., S, add (96, 20). 
triplicacio (-tio), f., tripling (66, 16). 
tripliciter, adv., triply (70, 22). 
triplico, v., triple (66, 15). 
tunc, adv., then, immediately (84, 26). 

uinco, v., exceed, surpass (84, 1, n). 

ullus, adj., any (68, 2). 

ultimus, adj., extreme, last (120, 26). 

unitas, f, unity, unit (66, 12). 

unus, num. adj., one (68, 19). 

unusquisque, pronominal adj., each one, each 

(76, 25), (78,4). 
usque, adv., with ad, up to, as far as (66, 

14). 

venalis, adj., salable, to be sold (120, 21). 
venio, v., S, come, arrive (72. 1). 
vero, adv., truly, certainly (68, 5). 
versor, v., be situated, lie (102, 17). 
verus, adj., true (76, 20). 
vis, I., force, significance (66, 10). 



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Vol. V. Sources of the Synoptic Gospels. By Rev. Dr. 
Carl S. Patton, First Congregational Church, Columbus, Ohio. 
Pp. xiii + 263. $ 1.30 net. 



Size, 28 x 18.5 cm. 4to. 

Vol. VI. Athenian Lekythoi with Outline Drawing in 
Glaze Varnish on a White Ground. By Arthur Fair- 
banks, Director of the Museum of Fine Arts, Boston. 
With 15 plates, and 57 illustrations in the text. Pp. viii +371. 
Bound in cloth. $4.00 net. 



Vol. VII. Athenian Lekythoi with Outline Drawing in 
Matt Color on a White Ground, and an Appendix: 
Additional Lekythoi with Outline Drawing in Glaze 
Varnish on a White Ground. By Arthur Fairbanks. 
With 41 plates. Pp. x + 275. Bound in cloth. $3.50 net. 



Vol. VIII. The Old Testament Manuscripts in the Freer 
Collection. By Professor Henry A. Sanders, University of 
Michigan. 

Part I. The Washington Manuscript of Deuteronomy and 
Joshua. With 3 folding plates of pages of the Manuscript in 
facsimile. Pp. vi + 104. Paper covers. $1.00. 

Part II. The Washington Manuscript of the Psalms. {In 
Press.) 

THE MACMILLAN COMPANY 
Publishers 64-66 Fifth Avenue New York 



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