(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "The algebra of Mohammed ben Musa"

Digitized by the Internet Archive 

in 2007 with funding from 

IVIicrosoft Corporation 



http://www.archive.org/details/algebraofmohammeOOkhuwrich 



THE 



ALGEBRA 



MOHAMMED BEN MUSA. 



THE 



ALGEBRA 



OF 



MOHAMMED BEN MUSA. 



EDITED AND TRANSLATED 



FREDERIC ROSEN. 



LONDON: 
PRINTED FOR THE ORIENTAL TRANSLATION FUND 

AN D SOLO BY 

J. MURRAY, ALBEMARLE STREET; 
PARBURY, ALLEN, & CO., LEADENHALL STREET; 
THACKER & CO., CALCUTTA; TREUTTEL & WUERTZ, PARIS; 
AND E. FLEISCHER, LEIPZIG. 

1831. 



PRINTED BY 

J. L. COX, GREAT QUEEN STREET, 

LONDON. 



us 



PREFACE. 



In the study of history, the attention of the 
observer is drawn by a peculiar charm towards 
those epochs, at which nations, after having 
secured their independence externally, strive 
to obtain an inward guarantee for their power, 
by acquiring eminence as great in science and 
in every art of peace as they have already at- 
tained in the field of war. Such an epoch was, 
in the history of the Arabs, that of the Caliphs 
Al Mansur, Harun al Rashid, and Al 
Mam UN, the illustrious contemporaries of 
Charlemagne; to the glory of which era, 
in the volume now offered to the public, a new 
monument is endeavoured to be raised. 

Abu Abdallah Mohammed ben Musa, 
of Khowarezm, who it appears, from his pre- 
face, wrote this Treatise at the command of the 
Caliph Al Mamun, was for a long time consi- 
dered as the original inventor of Algebra. * ' Hcbc 
ars olim ^Mahomete, Mosis Arabisjilio, initi- 
umsumsit: etenim hiijus ret locuples testis Leo- 



0Aft09*> 



( vi ) 

NARDUs PisANUs." Sucli are the words with 
which HiERONYMUs Cardanus commences 
his Ars Magna, in which he frequently refers 
to the work here translated, in a manner to 
leave no doubt of its identity. 

That he was not the inventor of the Art, is 
now well established ; but that he was the first 
Mohammedan who wrote upon it, is to be found 
asserted in several Oriental writers. Haji 
Khalfa, in his bibliographical work, cites the 
initial words of the treatise now before us,* and 



* I am indebted to the kindness of my friend Mr. Gus- 
TAv Fluegel of Dresden, for a most interesting extract 
from this part of Haji Khalfa's work. Complete ma- 
nuscript copies of the ^y^\ 4^o*X are very scarce. The 
only two which I have hitherto had an opportunity of exa- 
mining (the one bought in Egypt by Dr. Ehrenberg, 
and now deposited in the Royal Library at Berlin — the other 
among Rich's collection in the British Museum) are only 
abridgments of the original compilation, in which the quo- 
tation of the initial words of each work is generally omitted. 
The prospect of an edition and Latin translation of the 
complete original work, to be published by Mr. Fluegel, 
under the auspices of the Oriental Translation Committee, 
must under such circumstances be most gratifying to all 
friends of Asiatic literature. 



( vii ) 

states, in two distinct passages, that its author, 
Mohammed ben Musa, was the first Mussul- 
man who had ever written on the solution of 
problems by the rules of completion and reduc- 
tion. Two marginal notes in the Oxford ma- 
nuscript — from which the text of the present 
edition is taken — and an anonymous Arabic 
writer, whose Bibliotheca Philosophorum is fre- 
quently quoted byCAsiRi,* likewise maintain 
that this production of Mohammed ben Musa 
was the first work written on the subjectf by a 
Mohammedan. 

* -^U.^^1 ^j^ , written in the twelfth century. Casiri 
Bibliotheca Ardbica Escurialensis, t. i. 426. 428. 

+ The first of these marginal notes stands at the top of 
the first page of the manuscript, and reads thus : Jj! ijjb 

^liillj^l ti J^iJl ^^ ^ i:;^ J^ " This is the first 
book written on (the art of calculating by) completion 
and reduction by a Mohammedan : on this account the 
author has introduced into it rules of various kinds, in 
order to render useful the very rudiments of Algebra." 
The other scholium stands farther on : it is the same to 
which I have referred in my notes to the Arabic text, 
p. 177. 



( viii ) 
From the manner in which our author, in 
his preface, speaks of the task he had under- 
takeuv we cannot infer that he claimed to be the 
inventor. He says that the Caliph Al Mamun 
encouraged him to write a popular work on Al- 
gebra: an expression which would seem to 
imply that other treatises were then already 
extant. From a formula for finding the circum- 
ference of the circle, which occurs in the work 
itself (Text p. 51, Transl. p. 72), I have, in a 
note, drawn the conclusion, that part of the in- 
formation comprised in this volume was derived 
from an Indian source ; a conjecture which is 
supported by the direct assertion of the author 
of the Bihliotheca Philosophorum quoted by Ca- 
siRi (1.426, 428). That Mohammed ben Musa 
was conversant with Hindu science, is further 
evident from the fact* that he abridged, at Al 
Mamun's request — but before the accession of 
that prince to the caliphat — the Sindhind, or 

* Related by Ebn al Ad ami in the preface to his astro- 
nomical tables. Casirj, i. 427, 428. Colebrookej Dis- 
sertation, &c. p. Ixiv. Ixxii. 



( ix ) 

astronomical tables, translated by Moham- 
med BEN Ibrahim al Fazari from the 
work of an Indian astronomer who visited the 
court of Almansur in the 156th year of the 
Hejira (A.D. 773). 

The science as taught by Mohammed ben 
MusA, in the treatise now before us, does not 
extend beyond quadratic equations, including 
problems with an affected square. These he 
solves by the same rules which are followed by 
DioPHANTUs*, and which are taught, though 
less comprehensively, by the Hindu mathemati- 
cians!. That he should have borrowed from 
DioPHANTUs is not at all probable ; for it does 
not appear that the Arabs had any knowledge 
ofDioPHANTus' work before the middle of the 
fourth century after the Hejira, when Abu'l- 
WAFA BuzjANi rendered it into ArabicJ. It 

* See DioPHANTus, Introd. § ii. and Book iv. pro- 
blems 32 and 33. 

+ Lilavatl, p. 29, Vijaganita, p. 347, of Mr. (ole- 
brooke's translation. 

I Casiui Bibl. Arab. Escur. i. 433. Colebrooke's 
Dissertation, &c. p. Ixxii. 

b 



( X ) 

is far more probable that the Arabs received 
their first knowledge of Algebra from the 
Hindus, who furnished them with the decimal 
notation of numerals, and with various im- 
portant points of mathematical and astrono- 
mical information. 

But under whatever obligation our author 
may be to the Hindus, as to the subject matter 
of his performance, he seems to have been in- 
dependent of them in the manner of digest- 
ing and treating it : at least the method which 
he follows in expounding his rules, as well as 
in showing their application, differs considerably 
from that of the Hindu mathematical writers. 
BnASKARAand Brahmagupta give dogmati- 
cal precepts, unsupported by argument, which, 
even by the metrical form in which they are 
expressed, seem to address themselves rather 
to the memory than to the reasoning faculty 
of the learner: Mohammed gives his rules 
in simple prose, and establishes their accuracy 
by geometrical illustrations. The Hindus give 
comparatively few examples, and are fond of 
investing the statement of their problems in 



( xi > 

rhetorical pomp : the Arab, on the contrary, 
is remarkably rich in examples, but he intro- 
duces them with the same perspicuous simpli- 
city of style which distinguishes his rules. In 
solving their problems, the Hindus are satisfied 
with pointing at the result, and at the principal 
intermediate steps which lead to it : the Arab 
shows the working of each example at full 
length, keeping his view constantly fixed upon 
the two sides of the equation, as upon the two 
scales of a balance, and showing how any 
alteration in one side is counterpoised by a cor- 
responding change in the other. 

Besides the few facts which have already 
been mentioned in the course of this preface, 
little or nothing is known of our Author's life. 
He lived and wrote under the caliphat of Al 
Mamun, and must therefore be distinguished 
from Abu Jafar Mohammed ben Musa*, 

* The father of the latter, Musa ben Shaker, whose 
native country I do not find recorded, had been a robber 
or bandit in the earlier part of his life, but had after- 
wards found means to attach himself to the court of the 
Caliph Al-Mamun ; who, after Musa's death, took care of 



( xii ) 

likewise a mathematician and astronomer, who 
flourished under the Caliph Al Motaded 
(who reigned A.H. 279-289, A.D. 892-902). 

the education of his three sons, Mohammed, Ahmed, and 
Al Hassan. (Abilfaragii Histor. Dyn. p. 280. Casiri, 
I. 386. 418). Each of the sons subsequently distinguished 
himself in mathematics and astronomy. We learn from 
Abulfaraj (/. c. p. 281) and from T-bn Khallikan 
(art. ^ ^^ '^i^^) that Thabet ben Korrah, the well- 
known translator of the Almagest, was indebted to Mo- 
hammed for his introduction to Al Motaded, and the 
men of science at the court of that caliph. Ebn Khalli- 

kan's words are: l^ (♦'^'J ^y^j^ tlpJ <J\/^ cT* "^J^ 

(Thabet ben Korrah) left Harran, and established 
himself at Kafratutha, where he remained till Mohammed 
BEN MusA arrived there, on his return from the Greek domi- 
nions to Bagdad. The latter became acquainted with Thabet 
and on seeing his skill and sagacity, invited Thabet to ac- 
company him to Bagdad, where Mohammed made him 
lodge at his own house, introduced him to the Caliph, and 
procured him an appointment in the body of astronomers." 
Ebn Khallikan here speaks of Mohammed ben Musa as 
of a well-known individual : he has however devoted 
no special article to an account of his life. It is possible 



( xiii ) 

The manuscript from whence the text of the 
present edition is taken — and which is the only 
copy the existence of which I have as yet been 
able to trace — is preserved in the Bodleian col- 
lection at Oxford. It is, together with three 
other treatises on Arithmetic and Algebra, 
contained in the volume marked cmxviii. 
Hunt. 214, foL, and bears the date of the 
transcription A.H. 743 (A. D. 1342). It is 
written in a plain and legible hand, but unfor- 
tunately destitute of most of the diacritical 
points : a deficiency which has often been very 
sensibly felt ; for though the nature of the sub- 
ject matter can but seldom leave a doubt as to 
the general import of a sentence, yet the true 
reading of some passages, and the precise in- 
terpretation of others, remain involved in ob- 
scurity. Besides, there occur several omissions 
of words, and even of entire sentences ; and 
also instances of words or short passages writ- 

that the tour into the provinces of the Eastern Roman Em- 
pire here mentioned, was undertaken in search of some 
ancient Greek works on mathematics or astronomy. 



( xiv ) 

ten twice over, or words foreign to the sense in- 
troduced into the text. In printing the Arabic 
part, I have included in brackets many of those 
words which I found in the manuscript, the 
genuineness of which I suspected, and also 
such as I inserted from my own conjecture, to 
supply an apparent hiatus. 

The margin of the manuscript is partially filled 
with scholia in a very small and almost illegible 
character, a few specimens of which will be found 
in the notes appended to my translation. Some 
of them are marked as being extracted from a 
commentary (^j--») by Al Mozaihafi*, pro- 
bably the same author, whose full name is Je- 

MALEDDIN AbU AbDALLAH MoHAMMED BEN 

Omar al Jaza'i-}- al Mozaihafi, and whose 
" Introduction to Arithmetic," (c->L^l ^ i«jJU) 
is contained in the same volume with Moham- 
med's work in the Bodleian library. 

Numerals are in the text of the work always 

* Wherever I have met with this name, it is written 
without the diacritical points j^Aac^^l , and my pronuncia- 
tion rests on mere conjecture. 

+ ^\A'^ ( J ) 



( XV ) 

expressed by words : figures are only used in 
some of the diagrams, and in a few marginal 
notes. 

The work had been only briefly mentioned in 
Uris' catalogue of the Bodleian manuscripts. 
Mr. H. T. Cole BROOKE first introduced it to 
more general notice, by inserting a full account 
of it, with an English translation of the direc- 
tions for the solution of equations, simple and 
compound, into the notes of the " Dissertation''' 
prefixed to his invaluable work, '* Algebra, with 
Ar^ithmetic and Mensuration, from the Sanscrit 
of Brahmegupta and Bhascara," (London, 1817, 
4to. pages Ixxv-lxxix.) 

The account of the work given by Mr. Cole- 
BROOKE excited the attention of a highly dis- 
tinguished friend of mathematical science, who 
encouraged me to undertake an edition and 
translation of the whole : and who has taken the 
kindest interest in the execution of my task. 
He has with great patience and care revised 
and corrected my translation, and has furnished 
the commentary, subjoined to the text, in the 
form of common algebraic notation. But my 



( xvi ) 

obligations to him are not confined to this only ; 
for his luminous advice has enabled me to over- 
come many difficulties, which, to my own limit- 
ed proficiency in mathematics, would have been 
almost insurmountable. 

In some notes on the Arabic text which are 
appended to my translation, I have endeavoured, 
not so much to elucidate, as to point out for 
further enquiry, a few circumstances connected 
with the history of Algebra. The comparisons 
drawn between the Algebra of the Arabs and 
that of the early Italian writers might perhaps 
have been more numerous and more detailed ; 
but my enquiry was here restricted by the 
want of some important works. Montucla, 
CossALi, HuTTON, and the Basil edition of 
Card AN us' Ars magna, were the only sources 
which I had the opportunity of consulting. 



THE AUTHOR'S PREFACE 



In the Name of God, gracious and merciful! 

This work was written by Mohammed ben Musa, of 
Khowarezm. He commences 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 Moham- 
med (on whom may the blessing 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 



( 2 ) 
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 ! 

The learned in times which have passed away, and 
among nations which have ceased to exist, were con- 
stantly employed in writing books on the several de- 
partments 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 endeavours would 
meet with acknowledgment, attention, and remem- 
brance — 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. 
(2) 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 



( 3 ) 
placed it more within reach ; others again discovered 
mistakes in preceding works, and arranged that which 
was confused, or adjusted what was irregular, and cor- 
rected the faults of their fellow-labourers, without arro- 
gance towards them, or taking pride in what they did 
themselves. 

That fondness for science, by which God has distin- 
guished 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 elucida- 
tion of obscurities and in the removal of difl&culties, 
— has encouraged me to compose a short work on Cal- 
culating by (the rules of) Completion and Reduction, 
confining it to what is easiest and most useful in arith- 
metic, 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 digging of canals, geo- 
metrical computation, and other objects of various 
sorts and kinds are concerned — relying on the good- 



( 4 ) 
ness 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 confi- 
dence rests with God, in this as in every thing, and 
in Him I put my trust. He is the Lord of the Sub- 
lime Throne. May His blessing descend upon all the 
prophets and heavenly messengers ! 



MOHAMMED BEN MUSA'S 
COMPENDIUM 



ON CALCULATING BY 



COMPLETION AND REDUCTION. 



When I considered what people generally want in (^) 
calculating, I found that it always is a number. 

I also observed that every number is composed of 
units, and that any number may be divided into units. 

Moreover, I found that every number, which may 
be expressed from one to ten, surpasses the preceding 
by one unit : afterwards the ten is doubled or tripled, 
just as before the units were : thus arise twenty, thirty, 
&c., until a hundred ; then the hundred is doubled and 
tripled in the same manner as the units and the tens, 
up to a thousand ; then the thousand can be thus re- 
peated at any complex number ; and so forth to the 
utmost limit of numeration. 

J observed that the numbers which are required 
in calculating by Completion and Reduction are of 
three kinds, namely, roots, squares, and simple numbers 
relative to neither root nor square. 



( 6 ) 

A root is any quantity which is to be multiplied by 
itself, consisting of units, or numbers ascending, or 
fractions descending.^ 

A square is the whole amount of the root multiplied 
by itself. 

A simple number is any number which may be pro- 
nounced without reference to root or square. 

A number belonging to one of these three classes 
may be equal to a number of another class; you 
may say, for instance, " squares are equal to roots," or 
" squares are equal to numbers," or " roots are equal to 
numbers."! 
/,l\ Of the case in which squares are equal to roots, this 

is an example. " A square is equal to five roots of the 
same ;"J the root of the square is five, and the square 
is twenty-five, which is equal to five times its root. 

So you say, " one third of the square is equal to four 
roots ;"§ then the whole square is equal to twelve 
roots; that is a hundred and forty-four; and its root 
is twelve. 

Or you say, " five squares are equal to ten roots ;" || 
then one square is equal to two roots; the root of 
the square is two, and its square is four. 

♦ By the word root, is meant the simple power of the 
unknown quantity. 

f cx'^ — bx cx^ = a bx=a 

X x^=5x .*. x-^ 

:4^ .•.^a:^=l2J? ,\ x=.\q. 



5f=. 



3 

5x2=100; /. 0:2 = 2X -^ x^2 



( 7 ) 

In this manner, whether the squares be many or few, 
{i, e. multiplied or divided by any number), they are 
reduced to a single square ; and the same is done with 
the roots, which are their equivalents ; that is to say, 
they are reduced in the same proportion as the squares. 

As to the case in which squares are eqtial to numbers ; 
for instance, you say, " a square is equal to nine ;"* 
then this is a square, and its root is three. Or " five 
squares are equal to eighty ; "f then one square is equal 
to one-fifth of eighty, which is sixteen. Or "the half 
of the square is equal to eighteen ;"J then the square is 
thirty-six, and its root is six. 

Thus, all squares, multiples, and sub-multiples of 
them, are reduced to a single square. If there be only 
part of a square, you add thereto, until there is a whole 
square; you do the same with the equivalent in numbers. 

As to the case in which roots are equal to numbers ; 
for instance, " one root equals three in number ; "§ then 
the root is three, and its square nine. Or " four roots (5) 
are equal to twenty ;" || then one root is equal to five, 
and the square to be formed of it is twenty-five. 
Or "half the root is equal to ten; "f then the 



* x^:=g ar=3 

t 5^2=80/. «2^^ = i6 

t ^= 18/. x^ = s6 /. *=6 

§ x=3 

II 4^=20 /. x=5 

f -|=io .-. X = 20 



( 8 ) 

whole root is equal to twenty, and the square which is 
formed of it is four hundred. 

I found that these three kinds ; namely, roots, 
squares, and numbers, may be combined together, and 
thus three compound species arise ;* that is, " squares 
and roots equal to numbers ;'* " squares and numbers 
equal to roots ,*" "roots and numbers equal to squares." 

Roots and Squares are equal to Numbers ;\ for in- 
stance, " one square, and ten roots of the same, amount 
to thirty-nine dirhems ;" that is to say, what must be 
the square which^ when increased by ten of its own 
roots, amounts to thirty-nine? The solution is this : you 
halve the numberj of the roots, which in the present 
instance yields five. This you multiply by itself; 
the product is twenty-five. Add this to thirty-nine; 
the sum is sixty- four. Now take the root of this, which 
is eight, and subtract from it half the number of the 
roots, which is five ; the remainder is three. This is 
the root of the square which you sought for; the 
square itself is nine. 



* The three cases considered are, 

1st. cx^-^bx=a 

2d. cx'-\- a —bx 

3d. cx^ — bx-\-a 

f 1 St case : cx"^ -^bx=za 

Example x' + 1 ox = 39 

= ^64 — 5 
= 8-5 = 3 
X i. e. the coefficient. 



( 9 ) 

The solution is the same when two squares or three, 
or more or less be specified ;* you reduce them to one 
single square, and in the same proportion you reduce 
also the roots and simple numbers which are connected 
therewith. 

For instance, " two squares and ten roots are equal 
to forty-eight dirhems ;"f that is to say, what must be '^ 
the amount of two squares which, when summed up and 
added to ten times the root of one of them, make up a 
sum of forty-eight dirhems ? You must at first reduce 
the two squares to one ; and you know that one square 
of the two is the moiety of both. Then reduce every 
thing mentioned in the statement to its half, and it will 
be the same as if the question had been, a square and 
five roots of the same are equal to twenty-four dirhems; 
or, what must be the amount of a square which, when 
added to five times its root, is equal to twenty-four dir- 
hems ? Now halve the number of the roots; the moiety 
is two and a half. Multiply that by itself; the pro- 
duct is six and a quarter. Add this to twenty-four ; the 
sum is thirty dirhems and a quarter. Take the root of 
this ; it is five and a half. Subtract from this the moiety 
of the number of the roots, that is two and a half; the 



* ex' -\-bx=a is to be reduced to the form x^-^—x. 
t 2aP+iox=4S 

^ = ^/[(|)''+24]-| 

= 5i - 2i = 3 
c 



( 10 ) 

remainder is three. This is the root of the square, and 
the square itself is nine. 

The proceeding will be the same if the instance be, 
" half of a square and five roots are equal to twenty-eight 
dirhems ;"* that is to say, what must be the amount of 
a square, the moiety of which, when added to the equi- 
valent of five of its roots, is equal to twenty-eight dir- 
hems ? Your first business must be to complete your 
square, so that it amounts to one whole square. This 
you effect by doubling it. Therefore double it, and dou- 
ble also that which is added to it, as well as what is equal 
to it. Then you have a square and ten roots, equal to 
fifty-six dirhems. Now halve the roots ; the moiety is 
five. Multiply this by itself; the product is twenty-five. 
Add this to fifty-six ; the sum is eighty-one. Extract 
the root of this; it is nine. Subtract from this the 
moiety of the number of roots, which is five ; the re- 
mainder is four. This is the root of the square which 
you sought for ; the square is sixteen, and half the 
(*7) square eight. 

Proceed in this manner, whenever you meet with 
squares and roots that are equal to simple numbers : for 
it will always answer. 



* 1-1-50:^28 
x--|-iox=56 



10\2 I ^fi-i 1 



= v^ 25 -}- 56 - 5 
= n/8i - 5 
-9-5 = 4 



( 11 ) 

Squares and Numbers are equal to Roots;* for 
instance, " a square and twenty-one in numbers are 
equal to ten roots of the same square." That is to say, 
what must be the amount of a square, which, when 
twenty-one dirhems are added to it, becomes equal to 
the equivalent of ten roots of that square? Solution : 
Halve the number of the roots; the moiety is five. 
Multiply this by itself; the product is twenty-five. 
Subtract from this the twenty-one which are connected 
with the square ; the remainder is four. Extract its 
root ; it is two. Subtract this from the moiety of the 
roots, which is five ; the remainder is three. This is the 
root of the square which you required, and the square 
is nine. Or you may add the root to the moiety of the 
roots ; the sum is seven ; this is the root of the square 
which you sought for, and the square itself is forty- 
nine. 

When you meet with an instance which refers you to 
this case, try its solution by addition, and if that do not 
serve, then subtraction certainly will. For in this case 
both addition and subtraction may be employed, which 
will not answer in any other of the three cases in which 



* '2d case. cx^-\-a-bx 

Example. a:'4-2i— lo^ 



5 — n/ 25 —21 
5=t2 



( 12 ) 

the number of the roots must be halved. And know, 
that, when in a question belonging to this case you 
have halved the number of the roots and multiplied 
the moiety by itself, if the product be less than the 
number of dirhems connected with the square, then the 
instance is impossible;* but if the product be equal to 
(8) the dirhems by themselves, then the root of the square 
is equal to the moiety of the roots alone, without either 
addition or subtraction. 

In every instance where you have two squares, or 
more or less, reduce them to one entire square, f as I 
have explained under the first case. 

Roots and Numbers are equal to Squares ;% for instance, 
*' three roots and four of simple numbers are equal 
to a square." Solution : Halve the roots ; the moiety 
is one and a half. Multiply this by itself; the product 
is two and a quarter. Add this to the four ; the sum is 



* If in an equation, of the form x^-^-azz-bx, (|.)2 z. a, 
the case supposed in the equation cannot happen. If 
(1)2= a, then^=^ 

f cx'^'\-a=bx is to be reduced to x^ +4=—^ 
J 3d case cx^ =:zbx+a 
Example x^ = 30; + 4 

= v/(i i)2 + 4 +ii 
= \/ 2j^+4 +ij 

= 2j +ij=4 



( 13 ) 

six and a quarter. Extract its root ; it is two and a 
half. Add this to the moiety of the roots, which was 
one and a half; the sum is four. This is the root of the 
square, and the square is sixteen. 

Whenever you meet with a multiple or sub-multiple 
of a square, reduce it to one entire square. 

These are the six cases which I mentioned in the 
introduction to this book. They have now been ex- 
plained. I have shown that three among them do not 
require that the roots be halved, and I have taught 
how they must be resolved. As for the other three, in 
which halving the roots is necessary, I think it expe- 
dient, more accurately, to explain them by separate 
chapters, in which a figure will be given for each 
case, to point out the reasons for halving. 

Demonstration of the Case : " a Square and ten Roots 
are equal to thirty-nine Dirhems"^ 

The figure to explain this a quadrate, the sides of 
which are unknown. It represents the square, the 
which, or the root of which, you wish to know. This is 
the figure A B, each side of which may be considered 
as one of its roots ; and if you multiply one of these (9) 
sides by any number, then the amount of that number 
may be looked upon as the number of the roots which 
are added to the square. Each side of the quadrate 
represents the root of the square; and, as in the instance, 

* Geometrical illustration of the case, x^+ ioa: = 39 



( 14 ) 

the roots were connected with the square, we may take 
one-fourth of ten, that is to say, two and a half, and 
combine it with each of the four sides of the figure. 
Thus with the original quadrate A B, four new paral- 
lelofrrams are combined, each having a side of the qua- 
drate as its length, and the number of two and a half as 
its breadth ; they are the parallelograms C, G, T, and 
K. We have now a quadrate of equal, though unknown 
sides ; but in each of the four corners of which a square 
piece of two and a half multiplied by two and a half is 
wanting. In order to compensate for this want and to 
complete the quadrate, we must add (to that which we 
have already) four times the square of two and a half, that 
is, twenty-five. We know (by the statement) that the first 
figure, namely, the quadrate representing the square, 
together with the four parallelograms around it, which 
represent the ten roots, is equal to thirty-nine of num- 
bers. If to this we add twenty-five, which is the equivalent 
of the four quadrates at the corners of the figure A B, 
by which the great figure D H is completed, then we 
know that this together makes sixty-four. One side 
of this great quadrate is its root, tliat is, eight. If we 
subtract twice a fourth of ten, that is five, from eight, 
as from the two extremities of the side of the great 
quadrate D H, then the remainder of such a side will 
be three, and that is the root of the square, or the side 
of the original figure A B. It must be observed, that 
we have halved the number of the roots, and added the 
product of the moiety multiplied by itself to the number 



( 15 ) 



thirty-nine, in order to complete the great figure in its 
four corners ; because the fourth of any number multi- 
plied by itself, and then by four, is equal to the product 
of the moiety of that number multiplied by itself.* 
Accordingly, we multiplied only the moiety of the roots 
by itself, instead of multiplying its fourth by itself, and 
then by four. This is the figure : 



(10) 



u 




a 






c 


A 
R 


K 






T 





The same may also be explained by another figure. 
We proceed from the quadrate A B, which represents 
the square. It is our next business to add to it the ten 
roots of the same. We halve for this purpose the ten, 
so that it becomes five, and construct two quadrangles 
on two sides of the quadrate A B, namely, G and D, 
the length of each of them being ^ve, as the moiety of 
the ten roots, whilst the breadth of each is equal to a 
side of the quadrate A B. Then a quadrate remains 
opposite the corner of the quadrate A B. This is equal 
to five multiplied by five : this five being half of the 
number of the roots which we have added to each of the 
two sides of the first quadrate. Thus we know that 



<b \- 



* M^r=(k) 



( 16 ) 

the first quadrate, which is the square, and the two 
quadrangles on its sides, which are the ten roots, make 
together thirty-nine. In order to complete the great 
quadrate, there wants only a square of five multiplied 
(11) by five, or twenty-five. This we add to thirty-nine, in 
order to complete the great square S H. The sum is 
sixty-four. We extract its root, eight, which is one of 
the sides of the great quadrangle. By subtracting from 
this the same quantity which we have before added, 
namely five, we obtain three as the remainder. This is 
the side of the quadrangle A B, which represents the 
square; it is the root of this square, and the square 
itself is nine. This is^ the figure : — 



G- 


B 


26 




D 



Demonstration of the Case : « a Square and twenty-me 
Dirhems are equal to ten Boots,'* *^ 
We represent the square by a quadrate A D, the 
length of whose side we do not know. To this we join a 
parallelogram, the breadth of which is equal to one of 
the sides of the quadrate A D, such as the side H N. 
This paralellogram is H B. The length of the two 



* Geometrical illustration of the case, a:' + 2 1 = 1 



ox 



( 17 ) 

figures together is equal to the line H C. We know 
that its length is ten of numbers ; for every quadrate 
has equal sides and angles, and one of its sides multi- 
plied by a unit is the root of the quadrate, or multiplied 
by two it is twice the root of the same. As it is stated, 
therefore, that a square and twenty-one of numbers are 
equal to ten roots, we may conclude that the length of 
the line H C is equal to ten of numbers, since the line 
C D represents the root of the square. We now divide 
the line C H into two equal parts at the point G : the 
line G C is then equal to H G. It is also evident that (12) 
the line G T is equal to the line C D. At present we 
add to the line G T, in the same direction, a piece 
equal to the difference between C G and G T, in order 
to complete the square. Then the line T K becomes 
equal to K M, and we have a new quadrate of equal 
sides and angles, namely, the quadrate M T. We 
know that the line T K is five ; this is consequently the 
length also of the other sides : the quadrate itself is 
twenty-five, this being the product of the multiplication 
of half the number of the roots by themselves, for five 
times five is twenty-five. We have perceived that the 
quadrangle H B represents the twenty-one of numbers 
which were added to the quadrate. We have then cut 
off a piece from the quadrangle H B by the line K T 
(which is one of the sides of the quadrate M T), so that 
only the part T A remains. At present we take from 
the line K M the piece K L, which is equal to G K; it 
then appears that the line T G is equal to M L ; more- 

D 



( 18 ) 

over, the line K L, which has been cut off from K M^ 
is equal to K G; consequently, the quadrangle MR is 
equal to T A. Thus it is evident that the quadrangle 
H T, augmented by the quadrangle M R, is equal to 
the quadrangle H B, which represents the twenty-one. 
The whole quadrate M T was found to be equal to 
twenty-five. If we now subtract from this quadrate, 
MT, the quadrangles HT and M R, which are equal 
to twenty-one, there remains a small quadrate K R, 
which represents the difference between twenty-five and 
twenty-one. This is four ; and its root, represented by 
the line R G, which is equal to G A, is two. If you 
(13) subtract this number two from the line C G, which is 
the moiety of the roots, then the remainder is the line 
A C ; that is to say, three, which is the root of the ori- 
ginal square. But if you add the number two to the 
line C G, which is the moiety of the number of the 
roots, then the sum is seven, represented by the line 
C R, which is the root to a larger square. However, 
if you add twenty-one to this square, then the sum will 
likewise be equal to ten roots of the same square. Here 
is the figure : — 

^^r L K 

A . C 



K G 



IT 



T B 



( 19 ) 

Demonstration of the Case : " three Roots and four of 
Simple Numbers are equal to a Square"^ 
Let the square be represented by a quadrangle, the 
sides of which are unknown to us, though they are equal 
among themselves, as also the angles. This is the qua- 
drate A D, which comprises the three roots and the four 
of numbers mentioned in this instance. In every qua- 
drate one of its sides, multiplied by a unit, is its root. 
We now cut off the quadrangle H D from the quadrate 
A D, and take one of its sides H C for three, which is 
the number of the roots. The same is equal to R D. 
It follows, then, that the quadrangle H B represents 
the four of numbers which are added to the roots. Now 
we halve the side C H, which is equal to three roots, at 
the point G ; from this division we construct the square 
H T, which is the product of half the roots (or one and (14) 
a half) multiplied by themselves, that is to say, two and 
a quarter. We add then to the line G T a piece equal 
to the line A H, namely, the piece T L ; accordingly 
the line G L becomes equal to A G, and the line K N 
equal to T L. Thus a new quadrangle, with equal 
sides and angles, arises, namely, the quadrangle G M ; 
and we find that the line A G is equal to M L, and the 
same line A G is equal to G L. By these means the 
line C G remains equal to N R, and the line M N 
equal to T L, and from the quadrangle H B a piece 
equal to the quadrangle K L is cut off. 

* Geometrical illustration of the 3d case, x- = 3^ + 4 



( 20 ) 

But we know that the quadrangle A R represents the 
four of numbers which are added to the three roots. 
The quadrangle A N and the quadrangle K L are to- 
gether equal to the quadrangle A R, which represents 
the four of numbers. 

We have seen, also, that the quadrangle G M com- 
prises the product of the moiety of the roots, or of one 
and a half, multiplied by itself; that is to say two and 
a quarter, together with the four of numbers, which are 
represented by the quadrangles A N and K L. There 
remains now from the side of the great original quadrate 
A D, which represents the whole square, only the moiety 
of the roots, that is to say, one and a half, namely, the 
line G C. If we add this to the line A G, which is 
the root of the quadrate G M, being equal to two and 
a half; then this, together with C G, or the moiety of 
the three roots, namely, one and a half, makes four, 
which is the line A C, or the root to a square, which 
is represented by the quadrate A D. Here follows 
the figure. This it was which we were desirous to 
explain. 

(1^) B M A 



R 



N 



( 21 ) 

We have observed that every question which requires 
equation or reduction for its solution, will refer you to 
one of the six cases which I have proposed in this 
book. I have now also explained their arguments. 
Bear them, therefore, in mind. 



ON MULTIPLICATION. 

I SHALL now teach you how to multiply the unknown 
numbers, that is to say, the roots, one by the other, if 
they stand alone, or if numbers are added to them, or if 
numbers are subtracted from them, or if they are sub- 
tracted from numbers ; also how to add them one to the 
other, or how to subtract one from the other. 

Whenever one number is to be multiplied by another, 
the one must be repeated as many times as the other 
contains units.* 

If there are greater numbers combined with units to 
be added to or subtracted from them, then four multi- 
plications are necessary ;f namely, the greater numbers 
by the greater numbers, the greater numbers by the 



* If or is to be multiplied by y, x is to be repeated as 
many times as there are units in t/. 

f If X zt « is to be multiplied by j/ =t b, x is to be mul- 
tiplied by y, X is to be multiplied by i, a is, to be multiplied 
by y, and a is to be multiplied by b. 



( 22 ) 

units, the units by the greater numbers, and the units 
by the units. 

If the units, combined with the greater numbers, are 

positive, then the last multiplication is positive ; if they 

are both negative, then the fourth multiplication is like- 

vi^ise positive. But if one of them is positive, and one 

(16) negative, then the fourth multiplication is negative.* 

For instance, " ten and one to be multiplied by ten 
and two."f Ten times ten is a hundred ; once ten is 
ten positive ; twice ten is twenty positive, and once two 
is two positive; this altogether makes a hundred and 
thirty-two. 

But if the instance is " ten less one, to be multiplied 
by ten less one,"t then ten times ten is a hundred ; the 



* In multiplying y^xzha) by (ydb^) 
-{■ax-\-b = -{-ab 

— ax —bz=-\-ab 
■\-aX'-b=—ab 

— ax +b=—ab 

t (io + i)x(io + 2) 

= 10X10.... 100 

+ 1 XIO 10 

4- 2X10 20 

+ 1X2 2 

+ 132 

X (10-1) (10-1) 

= 10X 10.. +100 

— IX 10.. — 10 

— IX 10.. — 10 

— IX -1.. + 1 



+ 81 



( 23 ) 

negative one by ten is ten negative ; the other negative 
one by ten is likewise ten negative, so that it becomes 
eighty : but the negative one by the negative one is 
one positive, and this makes the result eighty-one. 

Or if the instance be " ten and two, to be multipled 
by ten less one,"* then ten times ten is a hundred, and 
the negative one by ten is ten negative; the positive 
two by ten is twenty positive ; this together is a hun- 
dred and ten ; the positive two by the negative one 
gives two negative. This makes the product a hundred 
and eight. 

I have explained this, that it might serve as an intro- 
duction to the multiplication of unknown sums, when 
numbers are added to them, or when numbers are 
subtracted from them, or when they are subtracted from 
numbers. 

For instance : " Ten less thing (the signification of 
thing being root) to be multipled by ten."f You 
begin by taking ten times ten, which is a hundred ; less 
thing by ten is ten roots negative; the product is there- 
fore a hundred less ten things. 



* (10 + -2)X(10— l) = 




\ 


10X10.... 100 






— 1 xio —lO 






+ 10X 2 +20 






- IX 2.. . . — 2 






lOS 






f (10— jc)x io=iox 10 — iox=:ioo~ 


-loj;. 





( 24 ) 

If the instance be : " ten and thing to be multiplied 
by ten,"* then you take ten times ten, which is a hun- 
dred, and thing by ten is ten things positive ; so that the 
product is a hundred plus ten things. 

If the instance be : " ten and thing to be multiplied 
(17) by itself,"t then ten times ten is a hundred, and ten 
times thing is ten things ; and again, ten times thing is 
ten things ; and thing multiplied by thing is a square 
positive, so that the whole product is a hundred dir- 
hems and twenty things and one positive square. 

If the instance be : " ten minus thing to be multiplied 
by ten minus thing, "J then ten times ten is a hundred; 
and minus thing by ten is minus ten things; and 
again, minus thing by ten is minus ten things. But 
minus thing multiplied by minus thing is a positive 
square. The product is therefore a hundred and a 
square, minus twenty things. 

In like manner if the following question be proposed 
to you : " one dirhem minus one-sixth to be multiplied 
by one dirhem minus one-sixth ;"§ that is to say, five- 
sixths by themselves, the product is five and twenty 
parts of a dirhem, which is divided into six and thirty 
parts, or two-thirds and one-sixth of a sixth. Compu- 
tation : You multiply one dirhem by one dirhem, the 



*(io+x)x io=iox 10+ loor — 100+1 oj; 

f (lO+x) (I0+j;)=10 X lO+lO^+lO^ + X- — 100 + 20:r + X*'^ 

:j:(io— a:)x(i0-:r) r::iox 10— lox— lox+x-^ioo— 20a:+a:- 
*(i-*)x(i-J)--i-^ + ixi = |+Jxi;/.e.J^,.|+ixA 



{ 25 ) 

product is one clirhem ; then one dirhem by minus one- 
sixth, that is one-sixth negative ; then, again, one dir- 
hem by minus one-sixth is one-sixth negative : so far, 
then, the result is two-thirds of a dirhem : but there is 
still minus one-sixth to be multiplied by minus one-sixth, 
which is one-sixth of a sixth positive ; the product is, 
therefore, two- thirds and one sixth of a sixth. 

If the instance be, " ten minus thing to be multiplied 
by ten and thing," then you say,* ten times ten is a 
hundred ; and minus thing by ten is ten things negative; 
and thing by ten is ten things positive; and minus 
thing by thing is a square positive ; therefore, the 
product is a hundred dirhems, minus a square. 

If the instance be, " ten minus thing to be multiplied 
by thing,"t then you say, ten multiplied by thing is ten 
things; and minus thing by thing is a square negative ; (18) 
therefore, the product is ten things minus a square. 

If the instance be, " ten and thing to be multiplied 
by thing less ten,"| then you say, thing multiplied by 
ten is ten things positive ; and thing by thing is a square 
positive ; and minus ten by ten is a hundred dirhems 
negative ; and minus ten by thing is ten things nega- 
tive. You say, therefore, a square minus a hundred 
dirhems ; for, having made the reduction, that is to say, 
having removed the ten things positive by the ten things 

* (lo— x) (io + j;) = iox 10— lojr+iox— x'-' = ioo— x2 

f (lo— x) xa: = ioar— .r- 

X (lo + x) (x— io) = io^-|-i;'-— loo — ioa: ~a?-— 100 

E 



( 26 ) 

negative, there remains a square minus a hundred 
dirhems. 

If the instance be, ^' ten dirhems and half a thing to 
be multiplied by half a dirhem, minus five things,"* 
then you say, half a dirhem by ten is five dirhems posi- 
tive ; and half a dirhem by half a thing is a quarter of 
thing positive ; and minus five things by ten dirhems is 
fifty roots negative. This altogether makes five dir- 
hems minus forty-nine things and three quarters of 
thing. After this you multiply five roots negative by 
half a root positive : it is two squares and a half negative. 
Therefore, the product is five dirhems, minus two 
squares and a half, minus forty-nine roots and three 
quarters of a root. 

If the instance be, " ten and thing to be multiplied 
by thing less ten,"f then this is the same as if it were 
said thing and ten by thing less ten. You say, there- 
fore, thing multiplied by thing is a square positive ; and 
ten by thing is ten things positive ; and minus ten by 
thing is ten things negative. You now remove the 
positive by the negative, then there only remains a 
square. Minus ten multiplied by ten is a hundred, to 
be subtracted from the square. This, therefore, alto- 
gether, is a square less a hundred dirhems. 
(19) Whenever a positive and a negative factor concur in 



t( 1 o -f j:)(a;— 1 o) := (ar-f 1 o)(a;— 1 o) - jr- 4- 1 go:— 1 ox— 1 00 = ar2 - 1 00 



( 27 ) 

a multiplication, such as thing positive and minus thing, 
the last multiplication gives always the negative pro- 
duct. Keep this in memory. 



ON ADDITION and SUBTRACTION. 

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, dimi- 
nished 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. 

If you require to double the root of any known or 
unknown square, (the meaning of its duplication being 

* 20— V^200-j-('v/200— 10)=10 

.j. 20— v/200— (-v/sOO— 10)z=30— 2\/200 = 30— \/8oo 

t 50+ lox— 2x'^-f (loo+a?'-^— 20a;) = i50— 10*— ^2 
§ loo+o;'-- 20a;— [50— 2x'- + ioa:] =50 + 3x''^— 30* 



( 28 ) 

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 pro- 
duct 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.t 
(20) 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 : If you require to double the root 
of nine, II you multiply two by two, and then by nine: 
this gives thirty- six ; take the root of this, it is six, 
and this is double the root of nine. 



2v'9 = v/4X9 = v/36=6 



( 29 ) 

In the same manner, if you require to triple the root of 
nine,* you multiply three by three, and then by nine : 
the product is eighty-one ; take its root, it is nine, which 
becomes equal to thrice the root of nine. 

If you require to have the moiety of the root of nine,t 
you multiply a half by a half, which gives a quarter, and 
then this by nine ; the result is two and a quarter : take 
its root ; it is one and a half, which is the moiety of the 
root of nine. 

You proceed in this manner with every root, whether 
positive or negative, and whether known or unknown. 



ON DIVISION. 



If you will divide the root of nine by the root of four4 
you begin with dividing nine by four, which gives two 
and a quarter : the root of this is the number which you 
require — it is one and a half. 

If you will divide the root of four by the root of nine,§ 
you divide four by nine ; it is four-ninths of the unit : 
the root of this is two divided by three ; namely, two- 
thirds of the unit. 



* 3v/9 = v^9X9=>/8i=9 



( 30 ) 

If you wish to divide twice the root of nine by the 
root of four^ or of any other square*, you double the 
(21) 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 above pointed out. 

In like manner, if you wish to divide three roots 
of nine, or more, or one-half or any multiple or sub- 
multiple of ihe root of nine, the rule is always the 
same :t follow it, the result will be right. 

If you wish to multiply the root of nine by the root of 
four,+ multiply nine by four ; this gives thirty- six ; take 
its root, it is six ; this is the root of nine, multiplied by 
the root of four. 

Thus, if you wish to multiply the root of five by the 
root of ten,§ multiply five by ten : the root of the pro- 
duct is what you have required. 

If you wish to multiply the root of one- third by the 
root of a half, II you multiply one- third by a half: it is 
one- sixth : the root of one- sixth is equal to the root of 
one-third, multiplied by the root of a half. 

If you require to multiply twice the root of nine by 



* Sv^g 



= x/^ = v/9: 



V4 

X >/4Xv/9=\/4X9=v/36 = 6 



§ v'ioxv'5= n/5xio=v'5o 

II s/^y^s/h^s/W^^s/^ 



( 31 ) 

thrice the root of fom', " then take twice the root of nine, 
according to the rule above given, so that you may know 
the root of what square it is. You do the same with 
respect to the three roots of four in order to know what 
must be the square of such a root. You then multiply 
these two squares, the one by the other, and the root of 
the product is equal to twice the root of nine, multiplied 
by thrice the root of four. 

You proceed in this manner with all positive or ne- 
gative roots. 

Demomtratiom, (22) 

The argument for the root of two hundred, minus ten, 
added to twenty, minus the root of two hundred, may be 
elucidated by a figure : 

Let the line A B represent the root of two hundred ; 
let the part from A to the point C be the ten, then the 
remainder of the root of two hundred will correspond to 
the remainder of the line A B, namely to the line C B. 
Draw now from the point B a line to the point D, to 
represent twenty ; let it, therefore, be twice as long as 
the line A C, which represents ten; and mark a part of 
it from the point B to the point H, to be equal to the 
line A B, which represents the root of two hundred; 
then the remainder of the twenty will be equal to the 
part of the line, from the point H to the point D. As 



3\/4 X 2 ^9 = >v/9 X4 X ^/4 X 9 - v/36 X 36=36 



( .^s ) 

our object was to add the remainder of the root of two 
hundred, after the subtraction of ten, that is to say, the 
hne C B, to the line H D, or to twenty, minus the root 
of two hundred, we cut off from the line B H a piece 
equal to C B, namely, the line S H. We know already 
that the line A B, or the root of two hundred, is equal to 
the line B H, and that the line A C, which represents the 
ten, is equal to the line S B, as also that the remainder 
of the line A B, namely, the line C B is equal to the 
remainder of the line B H, namely, to S H. Let us 
add, therefore, this piece S H, to the line H D. We 
have already seen that from the line B D, or twenty, a 
piece equal to A C, which is ten, was cut off, namely, 
the piece B S. There remains after this the line S D, 
which, consequently, is equal to ten. This it was that 
we intended to elucidate. Here follows the figure. 
(23) AJ 



S BT S S 

The argument for the root of two hundred, minus ten, 
to be subtracted from twenty, minus the root of two 
hundred, is as follows. Let the line A B represent the 
root of two hundred, and let the part thereof, from A to 
the point C, signify the ten mentioned in the instance. 
We draw now from the point B, a line towards the point 
D, to signify twenty. Then we trace from B to the 



( 33 ) 

point H, the same lengtli as the leiigtli of the line which 
represents the root of two liundred ; that is of the line 
A B. We have seen that the line C B is the remainder 
from the twenty, after the root of two hundred has been 
subtracted. It is our purpose, therefore, to subtract 
the line C B from the line H D ; and we now draw from 
the point B, a line towards the point S, equal in length 
to the line A C, which represents the ten. Then the 
whole line S D is equal to S B, plus B D, and we per- 
ceive that all this added together amounts to thirty. 
We now cut off from the line H D, a piece equal to 
C B, namely, ,the line H G ; thus we find that the line 
G D is the remainder from the line S D, which signifies 
thirty. We see also that the line B H is the root of 
two hundred and that the line S B and B C is likewise 
the root of two hundred. Kow the line H G is equal 
to C B ; therefore the piece subtracted from the line 
S D, which represents thirty, is equal to twice the 
root of two hundred, or once the root of eight hundred. (^^) 
This it is that we wished to elucidate. 
Here follows the figure : 



1> G M B S 

As for the hundred and square minus twenty roots 
added to fifty, and ten roots minus two squares, this does 

F 



( 34 ) 

not admit of any figure, because there are three diffe- 
rent species, viz. squares, and roots, and numbers, and 
nothing corresponding to them by which they might 
be represented. We had, indeed, contrived to con- 
struct a figure also for this case, but it was not suffi- 
ciently clear. 

The elucidation by words is very easy. You know 
that you have a hundred and a square, minus twenty 
roots. When you add to this fifty and ten roots, it be- 
comes a hundred and fifty and a square, minus ten roots. 
The reason for these ten negative roots is, that from the 
twenty negative roots ten positive roots were subtracted 
by reduction. This being done, there remains a hun- 
dred and fifty and a square, minus ten roots. With the 
hundred a square is connected. If you subtract from 
this hundred and square the two squares negative con- 
nected with fifty, then one square disappears by reason 
of the other, and the remainder is a hundred and fifty, 
minus a square, and minus ten roots. 
This it was that we wished to explain. 



( 35 ) 



OF THE SIX PROBLEMS. 

Before the chapters on computation and the several (25) 
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 calculat- 
ing by completion and reduction must always neces- 
sarily lead you to one of these cases. I now subjoin 
these problems, which will serve to bring the sub- 
ject nearer to the understanding, to render its com- 
prehension easier, and to make the arguments more 
perspicuous. 

First Problem, 

I have divided ten into two portions ; I have multi- 
plied the one of the two portions by the other ; after 
this I have multiplied the one of the two by itself, 
and the product of the multiplication by itself is four 
times as much as that of one of the portions by the 
other.* 

Computation : Suppose one of the portions to be 
thing, and the other ten minus thing : you multiply 

* x2--4^jlo— a;)=:40a;— 4x2 
5*2=400: 

X =8; (10— a;)=2 



( 36 ) 

thing by ten minus thing ; it is ten things minus a 
square. Then multiply it by four, because the in- 
stance states " four times as much." The result will be 
four times the product of one of the parts multiplied by 
the other. This is forty things minus four squares. 
After this you multiply thing by thing, that is to say, 
one of the portions by itself. This is a square, which 
is equal to forty things minus foursquares. Reduce it 
now by the four squares, and add them to the one 
square. Then the equation is : forty things are equal 
to five squares ; and one square will be equal to eight 
roots, that is, sixty-four ; the root of this is eight, and 
this is one of the two portions, namely, that which is to 
(26) be multiplied by itself. The remainder from the ten 
is two, and that is the other portion. Thus the question 
leads you to one of the six cases, namely, that of 
" squares equal to roots." Remark this. 

Second Problem, 

I have divided ten into two portions : I have multi- 
plied each of the parts by itself, and afterwards ten by 
itself: the product often 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.* 



* \o^=x^ 


X2^ 


100 =rx2 


^ 9 


^^xioo 


=x' 


36=:i:2* 




6=x 





( 37 ) 

Computation : Suppose one of the parts to be thing, 
and the other ten minus thing. You multiply thing by 
itself, it is a square; then by two and seven-ninths, 
this makes it two squares and seven- ninths of a square. 
You afterwards multiply ten by ten ; it is a hundred, 
which much be equal to two squares aaid seven-ninths 
of a square. Reduce it to one square, through division 
by nine twenty- fifths ;^ this being its fifth and four- 
fifths of its fifth, take now also the fifth and four-fifths 
of the fifth of a hundred ; this is thirty-six, which is 
equal to one square. Take its root, it is six. This is 
one of the two portions ; and accordingly the other is 
four. This question leads you, therefore, to one of the 
six cases, namely, " squares equal to numbers." 

Third Problem, 

I have divided ten into two parts. I have afterwards 
divided the one by the other, and the quotient was four.f 

Computation : Suppose one of the two parts to be (27) 
thing, the other ten minus thing. Then you divide ten 
minus thing by thing, in order that four may be ob- 
tained. You know that if you multiply the quotient 
by the divisor, the sum which was divided is restored. 



X 

10 — XZZ.^ 

io=5x 
2=j: 



( 38 ) 

In the present question tlie quotient is four and the 
divisor is thing. Multiply, therefore, four by thing ; 
the result is four things, which are equal to the sum to 
be divided, which was ten minus thing. You now 
reduce it by thing, which you add to the four things. 
Then we have ^\e things equal to ten ; therefore one 
thing is equal to two, and this is one of the two portions. 
This question refers you to one of the six cases, 
namely, " roots equal to numbers." 

Fourth Problem. 

I have multiplied one- third of thing and one dirhem 
by one-fourth of thing and one dirhem, and the product 
was twenty.* 

Computation : You multiply one- third of thing by 
one- fourth of thing; it is one-half of a sixth of a square. 
Further, you multiply one dirhem by one-third of thing, 
it is one- third of thing ; and one dirhem by one-fourth 
of thing, it is one-fourth of thing ; and one dirhem by 
one dirhem, it is one dirhem. The result of this is : the 
moiety of one-sixth of a square, and one- third of thing, 
and one-fourth of thing, and one dirhem, is equal to 
twenty dirhems. Subtract now the one dirhem from 

* (J:c+i)(J;r+i)=20 



( 39 ) 

these twenty dirhems, there remain nineteen dirhems, 
equal to the moiety of one-sixth of a square, and one- 
third of thing, and one-fourth of thing. Now make your 
square a whole one : you perform this by multiplying all 
that you have by twelve. Thus you have one square 
and seven roots, equal to two hundred and twenty-eight 
dirhems. Halve the number of the roots, and multiply 
it by itself; it is twelve and one- fourth. Add this to 
the numbers, that is, to two hundred and twenty-eight ; (28) 
the sum is two hundred and forty and one quarter. Ex- 
tract the root of this; it is fifteen and a half. Subtract 
from this the moiety of the roots, that is, three and a 
half, there remains twelve, which is the square required. 
This question leads you to one of the cases, namely, 
" squares and roots equal to numbers." 

Fifth Problem. 

I have divided ten into two parts ; I have then multi- 
plied each of them by itself, and when I had added the 
products together, the sum was fifty-eight dirhems.* 

Computation : Suppose one of the two parts to be 
thing, and the other ten minus thing. Multiply ten 
minus thing by itself; it is a hundred and a square 
minus twenty things. Then multiply thing by thing ; it 



* «2-i-{lo-a:)2=58 
2 j;2 — 20*4-100 = 58 
jf-^ — 100:4-50=29 
jc24.2i = ioa: 
a;=5d=v/25~2i=:5d:2=7 or 3 



( 40 ) 

is a square. Add both together. The sum is a hun- 
dred, phis two squares minus twenty things, which are 
equal to fifty-eight dirhems. Take now the twenty 
negative things from the hundred and the two squares, 
and add them to fifty- eight ; then a hundred, plus two 
squares, are equal to fifty-eight dirhems and twenty 
things. Reduce this to one square, by taking the moiety 
of all you have. It is then: fifty dirhems and a square, 
which are equal to twenty-nine dirhems and ten things. 
Then reduce this, by taking twenty-nine from fifty ; 
there remains twenty-one and a square, equal to ten 
things. Halve the number of the roots, it is five; multiply 
this by itself, it is twenty-five; take from this the twenty- 
one which are connected with the square, the remainder 
^ ^ is four. Extract the root, it is two. Subtract this from 
the moiety of the roots, namely, from five, there remains 
three. This is one of the portions; the other is seven. 
This question refers you to one of the six cases, namely 
*' squares and numbers equal to roots." 

Sixth Problem. , 

I have multiplied one-third of a root by one-fourth 
of a root, and the product is equal to the root and 
twenty-four dirhems.* 



.3 4 






( 4.1 ) 

Computation : Call the root thing; then one- third of 
thing is multiplied by one-fourth of thing ; this is the 
moiety of one-sixth of the square, and is equal to thing 
and twenty -four dirhems. Multiply this moiety of one- 
sixth of the square by twelve, in order to make your 
square a whole one, and multiply also the thing by 
twelve, which yields twelve things ; and also four-and- 
twenty by twelve : the product of the whole will be two 
hundred and eighty-eight dirhems and twelve roots, 
which are equal to one square. The moiety of the roots 
is six. Multiply this by itself, and add it to two hun- 
dred and eighty-eight, it will be three hundred and 
twenty-four. Extract the root from this, it is eighteen; 
add this to the moiety of the roots, which was six ; the 
sum is twenty-four, and this is the square sought for. 
This question refers you to one of the six cases, 
namely, " roots and numbers equal to squares." 



VARIOUS QUESTIONS. 

If a person puts such a question to you as : "I have (30) 
divided ten into two parts, and multiplying one of 
these by the other, the result was twenty-one;"^ then 



* (lO — X)X=:21 


10X-X- = 21 


which is to be reduced to 


a;'H2i=;io^ 


x=5±v/25-2l=5d=2 


G 



( 42 ) 

you know that one of the two parts is thing, and the 
other ten minus thing. Multiply, therefore, thing by 
ten minus thing; then you have ten things minus 
a square, which is equal to twenty-one. Separate the 
square from the ten things, and add it to the twenty- 
one. Then you have ten things, which are equal to 
twenty-one dirhems and a square. Take away the 
moiety of the roots, and multiply the remaining five 
by itself; it is twenty-five. Subtract from this the 
twenty-one which are connected with the square ; the 
remainder is four. Extract its root, it is two. Sub- 
tract this from the moiety of the roots, namely, five ; 
there remain three, which is one of the two parts. 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 resolved by addition and subtraction. 

If the question be : "I have divided ten into two parts, 
and having multiplied each part by itself, I have sub- 
tracted the smaller from the greater, and the remainder 
was forty;"* then the computation is — ^you multiply ten 
(31) minus thing by itself, it is a hundred plus one square 
minus twenty things ; and you also multiply thing by 



100 — 200; =40 
100 = 200?+ 40 
60 = 20X 
3 = ^ 



( 43 ) 

thing, it is one square. Subtract this from a hundred 
and a square minus twenty things, and you have a 
hundred, minus twenty things, equal to forty dirhems. 
Separate now the twenty things from a hundred, and 
add them to the forty ; then you have a hundred, equal 
to twenty things and forty dirhems. Subtract now forty 
from a hundred ; there remains sixty dirhems, equal to 
twenty things: therefore one thing is equal to three, 
which is one of the two parts. 

If the question be : " I have divided ten into two parts, 
and having multiplied each part by itself, I have put 
them together, and have added to them the difference 
of the two parts previously to their multiplication, and 
the amount of all this is fifty-four;"^ then the compu- 
tation is this: You multiply ten minus thing by itself; 
it is a hundred and a square minus twenty things. 
Then multiply also the other thing of the ten by itself ; 
it is one square. Add this together, it will be a hun- 
dred plus two squares minus twenty things. It was 
stated that the difference of the two parts before multi- 
plication should be added to them. You say, therefore, 
the difference between them is ten minus two things. 



* (lO — a:)2-|-j;2 + (l0— x)— x~54 
1 GO — 20a; + 2 x- + 1 o •— 2a: = 54 
100 — 2 20;+ 2a:' =54 
55-iia; + a:2=:27 



X =U±^m_28=U^=7or4 



( 44 ) 

The result is a hundred and ten and two squares minus 
twenty-two things, which are equal to fifty-four dirhems. 
Having reduced and equalized this, you may say, a 
hundred and ten dirhems and two squares are equal to 
fifty-four dirhems and twenty-two things. Reduce now 
the two squares to one square, by taking the moiety of 
all you have. Thus it becomes fifty-five dirhems and a 
square, equal to twenty-seven dirhems and eleven things. 
Subtract twenty-seven from fifty-five, there remain 
(32) twenty-eight dirhems and a square, equal to eleven 
things. Halve now the things, it will be five and a 
half; multiply this by itself, it is thirty and a quarter. 
Subtract from it the twenty-eight which are combined 
with the square, the remainder is two and a fourth. 
Extract its root, it is one and a half. Subtract this 
from the moiety of the roots, there remain four, which 
is one of the two parts. 

If one say, "I have divided ten into two parts ; and 
have divided the first by the second, and the second by 
the first, and the sum of the quotient is two dirhemis 
and one-sixth ;"* then the computation is this : If you 
multiply each part by itself, and add the products 
together, then their sum is equal to one of the parts 



J. 10 — X X , 

* ' _L — o-L 

X ^lO-a; 6 

100-h 2ar^— 20a? = 4^^ -^) X 2^ = 2 ifx - 2^0;''* 
^=,5~v/25— 24 = 5— 1 = 4 or 6 



( 45 ) 

multiplied by the other, and again by the quotient 
which is two and one-sixth. Multiply, therefore, ten 
less thing by itself; it is a hundred and a square less 
ten things. Multiply thing by thing; it is one square. 
Add this together ; the sum is a hundred plus two 
squares less twenty things, which is equal to thing mul- 
tiplied by ten less thing ; that is, to ten things less a 
square, multiplied by the sum of the quotients arising 
from the division of the two parts, namely, two and 
one-sixth. We have, therefore, twenty-one things and 
two-thirds of thing less two squares and one-sixth, equal 
to a hundred plus two squares less twenty things. Re- 
duce this by adding the two squares and one-sixth to a 
hundred plus two squares less twenty things, and add 
the twenty negative things from the hundred plus the 
two squares to the twenty- one things and two -thirds of 
thing. Then you have a hundred plus four squares (33) 
and one-sixth of a square, equal to forty-one things and 
two- thirds of thing. Now reduce this to one square. 
You know that one square is obtained from four squares 
and one-sixth, by taking a fifth and one-fifth of a fifth.* 
Take, therefore, the fifth and one-fifth of a fifth of all 
that you have. Then it is twenty-four and a square, 
equal to ten roots ; because ten is one-fifth and one-fifth 
of the fifth of the forty-one things and two-thirds of a 
thing. Now halve the roots; it gives five. Multiply this 



4=^6^ and^\ = i+ixi 



( 46 ) 

by itself; it is five-and-twenty. Subtract from this 
the twenty-four, which are connected with the square ; 
the remainder is one. Extract its root; it is one. 
Subtract this from the moiety of the roots, which 
is five. There remains four, which is one of the two 
parts. 

Observe that, in every case, where any two quantities 
whatsoever are divided, the first by the second and the 
second by the first, if you multiply the quotient of the 
one division by that of the other, the product is always 
one.^ 

If some one say: "You divide ten into two parts; 
multiply one of the two parts by five, and divide it by 
the other : then take the moiety of the quotient, and 
add this to the product of the one part, multiplied by 
five ; the sum is fifty dirhems ;"t then the computation 
is this : Take thing, and multiply it by five. This is 
now to be divided by the remainder of the ten, that is, 
by ten less thing ; and of the quotient the moiety is to 
be taken. 
(34) You know that if you divide five things by ten less 
thing, and take the moiety of the quotient, the result is 



a b 
-X-= 1 
6 a 

5x 


2(10-.)'^^-^'^ 


2(10-x) '^-^^ 



( 47 ) 

the same as if you divide the moiety of five things by 
ten less thing. Take, therefore, the moiety of five 
things; it is two things and a half: and this you 
require to divide by ten less thing. Now these two 
things and a half, divided by ten less thing, give a 
quotient which is equal to fifty less five things : for the 
question states : add this (the quotient) to the one 
part multiplied by five, the sum will be fifty. You 
have already observed, that if the quotient, or the result 
of the division, be multiplied by the divisor, the divi- 
dend, or capital to be divided, is restored. Now, your 
capital, in the present instance, is two things and a 
half. Multiply, therefore, ten less thing by fifty less 
^we things. Then you have five hundred dirhems and 
five squares less a hundred things, which are equal to 
two things and a half. Reduce this to one square. 
Then it becomes a hundred dirhems and a square less 
twenty things, equal to the moiety of thing. Separate 
now the twenty things from the hundred dirhems and 
square, and add them to the half thing. Then you 
have a hundred dirhems and a square, equal to twenty 
things and a half. Now halve the things, multiply 
the moiety by itself, subtract from this the hundred, 
extract the root of the remainder, and subtract this 
from the moiety of the roots, which is ten and one- 
fourth : the remainder is eight ; and this is one of the 
portions. 

If some one say : " You divide ten into two parts : 
multiply the one by itself; it will be equal to the other 



( 48 ) 

taken eighty-one times." ^ Computation : You say, ten 
less thing, multiplied by itself, is a hundred plus a 
(35) square less twenty things, and this is equal to eighty- 
one things. Separate the twenty things from a hundred 
and a square, and add them to eighty-one. It will 
then be a hundred plus a square, which is equal to a 
hundred and one roots. Halve the roots ; the moiety is 
fifty and a half Multiply this by itself, it is two thou- 
sand five hundred and fifty and a quarter. Subtract 
from this one hundred ; the remainder is two thousand 
four hundred and fifty and a quarter. Extract the root 
from this; it is forty-nine and a half. Subtract this 
from the moiety of the roots, which is fifty and a half. 
There remains one, and this is one of the two parts. 

If some one say : " I have purchased two measures of 
wheat or barley, each of them at a certain price. I 
afterwards added the expences, and the sum was equal 
to the difference of the two prices, added to the diffe- 
rence of the measures. "t 



* (io-x)2=8ia: 
100— 20a;+a;^=8i5: 
a:2 + 100 = 101^7 



^ = i|i_v/'if'-ioo=5ol-49i = i 
f The purchaser does not make a clear enunciation of the 
terms of his bargain. He intends to say, " 1 bought m 
bushels of wheat, and n bushels of barley, and the wheat was 
r times dearer than the barley. The sum I expended was 
equal to the difference in the quantities, added to the diffe- 
rence in the prices of the grain." 



( 49 ) 

Computation : Take what numbers you please, for it 
is indifferent ; for instance, four and six. Then you 
say : I have bought each measure of the four for thing; 
and accordingly you multiply four by thing, which gives 
four things; and I have bought the six, each for the 
moiety of thing, for which I have bought the four ; or, 
if you please, for one-third, or one-fourth, or for any 
other quota of that price, for it is indifferent. Suppose 
that you have bought the six measures for the moiety of 
thing, then you multiply the moiety of thing by six ; 
this gives three things. Add them to the four things ; 
the sum is seven things, which must be equal to the 
difference of the two quantities, which is two measures, 
plus the difference of the two prices, which is a moiety 
of thing. You have, therefore, seven things, equal to 
two and a moiety of thing. Remove, now, this moiety 
of thing, by subtracting it from the seven things. 
There remain six things and a half, equal to two dir- (36) 
hems: consequently, one thing is equal to four-thir- 
teenths of a dirhem. The six measures were bought, 
each at one-half of thing; that is, at two-thirteenths of 
a dirhem. Accordingly, the expenses amount to eight- 
and-twenty thirteenths of a dirhem, and this sum is 
equal to the difference of the two quantities; namely. 



If X is the price of the barley, rx is the price of the 
wheat ; whence, mrx -\- nx zz (m — n) + (rx — x) ; ,\ x = 

m^n - , , , . (mr-^-n) X (m— w) 

— r— 7 and the sum expended is - ^^ . ^ . ^ — T"- 



( 50 ) 

the two measures, the arithmetical equivalent for which 
is six-and-twenty thirteenths, added to the difference of 
the two prices, which is two-thirteenths : both diffe- 
rences together being likewise equal to twenty-eight 
parts. 

If he say: "There are two numbers,* the difference 
of which is two dirhems. I have divided the smaller by 
the larger, and the quotient was just half a dirhem."f 
Suppose one of the two numbers* to be thing, and the 
other to be thing plus two dirhems. By the division 
of thing by thing plus two dirhems, half a dirhem 
appears as quotient. You have already observed, that 
by multiplying the quotient by the divisor, the capital 
which you divided is restored. This capital, in the 
present case, is thing. Multiply, therefore, thing and 
two dirhems by half a dirhem, which is the quotient; 
the product is half one thing plus one dirhem ; this is 
equal to thing. Remove, now, half a thing on account 



* In the original, ** squares." The word square is used 
in the text to signify either, ist, a square, properly so called, 
fractional or integral; 2d, a rational integer, not being a 
square number ; 3d, a rational fraction, not being a square ; 
4th, a quadratic surd, fractional or integral. 



x-j- 2 



h 



x-f 2__x 
2 ~"2 

=z 1 and or -f 2 = 4 



*=-i-=5 + > 



( 51 ) 

of the other half thing; there remains one dirhem, 
equal to half a thing. Double it, then you have one 
thing, equal to two dirhems. Consequently, the other 
number* is four. 

If some one say: "I have divided ten into two parts; 
I have multiplied the one by ten and the other by itself, 
and the products were the same;"f then the computa- 
tion is this : You multiply thing by ten ; it is ten things. 
Then multiply ten less thing by itself; it is a hundred (37) 
and a square less twenty things, which is equal to ten 
things. Reduce this according to the rules, which I 
have above explained to you. 

In like manner, if he say: " I have divided ten into 
two parts ; I have multiplied one of the two by the 
other, and have then divided the product by the diffe- 
rence of the two parts before their multiplication, and the 
result of this division is five and one- fourth :"J the com- 
putation will be this: You subtract thing from ten; there 
remain ten less thing. Multiply the one by the other, it 
is ten things less a square. This is the product of the 
multiplication of one of the two parts by the other. At 

* " Square " in the original, 
f ioa:=(io— ary-^rzioo— 20a;+a;2 



a:=:i5-\/225— 100=15— >/ 125 
xJlO-x) 
+ 10 — 2a: ^* 

1 ox-^x^ = 51 J — 1 o\x 
20jx=::a:--|-52j 
jr=:ioJ-7i=3 



( 52 ) 

present you divide this by the difference between the 
two parts, which is ten less two things. The quotient 
of this division is, according to the statement, five and 
a fourth. If, therefore, you muliply five and one-fourth 
by ten less two things, the product must be equal to the 
above amount, obtained by multiplication, namely, ten 
things less one square. Multiply now five and one- 
fourth by ten less two squares. The result is fifty-two 
dirhems and a half less ten roots and a half, which is 
equal to ten roots less a square. Separate now the ten 
roots and a half from the fifty-two dirhems, and add 
them to the ten roots less a square ; at the same time 
separate this square from them, and add it to the 
fifty-two dirhems and a half. Thus you find twenty 
roots and a half, equal to fifty- two dirhems and a half 
and one square. Now continue reducing it, conform- 
ably to the rules explained at the commencement of 
this book. 
(38) If the question be: "There is a square^ two-thirds of 
one-fifth of which are equal to one-seventh of its root;" 
then the square is equal to one root and half a seventh 
of a root; and the root consists of fourteen-fifteenths 
of the square.* The computation is this : You 



xix^ = l 



c^ = ^Ix± = i^j: 



X =:lJ 



Ti 






( 53 ) 

multiply two- thirds of one-fifth of the square by 
seven and a half, in order that the square may be com- 
pleted. Multiply that which you have already, namely, 
one-seventh of its root, by the same. The result will 
be, that the square is equal to one root and half a 
seventh of the root ; and the root of the square is one 
and a half seventh ; and the square is one and twenty- 
nine one hundred and ninety-sixths of a dirhem. Two- 
thirds of the fifth of this are thirty parts of the hundred 
and ninety-six parts. One-seventh of its root is like- 
wise thirty parts of a hundred and ninety-six. 

If the instance be : " Three-fourths of the fifth of a 
square are equal to four-fifths of its root,"* then the 
computation is this : You add one-fifth to the four- 
fifths, in order to complete the root. This is then equal 
to three and three-fourths of twenty parts, that is, to 
fifteen eightieths of the square. Divide now eighty by 
fifteen ; the quotient is five and one-third. This is the 
root of the square, and the square is twenty-eight and 
four-ninths. 

If some one say : " What is the amount of a square- 
rootjt which, when multiplied by four times itself, 



* ixK=l^ 



f " Square " in the original. 



( 54 ) 

amounts to twenty?*" the answer is this : If you mul- 
tiply it by itself it will be five : it is therefore the root 
of five. 

If somebody ask you for the amount of a square- 
root,t which when multiplied by its third amounts to 
ten, J the solution is, that when multiplied by itself it 
will amount to thirty ; and it is consequently the root 
of thirty. 
(39) If the question be : " To find a quantity t, which 
when multiplied by four times itself, gives one- third of 
the first quantity as product,"^ the solution is, that if 
you multiply it by twelve times itself, the quantity 
itself must re-appear : it is the moiety of one moiety of 
one-third. 

If the question be : "A square, which when multiplied 
by its root gives three times the original square as pro- 
duct," 1| then the solution is: that if you multiply the 
root by one-third of the square, the original square is 





* 4a:2 _ 20 






x =\/5 




t 


" Square " in the 

. t xxJ=io 
a:2=30 
a; =\/30 

§ XX4X=1 


original. 


' 


[| X^XX = ^X' 

X -s 


• 



( 55 ) 

restored ; its root must consequently be three, and the 
square itself nine. 

If the instance be : " To i&nd a square, four roots of 
which, multiplied by three roots, restore the square 
with a surplus of forty-four dirhems,*** then the solution 
is : that you multiply four roots by three roots, which 
gives twelve squares, equal to a square and forty-four 
dirhems. Remove now one square of the twelve on 
account of the one square connected with the forty- four 
dirhems. There remain eleven squares, equal to forty- 
four dirhems. Make the division, the result will be 
four, and this is the square. 

If the instance be : "A square, four of the roots of 
which multiplied by five of its roots produce twice the 
square, with a surplus of thirty-six dirhems ;"f then the 
solution is : that you multiply four roots by five roots, 
which gives twenty squares, equal to two squares and 
thirty -six dirhems. Remove two squares from the twenty 
on account of the other two. The remainder is eigh- 
teen squares, equal to thirty-six dirhems. Divide now 
thirty-six dirhems by eighteen; the quotient is two, 
and this is the square. 



1 1x2 __ ^4 

X- = 4 

X r: 2 

f 4xX5a:=2x2-j-36 
i8x-=36 

x2= 2 



( 56 ) 

(40) In the same manner, if the question be : "A square, 
multiply its root by four of its roots, and the product 
will be three times the square, with a surplus of fifty 
dirhems."t Computation : You multiply the root by four 
roots, it is four squares, which are equal to three squares 
and fifty dirhems. Remove three squares from the four ; 
there remains one square, equal to fifty dirhems. One 
root of fifty, multiplied by four roots of the same, gives 
two hundred, which is equal to three times the square, 
and a residue of fifty dirhems. 

If the instance be: "A square, which when added to 
twenty dirhems, is equal to twelve of its roots,"+ then 
the solution is this : You say, one square and twenty 
dirhems are equal to twelve roots. Halve the roots and 
multiply them by themselves; this gives thirty-six. 
Subtract from this the twenty dirhems, extract the 
root from the remainder, and subtract it from the 
moiety of the roots, which is six. The remainder is 
the root of the square : it is two dirhems, and the square 
is four. 

If the instance be : " To find a square, of which if 
one-third be added to three dirhems, and the sum be 
subtracted from the square, the remainder multiplied by 



* 4x2= 3^9^. ^Q 
x^= 50 

f ^2.^20 =12X 

:r=6=t\/36~2o = 6±4= lo or 2 



( 57 ) 

itself restores tlie square;"^ then the computation 
is this: If you subtract one- third and three dirhems 
from tlie square, there remain two-thirds of it less three 
dirhems. This is the root. Multiply therefore two- thirds 
of thing less three dirhems by itself. You say two- 
thirds by two-thirds is four ninths of a square ; and less 
two- thirds by three dirhems is two roots : and again, 
two-thirds by three dirhems is two roots; and less three 
dirhems by less three dirhems is nine dirhems. You (41) 
have, therefore, four-ninths of a square and nine dirhems 
less four roots, which are equal to one root. Add the 
four roots to the one root, then you have five roots, 
which are equal to four-nintlis of a square and nine 
dirhems. Complete now your square ; that is, multiply 
the four-ninths of a square by two and a fourth, which 
gives one square ; multiply likewise the nine dirhems 
by two and a quarter; this gives twenty and a quarter ; 
finally, multiply the five roots by two and a quarter; 
this gives eleven roots and a quarter. You have, there- 
fore, a square and twenty dirhems and a quarter, equal 
to eleven roots and a quarter. Reduce this according to 
what I taught you about halving the roots. 



* [*-(f+3)r=x 

or [if-3]'^=» 
X - 9, or 2A 



( 58 ) 

If the instance be : " To find a number,* one-third 
of which, when multiplied by one-fourth of it, restores 
the *number,"f then the computation is : You multiply 
one-third of thing by one-fourth of thing, this gives 
one-twelfth of a square, equal to thing, and the square 
is equal to twelve things, which is the root of one 
hundred and forty-four. 

If the instance be : "A number,* one-third of which 
and one dirhem multiplied by one-fourth of it and two 
dirhems restore the number,* with a surplus of thirteen 
dirhems ;"J then the computation is this : You multiply 
one- third of thing by one-fourth of thing, this gives 
half one-sixth of a square; and you multiply two 
dirhems by one-third of thing, this gives two-thirds 
of a root; and one dirhem by one-fourth of thing 
gives one-fourth of a root ; and one dirhem by two 
dirhems gives two dirhems. This altogether is one- 
twelfth of a square and two dirhems and eleven- 
(42) twelfths of a thing, equal to thing and thirteen dir- 



» « 



Square " in the original. 

X = 12 

12 12 



( 59 ) 

hems. Remove now two dirhems from thirteen, on 
account of the other two dirhems, the remainder is 
eleven dirhems. Remove then the eleven- twelfths of a 
root from the one (root on the opposite side), there 
remains one-twelfth of a root and eleven dirhems, equal 
to one-twelfth of a square. Complete the square: that 
is, multiply it by twelve, and do the same with alj you 
have. The product is a square, which is equal to a 
hundred and thirty-two dirhems and one root. Reduce 
this, according to what I have taught you, it will be 
right. 

If the instance be: "A dirhem and a half to be di- 
vided among one person and certain persons, so that the 
share of the one person be twice as many dirhems as 
there are other persons;''* then the Computation is 
this :f You say, the one person and some persons are 
one and thing : it is the same as if the question had 
been one dirhem and a half to be divided by one and 
thing, and the share of one person to be equal to two 
things. Multiply, therefore, two things by one and 



* The enunciation in the original is faulty, and I have 
altered it to correspond with the computation. But in the 
computation, x, the number of persons, is fractional f I am 
unable to correct the passage satisfactorily. 

X =1—^ 



( 60 ) 

thing ; it is two squares and two things, equal to one 
dirhem and a half. Reduce them to one square : that 
is, take the moiety of all you have. You say, there- 
fore, one square and one thing are equal to three- 
fourths of a dirhem. Reduce this, according to what 
I have taught you in the beginning of this work. 

If the instance be: "A number,* you remove one- 
third of it, and one-fourth of it, and four dirhems : then 
you multiply the remainder by itself, and the number,* 
is restored, with a surplus of twelve dirhems :"t then 
the computation is this : You take thing, and subtract 
from it one-third and one-fourth; there remain five- 
twelfths of thing. Subtract from this four dirhems: 
(43) the remainder is five-twelfths of thing less four dirhems. 
Multiply this by itself. Thus the five parts become 
five-and-twenty parts ; and if you multiply twelve by 
itself, it is a hundred and forty-four. This makes, 
therefore, five and twenty hundred and forty-fourths 
of a square. Multiply then the four dirhems twice by 
the five-twelfths ; this gives forty parts, every twelve of 
which make one root (forty-twelfths) ; finally, the four 

* " Square" in the original. 

tV4-^H4 = 4J^ 
^' + 232V=24ifx 

ili-J-M:2|f- 24_-a; 



( 61 ) 

dirhems, multiplied by four dirhems, give sixteen dir- 
hems to be added. The forty-twelfths are equal to 
three roots and one-third of a root, to be subtracted. 
The whole product is, therefore, twenty-five-hundred- 
and-forty-fourths of a square and sixteen, dirhems less 
three roots and one-third of a root, equal to the original 
number,* which is thing and twelve dirhems. Reduce 
this, by adding the three roots and one-third to the 
thing and twelve dirhems. Thus you have four roots 
and one-third of a root and twelve dirhems. Go on 
balancing, and subtract the twelve (dirhems) from six- 
teen ; there remain four dirhems and five-and-twenty- 
hundred-and-forty-fourths of a square, equal to four 
roots and one-third. Now it is necessary to complete 
the square. This you can accomplish by multiply- 
ing all you have by five and nineteen twenty-fifths. 
Multiply, therefore, the twenty-five-one-hundred-and- 
forty-fourths of a square by five and nineteen twenty- 
fifths. This gives a square. Then multiply the four (44) 
dirhems by five and nineteen twenty-fifths ; this gives 
twenty-three dirhems and one twenty-fifth. Then 
multiply four roots and one- third by five and nineteen 
twenty-fifths ; this gives twenty-four roots and twenty- 
four twenty-fifths of a root. Now halve the number of 
the roots : the moiety is twelve roots and twelve twenty- 
fifths of a root. Multiply this by itself. It is one 
hundred -and- fifty-five dirhems and four hundred-and- 



Square " in the original. 



( 62 ) 

sixty-nine six-hundred-and- twenty-fifths. Subtract 
from this the twenty-three dirhems and the one twenty- 
fifth connected with the square. The remainder is 
one-hundred-and-thirty-two and four-hundred-and- 
forty six-hundred- and-twenty-fifths. Take the root of 
this : it is eleven dirhems and thirteen twenty-fifths. 
Add this to the moiety of the roots, which was twelve 
dirhems and twelve twenty-fifths. The sum is twenty- 
four. It is the number* which you sought. When 
you subtract its third and its fourth and four dirhems, 
and multiply the remainder by itself, the number * is 
restored, with a surplus of twelve dirhems. 

If the question be : " To find a square-root,* which, 
when multiplied by two-thirds of itself, amounts to 
(45) five;"f then the computation is this : You multiply 
one thing by two- thirds of thing; the product is two- 
thirds of square, equal to five. Complete it by adding 
its moiety to it, and add to five likewise its moiety. 
Thus you have a square, equal to seven and a half. 
Take its root ; it is the thing which you required, and 
which, when multiplied by two-thirds of itself, is equal 
to five. 

If the instance be: "Two numbers, J the difference 

* " Square " in the original, 
t ^ ^ 1^ = 5 

x2 = 7j 
X " Squares " in the original. 



( 63 ) 

of which is two dirhems ; you divide the small one by 
the great one, and the quotient is equal to half a dir- 
hem ;* then the computation is this : Multiply thing 
and two dirhems by the quotient, that is a half. The 
product is half a thing and one dirhem, equal to thing. 
Remove now half a dirhem on account of the half dir- 
hem on the other side. The remainder is one dir- 
hem, equal to half a thing. Double it: then you have 
thing, equal to two dirhems. This is one of the two 
numbers,f and the other is four. 

Instance : *' You divide one dirhem amongst a cer- 
tain number of men, which number is thing. Now you 
add one man more to them, and divide again one dir- 
hem amongst them; the quota of each is then one-sixth 
of a dirhem less than at the first time."t Computation: 
You multiply the first number of men, which is thing, 
by the difference of the share for each of the other 
number. Then multiply the product by the first and 
second number of men, and divide the product by the 



— X 



x-\-2 - ^ 


lx+ 1 ==x 


i:. = l 


a:=;2, x -{- 2 = 4. 


' Squares " in the original 


t X "" 2 -f 1 = 6 


6 


X^ + X = 6 


v/[Ap+6-4 = a;=2 



( 64 ) 

difference of these two numbers. Thus you obtain the 
sum which shall be divided. Multiply, therefore, the 
first number of men, which is thing, by the one- 
sixth, which is the difference of the shares; this gives 
one-sixth of root. Then multiply this by the original 
number of the men, and that of the additional one, 
that is to say, by thing plus one. The product is one- 
sixth of square and one- sixth of root divided by one 
(46) dirhem, and this is equal to one dirhem. Complete the 
square which you have through multiplying it by six. 
Then you have a square and a root equal to six dir- 
hems. Halve the root and multiply the moiety by 
itself, it is one-fourth. Add this to the six; take the 
root of the sum and subtract from it the moiety of the 
root, which you have multiplied by itself, namely, a 
half. The remainder is the first number of men ; which 
in this instance is two. 

If the instance be : " To find a square-root,* which 
when multiplied by two-thirds of itself amounts to 
five :"f then the computation is this : If you multiply 
it by itself, it gives seven and a half. Say, therefore. 



* " Square " in the original, 
t f ^'^ = 5 

_^ = A/7i 



( 65 ) 

it is the root of seven and a half multiplied by two- 
thirds of the root of seven and a half. Multiply then 
two-thirds by two-thirds, it is four-ninths ; and four- 
ninths multiplied by seven and a half is three and a 
third. The root of three and a third is two- thirds of 
the root of seven and a half Multiply three and a 
third by seven and a halt ; the product is twenty-five, 
and its root is five. 

If the instance be : "A square multiplied by three of 
its roots is equal to five times the original square;"* 
then this is the same as if it had been said, a square, 
which when multiplied by its root, is equal to the first 
square and two- thirds of it. Then the root of the 
square is one and two-thirds, and the square is two 
dirhems and seven-ninths. 

If the instance be : " Remove one-third from a 
square, then multiply the remainder by three roots of 
the first square, and the first" square will be restored."f 
Computation : If you multiply the first square, before (47) 
removing two-thirds from it, by three roots of the 
same, then it is one square and a half; for according 
to the statement two-thirds of it multiplied by three 

* af^ X sx = 5x^ 

x^ X X = i^x^ 

* = i| 

^2 = 2j 

f (:iP-ix^) X 3^=«- .*. f x2 X 3x=x^' 
x^X3^='^i^'^ 

X—2 . . **> —4 
K 



( 66 ) 

roots give one square ; and, consequently, the whole of 
it multiplied by three roots of it gives one square and a 
half. This entire square, when multiplied by one 
root, gives half a square ; the root of the square must 
therefore be a half, the square one-fourth, two- thirds 
of the square one-sixth, and three roots of the square 
one and a half. If you multiply one-sixth by one and 
a half, the product is one-fourth, which is the square. 

Instance : " A square; you subtract four roots of the 
same, then take one-third of the remainder; this is 
equal to the four roots." The square is two hundred 
and fifty-six.* Computation: You know that one-third 
of the remainder is equal to four roots ; consequently, 
the whole remainder must be twelve roots ; add to this 
the four roots ; the sum is sixteen, which is the root of 
the square. 

Instance : " A square ; you remove one root from it; 
and if you add to this root a root of the remainder, the 
sum is two dirhems."f Then, this is the root of a 



3 


.^^ 


a;^-4x 


= I2ar 


x2= 


16* 


X = 16 .-. 


X^ =: 256 


t Vx^-^ 


X -j- X = 2 


Vx^-x 


= 2 - X 


r2— a?=44-a;2 — 42; 


0:^+3^ = 


= 4 + ^2 


3^ = 


= 4 


X = 


H 



( 67 ) 

square, which, when added to the root of the same 
square, less one root, is equal to two dirhenis. Sub- 
tract from this one root of the square, and subtract also 
from the two dirhems one root of the square. Then 
two dirhems less one root multiplied by itself is four 
dirhems and one square less four roots, and this is equal 
to a square less one root. Reduce it, and you find a 
square and four dirhems, equal to a square and three 
roots. Remove square by square ; there remain three 
roots, equal to four dirhems ; consequently, one root is 
equal to one dirhem and one-third. This is the root of 
the square, and the square is one dirhem and seven- 
ninths of a dirhem. (48) 

Instance : " Subtract three roots from a square, then 
multiply the remainder by itself, and the square is 
restored."* You know by this statement that the re- 
mainder must be a root likewise; and that the square 
consists of four such roots; consequently, it must be 
sixteen. 



» 


(x-' - 


- 3^)2 = 


X- 




x'^- 


■3^ = ^ 






x' 


= 4x 






X 


= 4 





( 68 ) 



ON MERCANTILE TRANSACTIONS. 

You know that all mercantile transactions of people, 
such as buying and selling, exchange and hire, com- 
prehend always two notions and four numbers, which 
are stated by the enquirer ; namely, measure and price, 
and quantity and sum. The number which expresses 
the measure is inversely proportionate to the number 
which expresses the sum, and the number of the price 
inversely proportionate to that of the quantity. Three 
of these four numbers are always known, one is un- 
known, and this is implied when the person inquiring 
says tww much ? and it is the object of the question. 
The computation in such instances is this, that you try 
the three given numbers ; two of them must necessarily 
be inversely proportionate the one to the other. Then 
you multiply these two proportionate numbers by each 
other, and you divide the product by the third given 
number, the proportionate of which is unknown. The 
quotient of this division is the unknown number, which 
the inquirer asked for ; and it is inversely proportionate 
to the divisor.* 

Examples. — For the first case : If you are told, " ten 
(49) for six, how much for four ?" then ten is the measure ; 



* If a is given for h, and A for B, then a i b :: A : B or 



aB=.bA.'.a — -- and ^=—7 . 
B A 



( 69 ) 

six is the price ; the expression how much implies the 
unknown number of the quantity; and four is the 
number of the sum. The number of the measure, 
which is ten, is inversely proportionate to the number 
of the sum, namely, four. Multiply, therefore, ten 
by four, that is to say, the two known proportionate 
numbers by each other ; the product is forty. Divide 
this by the other known number, which is that of the 
price, namely, six. The quotient is six and two- 
thirds; it is the unknown number, implied in the words 
of the question " how much f it is the quantity, and 
inversely proportionate to the six, which is the price. 

For the second case : Suppose that some one ask this 
question : " ten for eight, what must be the sum for 
four ?" This is also sometimes expressed thus : " What 
must be the price of four of them ?" Ten is the number 
of the measure, and is inversely proportionate to the 
unknown number of the sum, which is involved in the 
expression how much of the statement. Eight is the 
number of the price, and this is inversely proportionate 
to the known number of the quantity, namely, four. 
Multiply now the two known proportionate numbers one 
by the other, that is to say, four by eight. The product 
is thirty-two. Divide this by the other known number, 
which is that of the measure, namely, ten. The quo- 
tient is three and one- fifth; this is the number of the 
sum, and inversely proportionate to the ten which was 
the divisor. In this manner all computations in matters 
of business may be solved. 



( ^0 ) 

If somebody says, " a workman receives a pay of ten 
(^^) dirhems per month ; how much must be his pay for six 
days?" Then you know that six days are one-fifth of 
the month; and that his portion of the dirhems must 
be proportionate to the portion of the month. You 
calculate it by observing that one month, or thirty 
days, is the measure, ten dirhems the price, six days 
the quantity, and his portion the sum. Multiply the 
price, that is, ten, by the quantity, which is propor- 
tionate to it, namely, six ; the product is sixty. Divide 
this by thirty, which is the known number of the mea- 
sure. The quotient is two dirhems, and this is the sum. 
This is the proceeding by which all transactions con- 
cerning exchange or measures or weights are settled. 



MENSURATION. 



Know that the meaning of the expression *' one by 
one'* is mensuration : one yard (in length) by one yard 
(in breadth) being understood. 

Every quadrangle of equal sides and angles, which 
has one yard for every side, has also one for its area. 
Has such a quadrangle two yards for its side, then the 
area of the quadrangle is four times the area of a qua- 
drangle, the side of which is one yard. The same takes 
place with three by three, and so on, ascending or 
descending : for instance, a half by a half, which gives 



( n ) 

a quarter, or other fractions, always following the same 
rule. A quadrate, every side of which is half a yard, is (51) 
equal to one-fourth of the figure which has one yard for 
its side. In the same manner, one-third by one-third, 
or one-fourth by one-fourth, or one-fifth by one-fifth, 
or two-thirds by a half, or more or less than this, al- 
ways according to the same rule. 

One side of an equilateral quadrangular figure, 
taken once, is its root ; or if the same be multiplied by 
two, then it is like two of its roots, whether it be small 
or great. 

If you multiply the height of any equilateral triangle 
by the moiety of the basis upon which the line marking 
the height stands perpendicularly, the product gives 
the area of that triangle. 

In every equilateral quadrangle, the product of one 
diameter multiplied by the moiety of the other will be 
equal to the area of it. 

In any circle, the product of its diameter, multiplied 
by three and one-seventh, will be equal to the peri- 
phery. This is the rule generally followed in practical 
life, though it is not quite exact. The geometricians 
have two other methods. One of them is, that you 
multiply the diameter by itself; then by ten, and 
hereafter take the root of the product ; the root will be 
the periphery. The other method is used by the astro- 
nomers among them : it is this, that you multiply the 
diameter by sixty-two thousand eight hundred and 
thirty- two and then divide the product by twenty 



( ^2 ) 

thousand ; the quotient is the periphery. Both methods 
come very nearly to the same effect.* 

If you divide the periphery by three and one-seventh, 
the quotient is the diameter. 

The area of any circle will be found by multiplying 
the moiety of the circumference by the moiety of the 
diameter; since, in every polygon of equal sides and 
(52) angles, such as triangles, quadrangles, pentagons, and 
so on, the area is found by multiplying the moiety of 
the circumference by the moiety of the diameter of the 
middle circle that may be drawn through it. 

If you multiply the diameter of any circle by itself, 
and subtract from the product one-seventh and half 
one-seventh of the same, then the remainder is equal 
to the area of the circle. This comes very nearly to the 
same result with the method given above, t 

Every part of a circle may be compared to a bow. 
It must be either exactly equal to half the circum- 
ference, or less or greater than it. This may be ascer- 
tained by the arrow of the bow. When this becomes 
equal to the moiety of the chord, then the arc is 



* The three formulas are, 

1st, 3\d=^p i.e. 3.1428 c? 



2d, \/iod'^=p i.e. 3.i622'7c? 

3d, -» I.e. 3.14166/ 

20000 ' 

+ The area of a circle whose diameter is c? is tt—-. 
' 4 



,icV=Ci-^-rx-,K^. 



( ^3 ) 

exactly the moiety of the circumference: is it shorter 
than the moiety of the chord, then the bow is less than 
half the circumference; is the arrow longer than half 
the chord, then the bow comprises more than half the 
circumference. 

If you want to ascertain the circle to which it be- 
longs, multiply the moiety of the chord by itself, divide 
it by the arrow, and add the quotient to the arrow, 
the sum is the diameter of the circle to which this bow 
belongs. 

If you want to compute the area of the bow, mul- 
tiply the moiety of the diameter of the circle by the 
moiety of the bow, and keep the product in mind. 
Then subtract the arrow of the bow from the moiety 
of the diameter of the circle, if the bow is smaller than 
half the circle ; or if it is greater than half the circle, 
subtract half the diameter of the circle from the arrow 
of the bow. Multiply the remainder by the moiety of 
the chord of the bow, and subtract the product from 
that which you have kept in mind if the bow is smaller (53) 
than the moiety of the circle, or add it thereto if the 
bow is greater than half the circle. The sum after the 
addition, or the remainder after the subtraction, is the 
area of the bow. 

The bulk of a quadrangular body will be found by 
multiplying the length by the breadth, and then by the 
height. 

If it is of another shape than the quadrangular (for 
instance, circular or triangular), so, however, that a 

L 



( ^4 ) 

line representing its height may stand perpendicularly 
on its basis, and yet be parallel to the sides, you must 
calculate it by ascertaining at first the area of its basis. 
This, multiplied by the height, gives the bulk of the 

body. 

Cones and pyramids, such as triangular or quadran- 
gular ones, are computed by multiplying one- third of 
the area of the basis by the height. 

Observe, that in every rectangular triangle the two 
short sides, each multiplied by itself and the products 
added together, equal the product of the long side mul- 
tiplied by itself. 

The proof of this is the follovi^ing. We draw a qua- 
drangle, with equal sides and angles A B C D. We 
divide the line A C into two moieties in the point H, 
from which we draw a parallel to the point R. Then 
we divide, also, the line A B into two moieties at the 
point T^ and draw a parallel to the point G. Then the 
quadrate A B C D is divided into four quadrangles of 
equal sides and angles, and of equal area ; namely, the 
squares AK, CK, BK, and DK. Now, we draw from 
f54) ^^^ point H to the point T a line which divides the 
quadrangle AK into two equal parts: thus there arise 
two triangles from the quadrangle, namely, the triangles 
A T H and H K T. We know that A T is the moiety 
of A B, and that A H is equal to it, being the moiety of 
AC; and the line TH joins them opposite the right 
angle. In the same manner we draw lines from T to 
R, and from R to G, and from G to H. Thus from 



( ^5 ) 

all the squares eight equal triangles arise, four of which 
must, consequently, be equal to the moiety of the great 
quadrate AD. We know that the line AT multiplied 
by itself is like the area of two triangles, and AK gives 
the area of two triangles equal to them ; the sum of 
them is therefore four triangles. But the line HT, 
multiplied by itself, gives likewise the area of four such 
triangles. We perceive, therefore, that the sum of AT 
multiplied by itself, added to AH multiplied by itself, 
is equal to TH multiplied by itself. This is the 
observation which we were desirous to elucidate. Here 
is the figure to it : 




Quadrangles are of five kinds : firstly, with right (55) 
angles and equal sides ; secondly, with right angles and 
unequal sides ; thirdly, the rhombus, with equal sides 
and unequal angles ; fourthly, the rhomboid, the length 
of which differs from its breadth, and the angles of 
which are unequal, only that the two long and the two 
short sides are respectively of equal length; fifthly, 
quadrangles with unequal sides and angles. 

First kind. — The area of any quadrangle with equal 
sides and right angles, or with unequal sides and right 



( 76 ) 

angles, may be found by multiplying the length by the 
breadth. The product is the area. For instance : a 
quadrangular piece of ground, every side of which has 
five yards, has an area of five-and- twenty square yards. 
Here is its figure. 




Second kind. — A quadrangular piece of ground, the 

two long sides of which are of eight yards each, while 

the breadth is six. You find the area by multiplying 

six by eight, which yields forty- eight yards. Here is 

(56) the figure to it : 




Third kind, the Rhombus. — Its sides are equal: let 
each of them be five, and let its diagonals be, the one 
eight and the other six yards. You may then compute 
the area, either from one of the diagonals, or from 
both. As you know them both, you multiply the one 
by the moiety of the other, the product is the area : 
that is to say, you multiply eight by three, or six by 
four ; this yields twenty-four yards, which is the area. 



( 77 ) 

If you know only one of the diagonals, then you are 
aware, that there are two triangles, two sides of each 
of which have every one five yards, while the third is 
the diagonal. Hereafter you can make the computa- 
tion according to the rules for the triangles.* This is 
the figure : 




The fourth kind, or Rhomboid, is computed in the 
same way as the rhombus. Here is the figure to it : 



/ 


4 


iii— 


4 


8 


3 




/3 







The other quadrangles are calculated by drawing a (57) 
diagonal, and computing them as triangles. 

Triangles are of three kinds, acute-angular, obtuse- 
angular, or rectangular. The peculiarity of the rec- 
tangular triangle is, that if you multiply each of its 
two short sides by itself, and then add them together, 
their sum will be equal to the long side multiplied by 
itself. The character of the acute-angled triangle is 



^ If the two diagonals are d and d\ and the side 5, the 
area of the rhombus is _ = ^ x v/*"— — • 

2 V 4 



( 78 ) 

this : if you multiply every one of its two short sides 
by itself, and add the products, their sum is more 
than the long side alone multiplied by itself. The 
definition of the obtuse-angled triangle is this : if you 
multiply its two short sides each by itself, and then add 
the products, their sum is less than the product of the 
long side multiplied by itself. 

The rectangular triangle has two cathetes and an 
hypotenuse. It may be considered as the moiety of a 
quadrangle. You find its area by multiplying one of 
its cathetes by the moiety of the other. The product 
is the area. 

Examples. — A rectangular triangle; one cathete being 
(58) six yards, the other eight, and the hypotenuse ten. 
You make the computation by multiplying six by four : 
this gives twenty-four, which is the area. Or if you 
prefer, you may also calculate it by the height, which 
rises perpendicularly from the longest side of it : for 
the two short sides may themselves be considered as 
two heights. If you prefer this, you multiply the 
height by the moiety of the basis. The product is the 
area. This is the figure : 




Second kind. — Kri equilateral triangle with acute 
angles, every side of which is ten yards long. Its area 



( T9 ) 

may be ascertained by the line representing its height 
and the point from which it rises.* Observe, that in 
every isosceles triangle, a line to represent the height 
drawn to the basis rises from the latter in a right 
angle, and the point from which it proceeds is always 
situated in the midst of the basis ; if, on the contrary, 
the two sides are not equal, then this point never lies 
in the middle of the basis. In the case now before us 
we perceive, that towards whatever side we may draw 
the line which is to represent the height, it must 
necessarily always fall in the middle of it, where the 
length of the basis is five. Now the height will be 
ascertained thus. You multiply five by itself; then 
multiply one of the sides, that is ten, by itself, which 
gives a hundred. Now you subtract from this the 
product of five multiplied by itself, which is twenty-five. (59) 
The remainder is seventy-five, the root of which is the 
height. This is a line common to two rectangular tri- 
angles. If you want to find the area, multiply the 
root of seventy-five by the moiety of the basis, which is 
five. This you perform by multiplying at first five by 
itself; then you may say, that the root of seventy-five is 
to be multiplied by the root of twenty-five. Multiply 
seventy-five by twenty-five. The product is one thou- 
sand eight hundred and seventy-five ; take its root, it is 



* The height of the equilateral triangle whose side is lo, 
is s/ 10'^ — 5^ rt v/75, and the area of the triangle is 
5 v/75 = 25 \/3 



( 80 ) 

the area : it is forty-three and a little. * Here is the 
figure : 




There are also acute-angled triangles, with different 
sides. Their area will be found by means of the line 
representing the height and the point from which it 
proceeds. Take, for instance, a triangle, one side of 
which is fifteen yards, another fourteen, and the third 
thirteen yards. In order to find the point from which 
the line marking the height does arise, you may take 
for the basis any side jou choose ; e. g. that which is 
fourteen yards long. The point from which the line 
(60) representing the height does arise, lies in this basis at 
an unknown distance from either of the two other 
sides. Let us try to find its unknown distance from 
the side which is thirteen yards long. Multiply this 
distance by itself; it becomes an [unknown] square. 
Subtract this from thirteen multiplied by itself; that is, 
one hundred and sixty-nine. The remainder is one 
hundred and sixty-nine less a square. The root from 
this is the height. The remainder of the basis is four- 
teen less thing. We multiply this by itself; it becomes 
one hundred and ninety-six, and a square less twenty- 

* The root is 43. 3 + 



( 81 ) 

eight things. We subtract this from fifteen multiplied 
by itself; the remainder is twenty-nine dirhems and 
twenty-eight things less one square. The root of this 
is the height. As, therefore, the root of this is the 
height, and the root of one hundred and sixty-nine less 
square is the height likewise, we know that they both are 
the same.* Reduce them, by removing square against 
square, since both are negatives. There remain twenty- 
nine [dirhems] plus twenty-eight things, which are 
equal to one hundred and sixty-nine. Subtract now 
twenty-nine from one hundred and sixty-nine. The 
remainder is one hundred and forty, equal to twenty- 
eight things. One thing is, consequently, five. This is 
the distance of the said point from the side of thirteen 
yards. The complement of the basis towards the other 
side is nine. Now in order to find the height, you 
multiply five by itself, and subtract it from the conti- 
guous side, which is thirteen, multiplied by itself. The 
remainder is one hundred and forty-four. Its root is the 
height. It is twelve. The height forms always two (gi) 
right angles with the basis, and it is called the column^ 
on account of its standing perpendicularly. Multiply 
the height into half the basis, which is seven. The 



* \/T69 — «2 = 29 -h 28a: — a?2 

1G3 = 29 + 28a; 

140 =z 28x 

5 =^ 

M 



( 82 ) 

product is eighty-four, which is the area. Here is the 
figure : 




The third species is that of the obtuse-angled triangle 
with one obtuse angle and sides of different length. 
For instance, one side being six, another five, and the 
third nine. The area of such a triangle will be found 
by means of the height and of the point from which a 
line representing the same arises. This point can, 
within such a triangle, lie only in its longest side. Take 
therefore this as the basis : for if you choose to take one 
of the short sides as the basis, then this point would 
fall beyond the triangle. You may find the distance 
of this point, and the height, in the same manner, 
which I have shown in the acute- angled triangle; the 
whole computation is the same. Here is the figure : 




We have above treated at length of the circles, of 

their qualities and their computation. The following 

(62) is an example : If a circle has seven for its diameter, 

then it has twenty-two for its circumference. Its area 

you find in the following manner : Multiply the moiety 



( 83 ) 

of the diameter, which is three and a half, by the moiety 
of the circumference, which is eleven. The product is 
thirty-eight and a half, which is the area. Or you may 
also multiply the diameter, which is seven, by itself: this 
is forty-nine; subtracting herefrom one-seventh and half 
one-seventh, which is ten and a half, there remain thirty- 
eight and a half, which is the area. Here is the figure: 




If some one inquires about the bulk of a pyramidal 
pillar, its base being four yards by four yards, its 
height ten yards, and the dimensions at its upper ex- 
tremity two yards by two yards ; then we know already 
that every pyramid is decreasing towards its top, and 
that one-third of the area of its basis, multiplied by the 
height, gives its bulk. The present pyramid has no top. 
We must therefore seek to ascertain what is wanting 
in its height to complete the top. We observe, that the 
proportion of the entire height to the ten, which we 
have now before us, is equal to the proportion of four 
to two. Now as two is the moiety of four, ten must 
likewise be the moiety of the entire height, and the 
whole height of the pillar must be twenty yards. At 
present we take one-third of the area of the basis, 
that is, five and one-third, and multiply it by the 
length, which is twenty* The product is one hundred (6^) 



( 84 ) 

and six yards and two-thirds. Herefrom we must then 
subtract the piece, which we have added in order to 
complete the pyramid. This we perform by multiply- 
ing one and one-third, which is one-third of the pro- 
duct of two by two, by ten : this gives thirteen and a 
third. This is the piece which we have added in order 
to complete the pyramid. Subtracting this from one 
hundred and six yards and two-thirds, there remain 
ninety- three yards and one-third : and this is the bulk 
of the mutilated pyramid. This is the figure : 




If the pillar has a circular basis, subtract one-seventh 
and half a seventh from the product of the diameter 
multiplied by itself, the remainder is the basis. 

If some one says : " There is a triangular piece of 
land, two of its sides having ten yards each, and the 
basis twelve ; 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 trian- 
gle, 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 



( 85 ) 

sixty-four: take the root from this; it is eight. This (64) 
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 qua- 
drate 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 : 




•s| ^^ 3i 



( 86 ) 



ON LEGACIES. 

On Capital^ and Money lent. 

(65) " 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 dir- 
hems 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 pro- 
perty, that is, three dirhems and one-third of thing. 
The remainder is six dirhems 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 

* If a father dies, leaving n sons, one of whom owes the 
father a sum exceeding an wth part of the residue of the 
father's estate, after paying legacies, then such son retains 
the whole sum which he owes the father : part, as a set-off 
against his share of the residue, the surplus as a gift from 
the father. 

In the present example, let each son's share of the residue 
be equal to x, 

§ [io-|-^] =2.1; /, i+a: — 30? /, 10 = 2j: ,\x — ^. 
The stranger receives 5 ; and the son, who is not indebted 
to the father, receives 5. 



( 87 ) 

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. 

If he leaves two sons and ten dirhems of capital and 
a demand of ten dirhems against one of the sons, and 
bequeaths one-fifth of his property and one dirhem to 
a stranger, the computation is this : Call the sum which 
is taken out of the debt, thing. Add this to the pro- 
perty ; the sum is thing and ten dirhems. Subtract 
one-fifth of this, since he has bequeathed one-fifth of (66) 
his capital, that is, two dirhems and one-fifth of thing ; 
the remainder is eight dirhems and four-fifths of thing. 
Subtract also the one dirhem which he has bequeathed; 
there remain seven dirhems and four-fifths of thing. 
Divide this between the two sons ; there will be for each 
of them three dirhems and a half plus two-fifths of 
thing ; and this is equal to one thing.^ Reduce it by 
subtracting two-fifths of thing from thing. Then you 
have three-fifths of thing, equal to three dirhems and a 
half. Complete the thing by adding to it two-thirds of 
the same : add as much to the three dirhems and a half, 



I [lo-f-a;] — 1=20; ,'. flio+a:2—i=x 
The stranger receives ;^[io + \^ ] + 1 ~4^ 



( 88 ) 

namely, two dirhems and one-third ; the sum is five and 
five-sixths. This is the thing, or the amount which is 
taken from the debt. 

If he leaves three sons, and bequeaths one- fifth of his 
property less one dirhem, leaving ten dirhems of capital 
and a demand of ten dirhems against one of the sons, 
the computation is this : You call the sum which is 
taken from the debt thing. Add this to the capital ; 
it gives ten and thing. Subtract from this one-fifth of 
it for the legacy : it is two dirhems and one-fifth of 
thing. There remain eight dirhems and four-fifths of 
thing ; add to this one dirhem, since he stated " less 
one dirhem." Thus you have nine dirhems and four- 
fifths of thing. Divide this between the three sons. 
There will be for each son three dirhems, and one- 
fifth and one- third and one-fifth of thing. This equals 
one thing. ^ Subtract one-fifth and one- third of one- 
(6T) fifth of thing from thing. There remain eleven- 
fifteenths of thing, equal to three dirhems. It is now 
required to complete the thing. For this purpose, add 
to it four-elevenths, and do the same with the three 
dirhems, by adding to them one dirhem and one- 
eleventh. Then you have four dirhems and one- 
eleventh, which are equal to thing. This is the sum 
which is taken out of the debt. 



JThe stranger receives y[xo-|-^x] — i = i-3^t 



( 89 ) 

Ow another Species of Legacy. 

*' A man dies, leaving his mother, his wife, and two 
brothers and two sisters by the same father and mother 
with himself ; and he bequeaths to a stranger one-ninth 
of his capital." 

Computation:* You constitute their shaifes by taking 
them out of forty-eight parts. You know that if you 
take one-ninth from any capital, eight-ninths of it will 
remain. Add now to the eight-ninths one-eighth of the 
same, and to the forty-eight also one-eighth of them, 
namely, six, in order to complete your capital. This 
gives fifty-four. The person to whom one-ninth is 
bequeathed receives six out of this, being one-ninth of 
the whole capital. The remaining forty-eight will be 
distributed among the heirs, proportionably to their 
legal shares. 

If the instance be: "A woman dies, leaving her 
husband, a son, and three daughters, and bequeathing 

* It appears in the sequel (p. 96) that a widow is enti- 
tled to l^th, and a mother to ^th of the residue ; J + e^iJ? 
leaving ^ of the residue to be distributed between two bro- 
thers and two sisters ; that is, '^ between a brother and a 
sister; but in what proportion these 17 parts are to be 
divided between the brother and sister does not appear in 
the course of this treatise. 

Let the whole capital of the testator = 1 
and let each 48th share of the residue =x 

8:^48^ ... ^=Qx :, ^=x 

that is, each 48th part of the residue -^^th of the whole 
capital. 



( 90 ) 

to a stranger one-eighth and one- seventh of her capi- 
(68)tal;" then you constitute the shares of the heirs, by 
taking them out of twenty.* Take a capital, and sub- 
tract from it one -eighth and one-seventh of the same. 
The remainder is, a capital less one-eighth and one- 
seventh. Complete your capital by adding to that 
vi^hich you have already, fifteen forty-one parts. Mul- 
tiply the parts of the capital, which are twenty, by 
forty-one ; the product is eight hundred and twenty. 
Add to it fifteen forty-one parts of the same, which are 
three hundred : the sum is one thousand one hundred 
and twenty parts. The person to whom one-eighth 
and one-seventh were bequeathed, receives one-eighth 
and one-seventh of this. One seventh of it is one hun- 
dred and sixty, and one-eighth one hundred and forty. 
Subtracting this, there remain eight hundred and 
twenty parts for the heirs, proportionably to their legal 
shares. 

* A husband is entitled to :ith of the residue, and the 
sons and daughters divide the remaining |ths of the residue 
in such proportion, that a son receives twice as much as a 
daughter. In the present instance, as there are three daughters 
and one son, each daughter receives } of |, = 2%» ^^ ^^^ 
residue, and the son, /^. Since the stranger takes 1-+^ = 
Jf of the capital, the residue =41- of the capital, and each 
^^th share of the residue=Jj^ X^=^^ of the capital. 
The stranger, therefore, receives i| =t^~, =T¥o^ff of the 
capital. 



( 91 ) 

On another Species of Legacies^'^ viz. 

If nothing has been imposed on some of the heirs,t 
and something has been imposed on others ; the legacy 
amounting to more than one-third. It must be known, 
that the law for such a case is, that if more than one- 
third of the legacy has been imposed on one of the 
heirs, this enters into his share ; but that also those on 
whom nothing has been imposed must, nevertheless, 
contribute one-third. 

Example: " A woman dies, leaving her husband, a 
son, and her mother. She bequeaths to a person two- 
fifths, and to another one- fourth of her capital. She 
imposes the two legacies together on her son, and on 
her mother one moiety (of the mother's share of the 
residue) ; on her husband she imposes nothing but one- 
third, (which he must contribute, according to the 

* The problems in this chapter may be considered as 
belonging rather to Law than to Algebra, as they contain 
little more than enunciations of the law of inheritance in 
certain complicated cases. 

f If some heirs are, by a testator, charged with payment 
of bequests, and other heirs are not charged with payment 
of any bequests whatever : if one bequest exceeds in amount 
Jd of the testator's whole property ; and if one of his heirs 
is charged with payment of more than Jd of such bequest ? 
then, whatever share of the residue such heir is entitled to 
receive, the like share must he pay of the bequest where- 
with he is charged, and those heirs whom the testator has 
not charged with any payment, must each contribute towards 
paying the bequests a third part of their several shares of 
the residue. 



( 92 ) 

law)."* Computation; You constitute the shares of the 
(^9) heritage, by taking them out of twelve parts : the son 
receives seven of them, the husband three, and the 
mother two parts. You know that the husband must 
give up one- third of his share; accordingly he retains 
twice as much as that which is detracted from his share 
for the legacy. As he has three parts in hand, one of 
these falls to the legacy, and the remaining two parts 
he retains for himself. The tw^o legacies together are 
imposed upon the son. It is therefore necessary to 
subtract from his share two-fifths and one-fourth of the 
same. He thus retains seven twentieths of his entire 
original share, dividing the whole of it into twenty 
equal parts. The mother retains as much as she con- 
tributes to the legacy ; this is one (twelfth part), the 
entire amount of what she had received being two parts. 

* If the bequests stated in the present example were charged 
on the heirs collectively, the husband would be entitled to ^, 
the mother to J of the residue : ^-f ^— x%; ^^^ remainder J_ 
would be the son's share of the residue ; but since the 
bequests, J+i — 1-| of the capital, are charged upon the son 
and mother, the law throws a portion of the charge on the 
husband. 
TheHusband contributes J x §■ =20X2^0, and retains i X ^ =40X^i(y 

The Mother ^ x h =2ox-^lj^, ^ x i =20x^^0 

The Son • • .tVxM = 91 x^, 1^X2^=49x^1^- 

Total contributed = if J Total retained = i£g 



I- 4- J^ — JL4._5_ — JLA 



The Legatee, to whom the f are bequeathed, receives -^^ x i|i = 
The Legatee, to whom J is bequeathed, receives /^ x Jf^ = •'^ ^^ — - 



3 120 



( 93 ) 

Take now a sum, one -fourth of which may be di- 
vided into thirds, or of one-sixth of which the moiety 
may be taken ; this being again divisible by twenty. 
Such a capital is two hundred and forty. The mother 
receives one-sixth of this, namely, forty ; twenty from 
this fall to the legacy, and she retains twenty for her- 
self. The husband receives one-fourth, namely, sixty ; 
from which twenty belong to the legacy, so that he 
retains forty. The remaining hundred and forty belong 
to the son ; the legacy from this is two-fifths and one- 
fourth, or ninety-one; so that there remain fortyr 
nine. The entire sum for the legacies is, therefore, 
one hundred and thirty-one, which must be divided 
among the two legatees. The one to whom two-fifths 
were bequeathed, receives eight-thirteenths of this; 
the one to whom one-fourth was devised, receives five- 
thirteenths. If you wish distinctly to express the 
shares of the two legatees, you need only to multiply (70) 
the parts of the heritage by thirteen, and to take them 
out of a capital of three thousand one hundred and 
twenty. 

But if she had imposed on her son (payment of) the 
two-fifths to the person to whom the two-fifths were 
bequeathed, and of nothing to the other legatee ; and 
upon her mother (payment of) the one-fourth to the 
person to whom one-fourth was granted, and of nothing 
to the other legatee; and upon her husband nothing 
besides the one- third (which he must according to law 
contribute) to both ; then you know that this one-third 



( 94, ) 

comes to the advantage of the heirs collectively ; and 
the legatee of the two-fifths receives eight- thirteenths, 
and the legatee of the one-fourth receives five-thir- 
teenths from it. Constitute the shares as I have shown 
above, by taking twelve parts ; the husband receives 
one-fourth of them, the mother one-sixth, and the son 
that which remains.^ Computation : You know that at 
all events the husband must give up one-third of his 
share, which consists of three parts. The mother must 
likewise give up one-third, of which each legatee par- 
takes according to the proportion of his legacy. Be- 
sides, she must pay to the legatee to whom one- fourth is 
bequeathed, and whose legacy has been imposed on her, 
as much as the difference between the one-fourth and his 



5-t-4 -g— 20 

The Husband, who would be entitled to j^ of the residue, is 
not charged by the Testator with any bequest. 

The Mother who would be entitled to J of the residue, is 
charged with the payment of |^ to the Legatee A. 

The Son, who would be entitled to yV of the residue, is 

charged with payment off to the Legatee B. 

The Husband! , , o i • , « , 

contributes I 4 X i = 780x^3^^ ; retamsix|=4|65. 

The Mother . . . i [i + _8^ X ^] = 7 1 o x ^^-V^ ; retains ^^^o^ 

The Son t'itH + A x*] - 2884X^ ; retains ff Jf 

Total contributed = 437 *; Total retained =4|:8 6 

The Legatee A, to whom 1 is") 5 ^ 4374 _5 X4J74 
bequeathed, receives J 1^ ^ tj^^ — ^b 4 8 (J 

, The Legatee B, to whom f are 1 j, ^ 4" 4 - b x 4 , , 4 
bequeathed, receives / ^j »37" ~" JsiOJS 



( 95 ) ' 

poftion of the one-third, namely, nineteen one hundred 
and fifty-sixths of her entire share, considering her share 
as consisting of one hundred and fifty-six parts. His 
portion of the one-third of her share is twenty parts. 
But what she gives him is one-fourth of her entire share, 
namely, thirty-nine parts. One third of her share is 
taken for both legacies, and besides nineteen parts 
which she must pay to him alone. The son gives to the 
legatee to whom two- fifths are bequeathed as much as 
the difference between two-fifths of his (the son's) share CTl) 
and the legatee's portion of the one-third, namely, 
thirty-eight one hundred and ninety-fifths of his (the 
son's) entire share^ besides the one-third of it which is 
taken off from both legacies. The portion which he 
(the legatee) receives from this one-third, is eight- 
thirteenths of it, namely, forty (one hundred and ninety- 
fifths); and what the son contributes of the two-fifths 
from his share is thirty-eight. These together make 
seventy-eight. Consequently, sixty-five will be taken 
from the son, as being one-third of his share, for both 
legacies, and besides this he gives thirty-eight to the 
one of them in particular. If you wish to express the 
parts of the heritage distinctly, you may do so with 
nine hundred and sixty-four thousand and eighty. 



On another Species of Legacies, 
" A man dies, leaving four sons and his wife ; and 
bequeathing to a person as much as the share of one 



( 96 ) 

of the sons less the amount of the share of the widow." 
Divide the heritage into thirty-two parts. The widow 
receives one-eighth,* namely, four; and each son seven. 
Consequently the legatee must receive three- sevenths of 
the share of a son. Add, therefore, to the heritage 
three-sevenths of the share of a son, that is to say> 
three parts, which is the amount of the legacy. This 
gives thirty-five, from which the legatee receives three; 
and the remaining thirty-two are distributed among 
the heirs proportionably to their legal shares. 

If he leaves two sons and a daughter, f and bequeaths 

to some one as much as would be the share of a third 

son, if he had one; then you must consider, what 

(72) would be the share of each son, in case he had three. 

Assume this to be seven, and for the entire heritage 



* A widow is entitled to Jth of the residue ; therefore 
Jths of the residue are to be distributed among the sons of 
the testator. Let x be the stranger's legacy. The widow's 
share =lnf; each son's share =ix§[i~a?]; and a son's 
share, minus the widow's share = [1 — i] Lll5 : 



1 X 



/. a;=|.i_^ .-. x=^-^; i-^^lf A son's share ■:^^; 
the widow's share = ^. 

f A son is entitled to receive twice as much as a daughter. 
Were there three sons and one daughter, each son would 
receive f ths of the residue. Let x be the stranger's legacy. 
.'. f[i— a:]=ar .*. a; = f, andi— a; = -^ 

Each Son's share = | {\—x\ = | x J = J^ 

The Daughter's share =;!■ [i — x] =^-g 

The Stranger's legacy =f =i^ 



( 97 ) 

take a number, one-fifth of which may be divided into 
sevenths, and one-seventh of which may be divided into 
fifths. Such a number is thirty-five. Add to it two- 
sevenths of the same, namely, ten. This gives forty- 
five. Herefrom the legatee receives ten, each son four- 
teen, and the daughter seven. 

If he leaves a mother, three sons, and a daughter, 
and bequeaths to some one as much as the share of one 
of his sons less the amount of the share of a second 
daughter, in case he had one ; then you distribute the 
heritage into such a number of parts as may be divided 
among the actual heirs, and also among the same, if a 
second daughter were added to them.* Such a number 
is three hundred and thirty-six. The share of the 
second daughter, if there were one, would be thirty- 
five, and that of a son eighty ; their difference is forty- 
five, and this is the legacy. Add to it three hundred 
and thirty-six, the sum is three hundred and eighty- 
one, which is the number of parts of the entire heritage. 



* Let X be the stranger's legacy ; i — x is the residue. 
A widow's share of the residue is ^th ; there remains 
J [i— j:], to be distributed among the children. 

Since there are 3 sons, and 1 daughter, 1 2 ^ 5r-__ i 
a son's share is / t ^ ^L J 

Were there 3 sons and 2 daughters, ^lixsri— j:1 
daughter's share would be J s ^l J 

The difference = ^^ Kf[i -x] 

1 _a;rj:|^|; the widow's share - ^ 
the daughter's share — ^^ 



( 98 ) 

If he leaves three sons, and bequeaths to some one 
as much as the share of one of his sons, less the share 
of a daughter, supposing he had one, plus one-third 
of the remainder of the one-third; the computation 
will be this :* distribute the heritage into such a number 
of parts as may be divided among the actual heirs, and 
also among them if a daughter were added to them. 
Such a number is twenty-one. Were a daughter among 
the heirs, her share would be three, and that of a son 
seven. The testator has therefore bequeathed to the 
(73) legatee four-sevenths of the share of a son, and one- 
third of what remains from one-third. Take therefore 
one-third, and remove from it four-sevenths of the 
share of a son. There remains one- third of the capital 
less four-sevenths of the share of a son. Subtract now 
one-third of what remains of the one-third, that is to 
say, one-ninth of the capital less one-seventh and one- 
third of the seventh of the share of a son ; the remainder 



* Since there are 3 sons, each son's share of the residue n J. 
Were there 3 sons and a daughter, the daughter's share 



would be \. 



i-i=4 



3 

Let X be the stranger's legacy, and v a son's share 
Then i—x = ^v 

and i^a:^%-{.i-±v-i[i^^v]=^3v 

.-. f + frr3AxV' orf=J^« 
.*. f=V '^ •*• *' = 2TT = ^ ^°^'^ share 

X zz -^^ = the stranger's legacy. 



( 99 ) 

is two-ninths of the capital less two-sevenths and two- 
thirds of a seventh of the share of a son. Add this to 
the two-thirds of the capital ; the sum is eight- ninths 
of the capital less two-sevenths and two thirds of a 
seventh of the share of a son, or eight twenty-one parts 
of that share, and this is equal to three shares. Re- 
duce this, you have then eight-ninths of the capital, 
equal to three shares and eight twenty-one parts of a 
share. Complete the capital by adding to eight-ninths 
as much as one- eighth of the same, and add in the 
same proportion to the shares. Then you find the 
capital equal to three shares and forty-five fifty-sixth 
parts of a share. Calculating now each share equal to 
fifty-six, the whole capital is two hundred and thirteen, 
the first legacy thirty-two, the second thirteen, and of 
the remaining one hundred and sixty-eight each son 
takes fifty-six. 

On another Species of Legacies. 
" A woman dies, leaving her daughter, her mother, 
and her husband, and bequeaths to some one as much 
as the share of her mother, and to another as much as 
one-ninth of her entire capital."* Computation : You 
begin by dividing the heritage into thirteen parts, two 

* In the former examples (p. 90) when a husband and a 
mother were among the heirs, a husband was found to be 
entitled to ^=^~^ and a mother to 6=y% of the residue. 
Here a husband is stated to be entitled to -f.^ , and a mother 
to -f^ of the residue. 



( 100 ) 

of which the mother receives. Now you perceive that the 
C^^) legacies amount to two parts plus one-ninth of the en- 
tire capital. Subtracting this, there remains eight-ninths 
of the capital less two parts, for distribution among 
the heirs. Complete the capital, by making the eight- 
ninths less two parts to be thirteen parts, and adding 
two parts to it, so that you have fifteen parts, equal 
to eight-ninths of capital; then add to this one- 
eighth of the same, and to the fifteen parts add like- 
wise one- eighth of the same, namely, one part and 
seven-eighths ; then you have sixteen parts and seven- 
eighths. The person to whom one-ninth is bequeathed, 
receives one-ninth of this, namely, one part and seven- 
eighths ; the other, to whom as much as the share of 
the mother is bequeathed, receives two parts. The 
remaining thirteen parts are divided among the heirs, 
according to their legal shares. You best determine 
the respective shares by dividing the whole heritage 
into one hundred and thirty-five parts. 

If she has bequeathed as much as the share of the 
husband and one- eighth and one-tenth of the capital,* 

Let -f^ of the residue =zv 

.'. urr-jf^ of the capital 
A mother's share— ^i^^^ 

A husband's share of the residue is -^^ 

.•. vrrg3j^; a husband's share =^^-^q 
The stranger's legacy = ||^ 



( 101 ) 

then you begin by dividing the heritage into thirteen 
parts. Add to this as much as the share of the hus- 
band, namely, three; thus you have sixteen. This is 
what remains of the capital after the deduction of one- 
eighth and one-tenth, that is to say, of nine-fortieths. 
The remainder of the capital, after the deduction of 
one-eighth and one- tenth, is thirty-one fortieths of the 
same, v^^hich must be equal to sixteen parts. Complete 
your capital by adding to it nine thirty-one parts of the 
same, and multiply sixteen by thirty-one, which gives 
four hundred and ninety-six ; add to this nine thirty- 
one parts of the same, which is one hundred and forty- ^75) 
four. The sum is six hundred and forty. Subtract 
one-eighth and one-tenth from it, which is one hun- 
dred and forty-four, and as much as the share of the 
husband, which is ninety- three. There remains four 
hundred and three, of which the husband receives 
ninety-three, the mother sixty-two, and every daughter 
one hundred and twenty-four. 

If the heirs are the same,* but that she bequeaths to 
a person as much as the share of the husband, less 
one-ninth and one-tenth of what remains of the capital, 

•••W [1-3] =13^ 

... 1^9= [13-1-1^9] « 

The husband's share =y^t 
The stranger's legacy =Yf§y 



( 102 ) 

after the subtraction of that share, the computation is 
this : Divide the heritage into thirteen parts. The 
legacy from the whole capital is three parts, after the 
subtraction of which there remains the capital less three 
parts. Now, one-ninth and one-tenth of the remain- 
ing capital must be added, namely, one-ninth and one- 
tenth of the whole capital less one- ninth and one-tenth 
of three parts, or less nineteen- thirtieths of a part ; this 
yields the capital and one-ninth and one -tenth less 
three parts and nineteen-thirtieths of a part, equal to 
thirteen parts. Reduce this, by removing the three 
parts and nineteen-thirtieths from your capital, and 
adding them to the thirteen parts. Then you have 
the capital and one-ninth and one-tenth of the same, 
equal to sixteen parts and nineteen-thirtieths of a part. 
Reduce this to one capital, by subtracting from it 
nineteen one-hundred-and-ninths. There remains a 
(76) capital, equal to thirteen parts and eighty one-hundred- 
and-ninths. Divide each part into one hundred and 
nine parts, by multiplying thirteen by one hundred 
and nine, and add eighty to it. This gives one thou- 
sand four hundred and ninety-seven parts. The share 
of the husband from it is three hundred and twenty- 
seven parts. 

If some one leaves two sisters and a wife,* and be- 
queaths to another person as much as the share of a 



* When the heirs are a wife, and 2 sisters, they each 
inherit ^ of the residue. 



Let 



( 103 ) 

sister less one- eighth of what remains of the capital 
after the deduction of the legacy, the computation is 
this : You consider the heritage as consisting of twelve 
parts. Each sister receives one-third of what remains 
of the capital after the subtraction of the legacy ; that 
is, of the capital less the legacy. You perceive that 
one-eighth of the remainder plus the legacy equals the 
share of a sister ; and also, one-eighth of the remainder 
is as much as one-eighth of the whole capital less one- 
eighth of the legacy ; and again, one-eighth of the 
capital less one-eighth of the legacy added to the legacy 
equals the share of a sister, namely, one-eighth of the 
capital and seven-eighths of the legacy. The whole 
capital is therefore equal to three- eighths of the capital 
plus three and five-eighth times the legacy. Subtract 
now from the capital three- eighths of the same. There 
remain five-eighths of the capital, equal to three and 
five-eighth times the legacy ; and the entire capital is 
equal to five and four-fifth times the legacy. Conse- 
quently, if you assume the capital to be twenty-nine, 
the legacy is five, and each sister's share eight. 

Let X be the stranger's iegacy. 

3 [^~""^]= ^ sister's share 

i[l-a;]-J[l-a:]=ar 

... /^[i~x]=ar .•.A=fl>^ 

and a sister's share =^ 



( 104 ) 

On another Species of Legacies. 
" A man dies, and leaves four sons, and bequeaths 
tx) some person as much as the share of one of his sons; 
and to another, one-fourth of what remains after the 
deduction of the above share from one-third." You 
perceive that this legacy belongs to the class of those 
V*^) which are taken from one-third of the capital.* Compu- 
tation : Take one-third of the capital, and subtract 
from it the share of a son. The remainder is one- 
third of the capital less the share. Then subtract from 
it one-fourth of what remains of the one-third, namely, 
one-fourth of one- third less one-fourth of the share. 
The remainder is one-fourth of the capital less three- 
fourths of the share. Add hereto two-thirds of the 
capital : then you have eleven-twelfths of the capital less 
three-fourths of a share, equal to four shares. Reduce 
this by removing the three-fourths of the share from the 
capital, and adding them to the four shares. Then you 
have eleven- twelfths of the capital, equal to four shares 
and three-fourths. Complete your capital, by adding 
to the four shares and three-fourths one-fourth of the 
same. Then you have five shares and two-elevenths, 



* Let the first bequest r= v; and the second =1/ 

Then i—v —y — 41; 

i.e.f+^-«-J[^-»]=4« 

•••|+f»-''] = 4» 

•••*+A=[4+J]» •••«=',?« 

.*. u=:i_i; the 2d bequest =3^ 



( 105 ) 

equal to the capital. Suppose, now, every share to be 
eleven ; then the whole square will be fifty-seven ; one- 
third of this is nineteen ; from this one share, namely, 
eleven, must be subtracted ; there remain eight. The 
legatee, to whom one-fourth of this remainder was be- 
queathed, receives two. The remaining six are re- 
turned to the other two-thirds, which are thirty-eight. 
Their sum is forty-four, which is to be divided amongst 
the four sons; so that each son receives eleven. 

If he leaves four sons, and bequeaths to a person as 
much as the share of a son, less one-fifth of what re- 
mains ffom one-third after the deduction of that share, 
then this is likewise a legacy, which is taken from one- 
third.^ Take one-third, and subtract from it one 
share ; there remains one-third less the share. Then 
return to it that which was excepted, namely, one-fifth 
of the one-third less one-fifth of the share. This gives 
one-third and one-fifth of one-third (or two-fifths) (78) 
less one share and one-fifth of a share. Add this to 
two-thirds of the capital. The sum is, the capital and 
one-third of one-fifth of the capital less one share and 
one-fifth of a share, equal to four shares. Reduce this 
by removing one share and one-fifth from the capital, 



or fx^-v-f^[i-v]=r4v 
^ , and the stranger's legacy = ^ 



( 106 ) 

and add to it the four shares. Then you have the 
capital and one -third of one-fifth of the capital, which 
are equal to five shares and one- fifth. Reduce this to 
one capital, by subtracting from what you have the 
moiety of one-eighth of it, that is to say, one-sixteenth. 
Then you find the capital equal to four shares and 
seven-eighths of a share. Assume now thirty-nine as 
capital; one- third of it will be thirteen, and one share 
eight ; what remains of one-third, after the deduction 
of that share, is five, and one-fifth of this is one. Sub- 
tract now the one, which was excepted from the legacy ; 
the remaining legacy then is seven ; subtracting this 
from the one- third of the capital, there remain six. 
Add this to the two-thirds of the capital, namely, to 
the twenty-six parts, the sum is thirty-two; which, 
when distributed among the four sons, yields eight for 
each of them. 

If he leaves three sons and a daughter,* and be- 
queaths to some person as much as the share of a 



* Since there are three sons and one daughter, the daugh- 
ter receives i, and each son |^ths of the residue. 

If the 1st legacy = 1), the 2d =:^, and therefore a daugh- 
ter's share = v, 

... QJ — ii-?i) . — 1B8 



The 2d legacy = ..^ = r?g,. 



( 107 ) 

daughter, and to another one-fifth and one-sixth of 
what remains of two-sevenths of the capital after the 
deduction of the first legacy ; then this legacy is to be 
taken out of two-sevenths of the capital. Subtract 
from two-sevenths the share of the daughter: there 
remain two -sevenths of the capital less that share. 
Deduct from this the second legacy, which comprises (T9) 
one-fifth and one-sixth of this remainder : there remain 
one-seventh and four-fifteenths of one-seventh of the 
capital less nine teen-thirtieths of the share. Add to 
this the other five-sevenths of the capital: then you 
have six-sevenths and four-fifteenths of one-seventh of 
the capital less nineteen thirtieths of the share, equal to 
seven shares. Reduce this, by removing the nineteen 
thirtieths, and adding them to the seven shares : then 
you have six- sevenths and four-fifteenths of one-seventh 
of capital, equal to seven shares and nineteen-thirtieths. 
Complete your capital by adding to every thing that 
you have eleven ninety-fourths of the same ; thus the 
capital will be equal to eight shares and ninety-nine 
one hundred and eighty-eighths. Assume now the 
capital to be one thousand six hundred and three ; then 
the share of the daughter is one hundred and eighty- 
eight. Take two- sevenths of the capital ; that is, four 
hundred and fifty-eight. Subtract from this the share, 
which is one hundred and eighty-eight ; there remain 
two hundred and seventy. Remove one-fifth and one- 
sixth of this, namely, ninety-nine ; the remainder is 
one hundred and seventy-one. Add thereto five- 



( 108 ) 

sevenths of the capital, which is one thousand one 
hundred and forty-five. The sum is one thousand three 
(80) hundred and sixteen parts. This may be divided into 
seven shares, each of one hundred and eighty-eight 
parts ; then this is the share of the daughter, whilst 
every son receives twice as much. 

If the heirs are the same, and he bequeaths to some 
person as much as the share of the daughter, and to 
another person one-fourth and one-fifth out of what 
remains from two-fifths of his capital after the deduc- 
tion of the share ; this is the computation :* You must 
observe that the legacy is determined by the two fifths. 
Take two-fifths of the capital and subtract the shares : 
the remainder is, two-fifths of the capital less the share. 
Subtract from this remainder one-fourth and one-fifth 
of the same, namely, nine- twentieths of two-fifths, less 
as much of the share. The remainder is one-fifth 
and one-tenth of one fifth of the capital less eleven- 
twentieths of the share. Add thereto three-fifths of the 



Let the ist legacy =v = a daughter's share 
Let the 2d legacy =y 

1— V — ^=7^ 

•••4+*-»'-A[f-«]=7« 

••4+M [#-"] =7" 

and the 2d legacy, y, =tA 



( 109 ) 

capital : the sum is four- fifths and one- tenth of one- 
fifth of the capital, less eleven-twentieths of the share, 
equal to seven shares. Reduce this by removing the 
eleven-twentieths of a share, and adding them to the 
seven shares. Then you have the same four-fifths and 
one-tenth of one-fifth of capital, equal to seven shares 
and eleven-twentieths. Complete the capital by adding 
to any thing that you have nine forty-one parts. Then 
you have capital equal to nine shares and seventeen 
eighty-seconds. Now assume each portion to consist 
of eighty- two parts ; then you have seven hundred and 
fifty-five parts. Two-fifths of these are three hundred (81) 
and two. Subtract from this the share of the daughter, 
which is eighty-two ; there remain two hundred and 
twenty. Subtract from this one-fourth and one-fifth, 
namely, ninety-nine parts. There remain one hun- 
dred and twenty- one. Add to this three-fifths of the^ 
capita], namely, four hundred and fifty-three. Then 
you have five hundred and seventy-four, to be divided 
into seven shares, each of eighty- two parts. This is 
the share of the daughter ; each son receives twice as 
much. 

If the heirs are the same, and he bequeaths to a 
person as much as the share of a son, less one-fourth 
and one-fifth of what remains of two-fifths (of the 
capital) after the deduction of the share; then you see 
that this legacy is likewise determined by two- fifths. 
Subtract two shares (of a daughter) from them, since 
every son receives two (such) shares; there remain 



( HO ) 

two-fifths of the capital less two (such) shares. Add 
thereto what was excepted from the legacy, namely, 
one-fourth and one-fifth of the two-fifths less nine- 
tenths of (a daughter's) share.* Then you have two- 
fifths and nine-tenths of one-fifth of the capital less two 
(daughter's) shares and nine-tenths. Add to this 
three- fifths of the capital. Then you have one -capital 
and nine- tenths of one-fifth of the capital less two 
(daughter's) shares and nine- tenths, equal to seven (such) 
shares. Reduce this by removing the two shares and 
nine-tenths and adding them to the seven shares. Then 
you have one capital and nine-tenths of one-fifth of the 
capital, equal to nine shares of a daughter and nine- 
(82) tenths. Reduce this to one entire capital, by deduct- 
ing nine fifty-ninths from what you have. There re- 
mains the capital equal to eight such shares and twenty- 
three fifty-ninths. Assume now each share (of a 
daughter) to contain fifty-nine parts. Then the whole 
heritage comprizes four hundred and ninety-five parts. 
Two-fifths of this are one hundred and ninety-eight 



* v = i. of the residue = a daughter's share. 
2v = a son's share 

»e-f+f-2^+A[f-H =7V 

.-. v = ^^; a son's share = i^f 
and the legacy to the stranger = -^^ 



( 111 ) 

parts. Subtract therefrom the two shares (of a daugh- 
ter) or one hundred and eighteen parts; there remain 
eighty parts. Subtract now that which was excepted, 
namely, one-fourth and one fifth of these eighty, or 
thirty-six parts ; there remain for the legatee eighty- 
two parts. Deduct this from the parts in the total 
number of parts in the heritage, namely, four hundred 
and ninety-five. There remain four hundred and thir- 
teen parts to be distributed into seven shares; the 
daughter receiving (one share or) fifty-nine (parts), and 
each son twice as much. 

If he leaves two sons and two daughters, and be- 
queaths to some person as much as the share* of a 



* Since there are two sons and two daughters, each son 
receives J, and each daughter ^ of the residue. Let 
V = a daughter's share. 

Let the ist legacy =a:=v— J [3— v] 

2d =3,=:i,_i[j_a;_i,] 

and 3d = ^ 

i-e. f_^V + J-.x~t; + J[i^-ar-T;] =61, 

or7 + B^L35a .'.v=:^ = k 
The 1 st Legacy =x = ^j 

The 2d =y = ij 

A son's share =:J 



( 112 ) 

daughter less one-fifth of what remains from one-third 
after the deduction of that share; and to another 
person as much as the share of the other daughter less 
one-third of what remains from one-third after the de- 
duction of all this ; and to another person half one- sixth 
of his entire capital ; then you observe that all these 
legacies are determined by the one-third. Take one- 
third of the capital, and subtract from it the share of a 
daughter ; there remains one-third of the capital less 
one share. Add to this that which was excepted, 
namely, one-fifth of the one-third less one-fifth of the 
share : this gives one-third and one-fifth of one-third of 
(83) the capital less one and one-fifth portion. Subtract 
herefrom the portion of the second daughter ; there 
remain one- third and one-fifth of one- third of the 
capital less two portions and one-fifth. Add to this 
that which was excepted; then you have one- third 
and three-fifths of one-third, less two portions and 
fourteen-fifteenths of a portion. Subtract herefrom 
half one-sixth of the entire capital : there remain 
twenty-seven sixtieths of the capital less the two 
shares and fourteen-fifteenths, which are to be sub- 
tracted. Add thereto two-thirds of the capital, and 
reduce it, by removing the shares which are to be sub- 
tracted, and adding them to the other shares. You 
have then one and seven-sixtieths of capital, equal to 
eight shares and fourteen-fifteenths. Reduce this to 
one capital by subtracting from every thing that you 
have seven-sixtieths. Then let a share be two hundred 



( 113 ) 

and one;* the whole capital will be one thousand six 
hundred and eight. 

If the heirs are the same, and he bequeaths to a 
person as much as the share of a daughter, and one- 
fifth of what remains from one- third after the deduction 
of that share ; and to another as much as the share of 
the second daughter and one-third of what remains 
from one-fourth after the deduction of that share; 
then, in the computation,! you must consider that the 
two legacies are determined by one-fourth and one- 
third. Take one- third of the capital, and subtract from 
it one share ; there remains one- third of the capital 
less one share. Then subtract one-fifth of the re- 
mainder, namely, one-fifth of one-third of the capital, 
less one-fifth of the share ; there remain four-fifths of 
one-third, less four-fifths of the share. Then take also 
one-fourth of the capital, and subtract from it one (84) 
share ; there remains one-fourth of the capital, less one 
share. Subtract one-third of this remainder : there 

The common denominator 1608 is unnecessarily great. 

f Let X be the 1st legacy ; 1/ the 2d ; v a daughter's share. 

1 — X — y—Qv 

Theni-i--Hi-^-i[i-"]+i-^-i[i-^J-6« 

. 51 — 112„ . 51 _ 153_ 
<r — 212 . ,/ — 214^ 



( 114 ) 
remain two-thirds of one-fourth of the capital, less two- 
thirds of one share. Add this to the remainder from 
the one-third of the capital ; the sum will be twenty- 
six sixtieths of the capital, less one share and twenty- 
eight sixtieths. Add thereto as much as remains of 
the capital after the deduction of one-third and one- 
fourth from it; that is to say, one-fourth and one- 
sixth; the sum is seven teen-twentieths of the capital, 
equal to seven shares and seven-fifteenths. Complete 
the capital, by adding to the portions which you have 
three-seventeenths of the same. Then you have one 
capital, equal to eight shares and one-hundred-and- 
twenty hundred-and-fifty-thirds. Assume now one share 
to consist of one-hundred-and- fifty-three parts, then 
the capital consists of one thousand three hundred and 
forty-four. The legacy determined by one- third, after 
the deduction of one share, is fifty-nine ; and the legacy 
determined by one-fourth, after the deduction of the 
share, is sixty- one. 

If he leaves six sons, and bequeaths to a person as 
much as the share of a son and one-fifth of what remains 
of one- fourth ; and to another person as much as the 
share of another son less one-fourth of what remains 
of one-third, after the deduction of the two first lega- 
cies and the second share; the computation is this:* 
You subtract one share from one-fourth of the capital ; 



* Let x be the legacy to the ist stranger 

and 7/ 2d ; v- a son's share 



( 115 ) 

there remains one-fourth less the share. Remove then (85) 
one-fifth of what remains of the one-fourth, namely, 
half one-tenth of the capital less one-fifth of the share. 
Then return to the one- third, and deduct from it half 
one-tenth of the capital, and four-fifths of a share, and 
one other share besides. The remainder then is one- 
third, less half one-tenth of the capital, and less one 
share and four-fifths. Add hereto one-fourth of the 
remainder, which was excepted, and assume the one- 
third to be eighty; subtracting from it half one-tenth of 
the capital, there remain of it sixty-eight less one 
share and four-fifths. Add to this one-fourth of it, 
namely, seventeen parts, less one-fourth of the shares 
to be subtracted from the parts. Then you have 
eighty-five parts less two shares and one- fourth. Add 
this to the other two- thirds of the capital, namely, one 
hundred and sixty parts. Then you have one and one- 
eighth of one-sixth of capital, less two shares and one- 
fourth, equal to six shares. Reduce this, by remov- 
ing the shares which are to be subtracted, and adding 



1 —x—y=^v 
i.e. §-f j-— jc — u-t-J [J— x— u]=6u . 

.-. x-v^^s, and 3^=©-^^ 



( 116 ) 

them to the other shares. Then you have one and one- 
eighth of one- sixth of capital, equal to eight shares 
and one-fourth. Reduce this to one capital, by sub- 
tracting from the parts as much as one forty-ninth of 
them. Then you have a capital equal to eight shares 
and four forty-ninths. Assume now every share to be 
forty-nine ; then the entire capital will be three hun- 
dred and ninety-six : the share forty-nine ; the legacy 
(86) determined by one-fourth, ten ; and the exception from 
the second share will be six. 



On the Legacy with a Dirhem. 

" A man dies, and leaves four sons, and bequeaths 

to some one a dirhem, and as much as the share of a 

son, and one-fourth of what remains from one-third 

after the deduction of that share." Computation :* Take 



* Let the capital = i ; a dirhem =^ ; 
the legacy —a:; and a son's share —v 

1 — X=:^>C 

•••f+i-«-i[*-v]-^=4t^ 

.-. H-^ = ¥^ 

.*. ii of the capital — Af of a dirhem —v 
and Jf of the capital +ff of a dirhem = j:, the legacy. 
If we assume the capital to be so many dirhems, or a 
dirhem to be such a part of the capital, we shall obtain the 



{ 117 ) 

one third of the capital and subtract from it one share; 
there remains one-third, less one share. Then sub- 
tract one-fourth of the remainder, namely, one-fourth 
of one-third, less one-fourth of the share ; then sub- 
tract also one dirhem ; there remain three-fourths of 
one- third of the capital, that is, one-fourth of the 
capital, less three-fourths of the share, and less one 
dirhem. Add this to two-thirds of the capital. The 
sum is eleven- twelfths of the capital, less three-fourths 
of the share and less one dirhem, equal to four shares. 
Reduce this by removing three-fourths of the share 
and one dirhem ; then you have eleven-twelfths of the 
capital, equal to four shares and three- fourths, plus 
one dirhem. Complete your capital, by adding to the 
shares and one dirhem one-eleventh of the same. Then 
you have the capital equal to five shares and two- 
elevenths and one dirhem and one-eleventh. If you (8*7) 
wish to exhibit the dirhem distinctly, do not complete 
your capital, but subtract one from the eleven on 
account of the dirhem, and divide the remaining ten by 
the portions, which are four and three-fourths. The 
quotient is two and two-nineteenths of a dirhem. 
Assuming, then, the capital to be twelve dirhems, each 



value of the son's share in terms of a dirhem, or of the 
capital only. 

Thus, if we assume the capital to be 1 2 dirhems, 

V - ;^f [ 1 1 — 1 ]5 = V"t ^ = 2y% dirhems, 
x=if [13 + 4] ^=W^ = 3ii dirhems. 



( 118 ) 

share will be two dirhems and two-nineteenths. Or, if 
you wish to exhibit the share distinctly, complete your 
square, and reduce it, when the dirhem will be eleven 
of the capital. 

If he leaves five sons, and bequeaths to some per- 
son a dirhem, and as much as the share of one of the 
sons, and one-third of what remains from one-third, 
and again, one-fourth of what remains from the one- 
third after the deduction of this, and one dirhem more ; 
then the computation is this:^ You take one-third, and 
subtract one share ; there remains one- third less one 
share. Subtract herefrom that which is still in your 
hands, namely, one-third of one-third less one- third of 
the share. Then subtract also the dirhem ; there re- 
main two- thirds of one-third, less two- thirds of the 
share and less one dirhem. Then subtract one -fourth 
of what you have, that is, one-eighteenth, less one- 
sixth of a share and less one-fourth of a dirhem, and 



* Let the legacy =a:; and a son's share =r 



•P=v« 



.*. I^f of the capital —f^^ of a dirhem —v 
.*. Jl of the capital + y^/ of a dirhem =x, the legacy. 
If the capital = %^ dirhems, or J of the capital =7 J dirhems, 
V =: f f dirhems = 3^1^ dirhems. 



( 119 ) 

subtract also the second dirhem ; the remainder is half 
one-third of the capital, less half a share and less one 
dirhem and three fourths ; add thereto two-thirdsof the 
capital, the sum is five-sixths of the capital, less one 
half of a share, and less one dirhem and three-fourths, 
equal to five shares. Reduce this, by removing the (88) 
half share and the one dirhem and three-fourths, 
and adding them to the (five) shares. Then you 
have five-sixths of capital, equal to five shares and a 
half plus one dirhem and three-fourths. Complete ) 
your capital, by adding to five shares and a half and 
to one dirhem and three- fourths, as much as one-fifth 
of the same. Then you have the capital equal to six 
shares and three-fifths plus two dirhems and one- 
tenth. Assume, now, each share to consist of ten 
parts, and one dirhem likewise of ten ; then the ca- 
pital is eighty-seven parts. Or, if you wish to exhibit 
the dirhem distinctly, take the one-third, and subtract 
from it the share; there remains one-third, less one 
share. Assume the one-third (of the capital) to be 
seven and a half (dirhems). Subtract one- third of what 
you have, namely, one-third of one-third;* there 
remain two- thirds of one- third, less two-thirds of the 
share : that is, five dirhems, less two- thirds of the 
share. Then subtract one, on account of the one 
dirhem, and you retain four dirhems, less two-thirds 

* There is an omission here of the words *' less one third 
of a share." 



( 120 ) , 

of the share. Subtract now one-fourth of what you 
have, namely, one part less one-sixth of ^ share ; 
and remove also one part on account of the one 
dirhem; the remainder, then, is two parts less half 
a share. Add this to the two-thirds of the capital, 
which is fifteen (dirhems). Then you have seventeen 
parts less half a share, equal to five shares. Reduce 
this, by removing half a share, and adding it to the 
five shares. Then it is seventeen parts, equal to 
/89) five shares and a half. Divide now seventeen by five 
and a half; the quotient is the value of one share, 
namely, three dirhems and one-eleventh ; and one- 
third (of the capital) is seven and a half (dirhems). 

If he leaves four sons, and bequeaths to some person 
as much as the share of one of his sons, less one- 
fourth of what remains from one-third after the deduc- 
tion of the share, and one dirhem; and to another 
one-third of what remains from the one-third, and one 
dirhem; then this legacy is determined by one- third.* 



* Let the ist legacy be x, the 2d y; and a son's share rr v 

1 —X — J/z:z4.V 

i.e. f H-v+i {i-v]-i-i \i-v+i (l-„)-Jj- J=4„ 
'•e-f+f[*-«+i(i-<')-JJ-J=4» 

••• l+A-f«-f ^=4» 



( 121 ) 

Take one-third of the capital, and subtract from it one 
share ; there remains one-third, less one share ; add 
hereto one-fourth of what you have : then it is one- 
third and one-fourth of one-third, less one share and 
one-fourth. Subtract one dirhem ; there remains one- 
third of one and one-fourth, less one dirhem, and less 
one share and one-fourth. There remains from the 
one-third as much as five-eighteenths of the capital, less 
two-thirds of a dirhem, and less five-sixths of a share. 
Now subtract the second dirhem, and you retain five- 
eighteenths of the capital, less one dirhem and two- 
thirds, and less five-sixths of a share. Add to this 
two-thirds of the capital, and you have seventeen- 
eighteenths of the capital, less one dirhem and two- 
thirds, and less five-sixths of a share, equal to four 
shares. Reduce this, by removing the quantities 
which are to be subtracted, and adding them to the 
shares; then you have seventeen-eighteenths of the 
capital, equal to four portions and five-sixths plus one 
dirhem and two-thirds. Complete your capital by (90) 
adding to the four shares and five-sixths, and one 
dirhem and two-thirds, as much as one- seventeenth of 
the same. Assume, then, each share to be seventeen, 
and also one dirhem to be seventeen.* The whole 
capital will then be one hundred and seventeen. 
If you wish to exhibit the dirhem distinctly, proceed 
with it as I have shown you. 



* Capital =;f|v + f^ J .-. ifv=i7, and5=:i7, capital=ii7 

R 



( 122 ) 

If he leaves three sons and two daughters, and 
bequeaths to some person as much as the share of a 
daughter plus one dirhem ; and to another one-fifth of 
what remains from one-fourth after the deduction of 
the first legacy, plus one dirhem ; and to a third per- 
son one-fourth of what remains from one- third after 
the deduction of all this, plus one dirhem ; and to a 
fourth person one-eighth of the whole capital, requiring 
all the legacies to be paid off by the heirs generally : 
then you calculate this by exhibiting the dirhems dis- 
tinctly, which is better in such a case.^ Take one-fourth 
of the capital, and assume it to be six dirhems ; the 
entire capital will be twenty-four dirhems. Subtract 
one share from the one-fourth; there remain six 
dirhems less one share. Subtract also one dirhem; 
there remain five dirhems less one share. Subtract 



* Let the legacies to the three first legatees be, severally, 
X, ^, z; the fourth legacy = J ; and let a daughters' share 

Then !i-^-^^-a:-.j^-l\l~x-,]-^ = Sv 
but ^-x-i/^i^i+l^a:-^ [l~^] -^ 

^=-h%hnm^> y=iUT+uu^> ^=^'^+iUP 



( 123 ) 

one-fifth of this remainder; there remain four dirhems, 
less four-fifths of a share. Now deduct the second 
dirhem, and you retain three dirhems, less four-fifths 
of a share. You know, therefore, that the legacy 
which was determined by one-fourth, is three dirhems, 
less four-fifths of a share. Return now to the one- 
third, which is eight, and subtract from it three dir- 
hems, less four-fifths of a share. There remain five ^ ^ 
dirhems, less four-fifths of a share. Subtract also one- 
fourth of this and one dirhem, for the legacy ; you then 
retain two dirhems and three-fourths, less three -fifths of 
a share. Take now one-eighth of the capital, namely, 
three ; after the deduction of one-third, you retain one- 
fourth of a dirhem, less three -fifths of a share. Return 
now to the two- thirds, namely, sixteen, and subtract 
from them one-fourth of a dirhem less three- fifths of a 
share ; there remain of the capital fifteen dirhems and 
three-fourths, less three-fifths of a share, which are 
equal to eight shares. Reduce this, by removing three- 
fifths of a share, and adding them to the shares, which 
are eight. Then you have fifteen dirhems and three- 
fourths, equal to eight shares and three-fifths. Make 
the division: the quotient is one share of the whole 
capital, which is twenty-four (dirhems). Every daugh- 
ter receives one dirhem and one-hundred- and- forty- 
three one-hundred-and-seventy-second parts of a dir- 
hem.* 



v=JQ^gL of the capital— ^/^\ of a dirhem. If we assume 



( 124 ) 

If you prefer to produce the shares distinctly, take 
one-fourth of the capital, and subtract from it one 
share; there remains one-fourth of the capital less 
one share. Then subtract from this one dirhem: 
then subtract one-fifth of the remainder of one-fourth, 
which is one-fifth of one-fourth of the capital, less one- 
fifth of the share and less one-fifth of a dirhem ; and 
subtract also the second dirhem. There remain four- 
fifths of the one-fourth less four-fifths of a share, and 
less one dirhem and four- fifths. The legacies paid out 
of one fourth amount to twelve two-hundred-and- 
(92) fortieths of the capital and four- fifths of a share, and 
one dirhem and four-fifths. Take one-third, which is 
eighty, and subtract from it twelve, and four- fifths of a 
share, and one dirhem and four-fifths, and remove 
one-fourth of what remains, and one dirhem. You 
retain, then, of the one-third, only fifty-one, less three- 
fifths of a share, less two dirhems and seven-twentieths. 
Subtract herefrom one-eighth of the capital, which is 
thirty, and you retain twenty- one, less three- fifths of 
a share, and less two dirhems and seven-twentieths, 
and two-thirds of the capital, being equal to eight 
shares. Reduce this, by removing that which is to 
be subtracted, and adding it to the eight shares. Then 
you have one hundred and eighty-one parts of the 



the capital to be equal to 24 dirhems 

=45-80 5=: ijll dirhems 



V = 18^X 24-564 dirhems =lAi ^rifAl ^ 

2064 2064 



( 123 ) 

capital, equal to eight shares and three-fifths, plus 
two dirhems and seven twentieths. Complete your 
capital, by adding to that which you have fifty-nine one- 
hundred-and-eighty-one parts. Let, then, a share be 
three hundred and sixty two, and a dirhem likewise 
three hundred and sixty-two.* The whole capital is 
then five thousand two hundred and fifty-six, and the 
legacy out of one-fourtht is one thousand two hundred 
and four, and that out of one-third is four hundred and 
ninety-nine, and the one- eighth is six hundred and 
fifty-seven. 

On Completement. 

" A woman dies and leaves eight daughters, a mo- (93) 
ther, and her husband, and bequeaths to some per- 
son as much as must be added to the share of a 
daughter to make it equal to one-fifth of the capital ; 
and to another person as much as must be added to the 
share of the mother to make it equal to one-fourth of 



* The capital = 2^4u +411 S 
If we assume v -362, and J = 362, the capital =5256 
Then 07=724; ^ = 480; 2 = 499; J^h of capital =657. 

f The text ought to stand " the two first legacies are 
instead of " the legacy out of one-fourth is." 

The first legacy is 724 

, The second 480 



the first + second legacy =r 1204 



( 126 ) 

the capital."* Computation: Determine the parts of 
the residue, which in the present instance are thir- 
teen. Take the capital, and subtract from it one-fifth 
of the same, less one part, as the share of a daugh- 
ter : this being the first legacy. Then subtract also 
one- fourth, less two parts, as the share of the mother : 
this being the second legacy. There remain eleven - 
twentieths of the capital, which, when increased by 
three parts, are equal to thirteen parts. Remove now 
from thirteen parts the three parts on account of the 
three parts (on the other side), and you retain eleven- 
twentieths of the capital, equal to ten parts. Complete 
the capital, by adding to the ten parts as much as nine- 
elevenths of the same ; then you find the capital equal 
to eighteen parts and two- elevenths. Assume now 
each part to be eleven ; then the whole capital is two 
hundred, each part is eleven ; the first legacy will be 
twenty-nine, and the second twenty-eight. 

If the case is the same, and she bequeaths to 

some person as much as must be added to the share 

(94) of the husband to make it equal to one-third, and to 

another person as much as must be added to the share 

of the mother to make it equal to one-fourth ; and to a 



* In this case, the modier has -fj ; and each daughter has 
J^ of the residue. 

i.e. i_i.-fv— i+2u = i3u 



( 127 ) 

third as much as must be added to the share of a 
daughter to make it equal to one-fifth ; all these lega- 
cies being imposed on the heirs generally : then you 
divide the residue into thirteen parts.* Take the 
capital, and subtract from it one-third, less three parts, 
being the share of the husband ; and one-fourth, less 
two parts, being the share of the mother ; and lastly, 
one-fifth less one part, being the share of a daughter. 
The remainder is thirteen-sixtieths of the capital, which, 
when increased by six parts, is equal to thirteen parts. 
Subtract the six from the thirteen parts: there re- 
main thirteen-sixtieths of the capital, equal to seven 
parts. Complete your capital by multiplying the seven 
parts by four and eight-thirteenths, and you have a 
capital equal to thirty-two parts and four- thirteenths. 
Assuming then each part to be thirteen, the whole 
capital is four hundred and twenty. 

If the case is the same, and she bequeaths to some 
person as much as must be added to the share of the 
mother to make it one-fourth of the capital; and to 
another as much as must be added to the portion of a 
daughter, to make it one-fifth of what remains of the 
capital, after the deduction of the first legacy; then 



* i-lJ-3^]-[i-2u]-[i-^]=i3v 
i.e. i-A_i-i = 7v 



( 1S8 ) 

you constitute the parts of the residue by taking them 
out of thirteen.* Take the capital, and subtract from 
it one-fourth less two parts; and again, subtract one- 
fifth of what you retain of the capital, less one part; 
then look how much remains of the capital after the 
deduction of the parts. This remainder, namely, three- 
fifths of the capital, when increased by two parts and 
three-fifths, will be equal to thirteen parts. Subtract 
two parts and three-fifths from thirteen parts, there 
remain ten parts and two- fifths, equal to three-fifths of 
capital. Complete the capital, by adding to the parts 
which you have, as much as two-thirds of the same. 
Then you have a capital equal to seventeen parts and 
one-third. Assume a part to be three, then the capital 
is fifty-two, each part three ; the first legacy will be 
seven, and the second six. 

If the case is the same, and she bequeaths to some 
person as much as must be added to the share of the 
mother to make it one-fifth of the capital, and to ano- 
ther one-sixth of the remainder of the capital ; then 



* 1— ^— ^=i3u 



( 129 ) 

the parts are thirteen.* Take the capital, and subtract 
from it one-fifth less two parts; and again, subtract 
one-sixth of the remainder. You retain two-thirds of 
the capital, which, when increased by one part and 
two-thirds, are equal to thirteen parts. Subtract the 
one part and two-thirds from the thirteen parts : there 
remain two thirds of the capital, equal to eleven parts 
and one -third. Complete your capital, by adding to 
the parts as much as their moiety ; thus you find the 
capital equal to seventeen parts. Assume now the 
capital to be eighty-five, and each part five ; then the 
first legacy is seven, and the second thirteen, and the 
remaining sixty-five are for the heirs. 

If the case is the same, and she bequeaths to some 
person as much as must be added to the share of the 
mother, to make it one-third of the capital, less that 
sum which must be added to make the share of a 
daughter equal to one-fourth of what remains of the 
capital after the deduction of the above complement ; 
then the parts are thirteen.f Take the capital, and (96) 

* \—x—y= 13U 
07 = ^-217; y = i\\-x\ 

.-. J[# + 2U] =13U 

... | = ^t, ... ^ = ^; x = ^; y^y^ 

t i-a:-f?^=i3v; and^ = J— 2u; y=\\\—x\-v 

.-. 1— a:+i[i— ar]-u=i3u 

.-. I f 1 — a:] = 14U .-. f [J+ 2uJ = Hu 

... 5^2^3„ ... ^^^5^. x^y = ^^ 

s 



( 130 ) 

subtract from it one-third less two parts, and add to 
the remainder one-fourth (of such remainder) less one 
part ; then you have five-sixths of the capital and one 
part and a half, equal to thirteen parts. Subtract 
one part and a half from thirteen parts. There re- 
main eleven parts and a half, equal to five-sixths of 
the capital. Complete the capital, by adding to the 
parts as much as one-fifth of them. Thus you find 
the capital equal to thirteen parts and four-fifths. 
Assume, now, a part to be five, then the capital is 
sixty-nine, and the legacy four. 

" A man dies, and leaves a son and five daughters, 
and bequeaths to some person as much as must be 
added to the share of the son to complete one-fifth 
and one-sixth, less one-fourth of what remains of one- 
third after the subtraction of the complement."* Take 
one-third of the capital, and subtract from it one-fifth 
and one-sixth of the capital, less two (seventh) parts ; 
so that you retain two parts less four one hundred and 
twentieths of the capital. Then add it to the excep- 
tion, which is half a part less one one hundred and 



* Since there are five daughters and one son, each 
daughter receives i, and the son f of the residue. 

••• l-HffeH-H-^7^ 



( 131 ) 

twentieth, and you have two parts and a half less five 
one hundred and twentieths of capital. Add hereto 
two-thirds of the capital, and you have seventy-five 
one hundred and twentieths of the capital and two 
parts and a half, equal to seven parts. Subtract, now, 
two parts and a half from seven, and you retain seventy- 
five one hundred and twentieths, or five-eighths, equal 
to four parts and a half. Complete your capital, by (9T) 
adding to the parts as much as three-fifths of the same, 
and you find the capital equal to seven parts and one- 
fifth part. Let each part be five ; the capital is then 
thirty-six, each portion five, and the legacy one. 

If he leaves his mother, his wife, and four sisters, 
and bequeaths to a person as much as must be added to 
the shares of the wife and a sister, in order to make them 
equal to the moiety of the capital, less two-sevenths of 
the sum which remains from one-third after the deduc- 
tion of that complement; the Computation is this :* If 



* From the context it appears, that when the heirs of the 
residue are a mother, a wife, and 4. sisters, the residue is to 
be divided into 13 parts, of which the wife and one sister, 
together, take 5 : therefore the mother and 3 sisters, toge- 
ther, take 8 parts. Each sister, therefore, must take not 
less than -^^, nor more than -f^. In the case stated at page 
102, a sister was made to inherit as much as a wife ; in the 
present case that is not possible ; but the widow must take 
not less than ^-^ ; and each sister not more than -fj. Proba- 
bly, in this case, the mother is supposed to inherit ^-j ; the 
wife y\ ; each sister ^■^, 



( 13a ) 

you take the moiety from one-third, there remains one- 
sixth. This is the sum excepted. It is the share of the 
wife and the sister. Let it be five (thirteenth) parts. 
What remains of the one-third is five parts less one- 
sixth of the capital. The two-sevenths which he has 
excepted are two-sevenths of five parts less two- 
sevenths of one-sixth of the capital. Then you have 
six parts and three-sevenths, less one-sixth and two- 
sevenths of one- sixth of the capital. Add hereto 
two-thirds of the capital; then you have nineteen 
forty-seconds of the capital and six parts and three- 
sevenths, equal to thirteen parts. Subtract herefrom 
the six parts and three-sevenths. There remain nine- 
teen forty-seconds of the capital, equal to six parts 
and four- sevenths. Complete your capital by adding 
to it its double and four-nineteenths of it. Then you 
find the capital equal to fourteen parts, and seventy 
(98) one hundred and thirty-thirds of a part. Assume one 
part to be one hundred and thirty-three; then the 
whole capital is one thousand nine hundred and thirty- 



.-. ^=^V and the residue -^^ 
The author unnecessarily takes 7x276=1932 for the 
-common denominator. 



( 133 ) 

two; each part is one hundred and thirty-three, the 
completion of it is three hundred and one, and the 
exception of one-third is ninety-eight, so that the re- 
maining legacy is two hundred and three. For the 
heirs remain one thousand seven hundred and twenty- 
nine. 



COMPUTATION OF RETURNS.* 

On Marriage in Illness, 

" A man, in his last illness, marries a wife, paying 
(^ marriage settlement of) one hundred dirhems, 
besides which he has no property, her dowry being 

* The solutions which the author has given of the remain- 
ing problems of this treatise, are, mathematically consider- 
ed, for the most part incorrect. It is not that the problems, 
when once reduced into equations, are incorrectly worked 
out ; but that in reducing them to equations, arbitrary as- 
sumptions are made, which are foreign or contradictory to 
the data first enounced, for the purpose, it should seem, of 
forcing the solutions to accord with the established rules of 
inheritance, as expounded by Arabian lawyers. 

The object of the lawyers in their interpretations, and of 
the author in his solutions, seems to have been, to favour 
heirs and next of kin ; by limiting the power of a testator, 
during illness, to bequeath property, or to emancipate slaves; 
and by requiring payment of heavy ransom for slaves whom 
a testator might, during illness, have directed to be eman- 
cipated. 



( 134 ) 

ten dirhems. Then the wife dies, bequeathing one- 
third of her property. After this the husband dies."* 
Computation : You take from the one hundred that 
which belongs entirely to her, on account of the 
dowry, namely, ten dirhems ; there remain ninety dir- 
hems, out of which she has bequeathed a legacy. Call 
the sum given to her (by her husband, exclusive of her 
dowry) thing; subtracting it, there remain ninety 
dirhems less thing. Ten dirhems and thing are al- 
ready in her hands; she has disposed of one -third of 
her property, which is three dirhems and one-third, 
and one-third of thing ; there remain six dirhems and 



* Let .9 be the sum, including the dowry, paid by the 
man, as a marriage settlement ; d the dowry ; x the gift to 
the wife, which she is empowered to bequeath if she pleases. 

She may bequeath, if she pleases, d-{-x\ she actually 
does bequath ^ [^+d;] ; the residue is f [o^+o;], of which 
one half, viz. i [^+j^] goes to her heirs, and the other half 
reverts to the husband 

.*. the husband's heirs have s — [^ + :r] + i [c? -f x] or 
«— f [ff+x] ; and since what the wife has disposed of, exclu- 
sive of the dowry, is x, twice which sum the husband is to 
receiwe, S'-^[d-\-x\='2x .'. ^['^s -<2d\=x. But 5=100; 
flf=io .•.07=35; </+a; = 45; ^[g?+o;]=15. Therefore 
the legacy which she bequeaths is 15, her husband receives 
15, and her other heirs, 15. The husband's heirs receive 

2X = 70. 

But had the husband also bequeathed a legacy, then, as 
we shall see presently, the law would have defeated, in part, 
the woman's intentions. 



( 135 ) 

two-thirds plus two-thirds of thing, the moiety of 
which, namely, three dirhems and one-third plus one- 
third of thing, returns as his portion to the husband.* 
Thus the heirs of the husband obtain (as his share) 
ninety-three dirhems and one-third, less two- thirds of 
thing ; and this is twice as much as the sum given to (99) 
the woman, which was thing, since the woman had 
power to bequeath one-third of all which the husband 
left;t and twice as much as the gift to her is two 
things. Remove now the ninety-three and one-third, 
from two-thirds of thing, and add these to the two 
things. Then you have ninety-three dirhems and one- 
third equal to two things and two- thirds. One thing is 
three-eighths of it, namely, as much as three-eighths 
of the ninety-three and one- third, that is, thirty- five 
dirhems. 

If the question is the same, with this exception only, 
that the wife has ten dirhems of debts, and that she 
bequeaths one-third of her capital ; then the Computar 



* In other cases, as appears from pages 92 and 93, a 
husband inherits one-fourth of the residue of his wife's es- 
tate, after deducting the legacies which she may have 
bequeathed. But in this instance he inherits half the re- 
sidue. If she die in debt, the debt is first to be deducted 
from her property, at least to the extent of her dowry (see 
the next problem.) 

f When the husband makes a bequest to a stranger, the 
third is reduced to one-sixth. Vide p. 137. 



( 136 ) 

tion is as follows :^ Give to the wife the ten dirhems of 
her dowry, so that there remain ninety dirhems, out of 
which she bequeaths a legacy. Call the gift to her 
thing ; there remain ninety less thing. At the disposal 
of the woman is therefore ten plus thing. From this 
her debts must be subtracted, which are ten dirhems. 
She retains then only thing. Of this she bequeaths 
one-third, namely, one-third of thing : there remains 
two-thirds of thing. Of this the husband receives by 
inheritance the moiety, namely, one-third of thing. 
The heirs of the husband obtain, therefore, ninety 
dirhems, less two-thirds of thing ; and this is twice as 
much as the gift to her, which was thing ; that is, two 
things. Reduce this, by removing the two-thirds of 
thing from ninety, and adding them to two things. 
Then you have ninety dirhems, equal to two things 
and two- thirds. One thing is three-eighths of this; 
that is to say, thirty-three dirhems and three-fourths, 
which is the gift (to the wife). 

If he has married her, paying (a marriage settle- 



* The same things being assumed as in the last example, 
s - [fZ-fx] remains with the husband ; d goes to pay the 
debts of the wife ; and | reverts from the wife to the hus- 
band. 

.'. s — d-^x-2x .'. ^[s-'d]—x 
.-. if 5= 100, and d= lo, x='33| ; she bequeaths iij; iij 
reverts to her husband; and her other heirs receive iii. 
The husband's heirs receive 2x = 67^. 



( 137 ) 

ment of one hundred dirhems, her dowry being ten (100) 
dirhems, and he bequeaths to some person one-third ot 
his property; then the computation is this:^ Pay to 
the woman her dowry, that is, ten dirhems ; there re- 
main ninety dirhems. Herefrom pay the gift to her, 
thing; then pay likewise to the legatee who is to 
receive one-third, thing : for the one-third is divided 

* This case is distinguished from that in page 133 by 
two circumstances ; first, that the woman does not make 
any bequest ; second, that the husband bequeaths one-third 
of his property. 

Suppose the husband not to make any bequest. Then, 
since the woman had at her disposal d-^-x, but did not make 
any bequest, ^ [c?+x] reverts to her husband ; and the like 
amount goes to her other heirs. 

.'. s -[d-\-x]-\-l[d-^a:]=2x .*. x = -^[2s — d] 

and since 5= 100, and rf= 10 ; a;=38; d-\-x = 4.S; 
^ [d^x] = 24 reverts to the husband, and the like sum goes 
to her other heirs ; and 2x^'j6, belongs to the husband's 
heirs. 

Now suppose the husband to bequeath one-third of his 
property. The law here interferes with the testator's right 
of bequeathing ; and provides that whatever sum is at the 
disposal of the wife, the same sum shall be at the disposal 
of the husband ; and that the sum to be retained by the 
husband's heirs shall be twice the sum which the husband 
and wife together may dispose of. 

.*. s — ^[d+x] — x=z4x 
... ^[2s-d]=x; if 5 = 100, andfl?=lo ; a: = iy\o = i7^; 
d'\-x=2'j^^ ; I [d'\-x]z=i^-^j reverts to the husband, and 
the like sum goes to the other heirs of the woman ; 1 7^^ is 
what the husband bequeaths ; and 69-3I3- = 4x goes to the 
husband's heirs. 

T 



( 138 ) 

into two moieties between them^ since the wife cannot 
take any thing, unless the husband takes the same. 
Therefore give, likewise, to the legatee who is to have 
one-third, thing. Then return to the heirs of the hus- 
band. His inheritance from the woman is five dirhems 
and half a thing. There remains for the heirs of the 
husband ninety-five less one thing and a half, which 
is equal to four things. Reduce this, by removing one 
thing and a half, and adding it to the four things. 
There remain ninety-five, equal to five things and a 
half. Make them all moieties ; there will be eleven 
moieties; and one thing will be equal to seventeen 
dirhems and three-elevenths, and this will be the 
legacy. 

" A man has married a wife paying (a marriage set- 
tlement of) one hundred dirhems, her dowry being ten 
dirhems; and she dies before him, leaving ten dirhems, 
and bequeathing one-third of her capital; afterwards 
the husband dies, leaving one hundred and twenty dir- 
hems, and bequeathing to some person one-third of his 
capital/' Computation r* Give to the wife her dowry, 



* Let c be the property which the wife leaves, besides d 
the dowry, and x the gift from the husband. She bequeaths 
J [c + G?+a;]; J [c-|-rf-{-^] goes to her husband; and ^ 
[c-f fl?+a:] to her other heirs. The husband leaves property 
5, out of which must be paid the dowry, d', the gift to the 
wife, X ; and the bequest he makes to the stranger, x ; and 
his heirs receive from the wife's heirs \ [c-\'d-\-x] 



( 139 ) 

namely, ten dirhems; then one hundred and ten dirhems 
remain for the heirs of the husband. From these the (101) 
gift to the wife is thing, so that there remain one 
hundred and ten dirhems less thing; and the heirs 
of the woman obtain twenty dirhems plus thing. She 
bequeaths one- third of this, namely, six dirhems and 
two-thirds, and one-third of thing. The moiety of the 
residue, namely, six dirhems and two-thirds plus one- 
third of thing, returns to the heirs of the husband : so 
that one hundred and sixteen and two-thirds, less two- 
thirds of thing, come into their hands. He has be- 
queathed one-third of this, which is thing. There 
remain, therefore, one hundred and sixteen dirhems 
and two-thirds less one thing and two-thirds, and this 
is twice as much as the husband's gift to the wife 
added to his legacy to the stranger, namely, four 
things. Reduce this, and you find one hundred and 
sixteen dirhems and two-thirds, equal to five things 
and two-thirds. Consequently one thing is equal to 



s —d—2x+^[c-{-d+x]= 4x, according to the law of inhe- 
ritance. 

.% ss + c-2d=i^x, and 3: = 31+^-2^ 

' 1 7 
If 5=120, ci=:io, and 0?= 10, x = ^^z=20\^ 

c+rf-f a; = 40i:?; J [c-^d+x] = i3t^t 
The wife bequeaths 1 3^7^ ; 1 3-ff go to her husband, and 

13^9^ to her other heirs. 
The husband bequeaths to the stranger 2oif ; he gives the 

same sum to the wife ; and 4x= 82j\ go to his heirs. 



( 140 ) 

twenty dirhems and ten-seventeenths ; and this is the 
legacy. 

On Emancipation in Illness. 
" Suppose that a man on his death-bed were to eman- 
cipate two slaves ; the master himself leaving a son and 
a daughter. Then one of the two slaves dies, leaving 
a daughter and property to a greater amount than his 
price.^" You take two-thirds of his price, and what the 
other slave has to return (in order to complete his 
(102) ransom). If the slave die before the master, then the 
son and the daughter of the latter partake of the heri- 
tage, in such proportion, that the son receives as much 
as the two daughters together. But if the slave die 
after the master, then you take two-thirds of his value 
and what is returned by the other slave, and distribute 



* From the property of the slave, who dies, is to be de- 
ducted and paid to the master's heirs, first, two-thirds of 
the original cost of that slave, and secondly what is wanting 
to complete the ransom of the other slave. Call the amount 
of these two sums p ; and the property which the slave 
leaves «. 

Next, as to the residue of the slaves' property : 

First. If the slave dies before the master, the master's 
son takes J [^^-rf; the master's daughter :J- [<»-7?], and 
the slave's daughter J [««—/?] • 

Second. If the slave dies after the master ; the master's 
son is to receive f p, and the master's daughter ^p ; and then 
the master's son takes J [«—/>]» and the slave's daughter 



( 141 ) 

it between the son and the daughter (of the master), in 
such a manner, that the son receives twice as much as 
the daughter ; and what then remains (from the heri- 
tage of the slave) is for the son alone, exclusive of the 
daughter; for the moiety of the heritage of the slave 
descends to the daughter of the slave, and the other 
moiety, according to the law of succession, to the son 
of the master, and there is nothing for the daughter (of 
the master). 

It is the same, if a man on his death-bed emancipates 
a slave, besides whom he has no capital, and then the 
slave dies before his master. 

If a man in his illness emancipates a slave, besides 
whom he possesses nothing, then that slave must ran- 
som himself by two-thirds of his price. If the master has 
anticipated these two-thirds of his price and has spent 
them, then, upon the death of the master, the slave 
must pay two- thirds of what he retains.^ But if the 
master has anticipated from him his whole price and 
spent it, then there is no claim against the slave, since 
he has already paid his entire price. 

" Suppose that a man on his death-bed emancipates 
a slave, whose price is three hundred dirhems, not 
having any property besides ; then the slave dies, leav- 
ing three hundred dirhems and a daughter." The 



* The slave retains one-third of his price ; and this 4ie 
must redeem at two-thirds of its value ; namely at f x ^ = | 
of his original price. 



( 142 ) 

computation is this :* Call the legacy to the slave thing. 
He has to return the remainder of his price, after the 
deduction of the legacy, or three hundred less thing. 
This ransom, of three hundred less thing, belongs to 
the master. Now the slave dies, and leaves thing and a 
(103) daughter. She must receive the moiety of this, namely, 
one half of thing ; and the master receives as much. 
Therefore the heirs of the master receive three hundred 
less half a thing, and this is twice as much as the le- 
gacy, which is thing, namely, two things. Reduce this 
by removing half a thing from the three hundred, and 
adding it to the two things. Then you have three 
hundred, equal to two things and a half. One thing 
is, therefore, as much as two-fifths of three hundred, 



* Let the slave's original cost be a ; the property which 
he dies possessed of, cc ; what the master bequeaths to the 
slave, in emancipating him, x. Then the net property 
which the slave dies possessed of is cc+x — a, \[ot-\-x — a\ 
belongs, by law, to the master; and ^[a+a:-a] to the 
slave's daughter. The master's heirs, therefore, receive the 
ransom, a — a;, and the inheritance, J [ct+or— a]; that is, 
J [i«-|-fl— a;]; and on the same principle as the slave, when 
emancipated, is allowed to ransom himself at two-thirds of 
his cost, the law of the case is that 2 are to be taken, 
where 1 is given. 

.*. ^Icc-^-a—x^^^ix .*. 0; = ;^ [eft-i. a] 
The daughter's share of the inheritance = J[3it--2a] 
The master's heirs receive |. [^-j- a\ 

If, as in the example, a, = a, j; = fa; the daughter's 
share = \a ; the heirs of the master receive fa. 



( 143 ) 

namely, one hundred and twenty. This is the legacy 
(to the slave,) and the ransom is one hundred and eighty. 
" Some person on his sick-bed has emancipated a 
slave, whose price is three hundred dirhems; the slave 
then dies, leaving four hundred dirhems and ten dir- 
hems of debt, and two daughters, and bequeathing to 
a person one -third of his capital ; the master has twenty 
dirhems debts." The computation of this case is the 
following:* Call the legacy to the slave thing; his ran- 
som is the remainder of his price, namely, three hun- 
dred less thing. But the slave, when dying, left four 
hundred dirhems; and out of this sum, his ransom, 
namely, three hundred less thing, is paid to the 



* Let the slave's original cost=a; the property he dies 
possessed of— « ; the debt he owes^g 

He leaves two daughters, and bequeaths to a stranger one- 
third of his capital. 

The master owes debts to the amount ^it; where a -300; 
c6 = 400; g = io; f^=20. 

Let what the master gives to the slave, in emancipating 
him =x. 

Slave's ransom = a— a:; slave's property— slave's ransom = 

eC'{-X — a 

Slave's property — ransom— debt =fl6-|-a;— a— g 
Legacy to stranger z=^[c6-\-x — a — g] 

Residue =f [x+x — a — g] 

The master, and each daughter, are, by law, severally 

entitled to^x^{x-\-x-a — i] 
The master's heirs receive altogether a— a; -\-^[x-\-x-a^i] 
or ^g\a—x]+% [oi—i], which, on the principle that 2 



( 144 ) 

master, so that one hundred dirhems and thing re- 
main in the hands of the slave's heirs. Herefrom are 
(first) subtracted the debts, namely, ten dirhems; 
there remain then ninety dirhems and thing. Of 
this he has bequeathed one- third, that is, thirty dir- 
hems and one- third of thing; so that there remain for 
the heirs sixty dirhems and two-thirds of thing. Of 
this the two daughters receive two- thirds, namely, 
forty dirhems and four-ninths of thing, and the master 
(104) receives twenty dirhems and two-ninths of thing, so 
that the heirs of the master obtain three hundred and 
twenty dirhems Iqss seven-ninths of thing. Of this the 
debts of the master must be deducted, namely, twenty 
dirhems; there remain then three hundred dirhems less 



are to be taken for i given, ought to be made equal 
to 2X, 
But the author directs that the equation for determining x be 

.-. .r = J3- [7a+2 [cc—i]-gu] =108 

Hence the slave receives, the debts which he owes, g =10 
+ the legacy to the stranger —-^-g[Q[u—i]--6a—Sf^]= 66 
+ the inheritance of 1 st daughter = ^Vl^[^~ ^1 — 4 a — 2^] = 44 
+ theinheritanceof 2ddaughter= J3.[6[«;— e] — 4a — 2^]= 44 

Total r; 2V[2 1 ^ + 4s— 1 4«— 7^14] = 1 64 

And the master takes ^-\-2x=-^[4.cc—4.i+i4.a—jfA'j = 2^6 
Had the slave died possessed of no property whatever, his 

ransom would have been 200. 
His ransom, here stated, exclusive of the sum which the 

master inherits from him, or «— x, — 192. 



( 11^ ) 

seven-ninths of thing ; and this sum is twice as much as 
the legacy of the slave, which was thing; or, it is equal 
to two things. Reduce this, by removing the seven- 
ninths of thing, and adding them to two things ; there 
remain three hundred, equal to two things and seven- 
ninths. One thing is as much as nine twenty-fifths of 
eight hundred, which is one hundred and eight ; and 
so much is the legacy to the slave. 

If, on his sick-bed, he emancipates two slaves, besides 
whom he has no property, the price of each of them 
being three hundred dirhems ; the master having anti- 
cipated and spent two-thirds of the price of one of 
them before he dies;^ then only one-third of the price 



* Were there the first slave only, who has paid off two- 
thirds of his original cost, the master having spent the 
money, that slave would have to complete his ransom by 
paying two-ninths of his original cost, that is 66^ (see page 
141). 

Were there the second slave only, who has paid off none 
of his original cost, he would have to ransom himself at two- 
thirds of his cost; that is by paying 200 (see also page 141). 

The master's heirs, in the case described in the text, are 
entitled to receive the same amount from the two slaves 
jointly, viz. o.SG^, as they would be entitled to receive, 
according to the rule of page 141, from the two slaves, sepa- 
rately ; but the payment of the sum is differently distributed; 
the slave who has paid two-thirds of his ransom being required 
to pay one- ninth only of his original cost; and the slave 
who has paid no ransom, being required to pay two-thirds of 
his own cost, and one-ninth of the cost of the first slave. 



( 146 ) 

of this slave, who has already paid off a part of his 
ransom, belongs to the master ; and thus the master's 
capital is the entire price of the one who has paid off 
nothing of his ransom, and one- third of the price of 
the other who has paid part of it ; the latter is one 
hundred dirhems ; the other three hundred dir- 
hems: one-third of the amount, namely, one hun- 
dred and thirty-three dirhems and one third, is divided 
into two moieties among them ; so that each of them 
receives sixty-six dirhems and two-thirds. The first 
slave, who has already paid two-thirds of his ran- 
som, pays thirty-three dirhems and one-third; for 
(\0^) sixty-six dirhems and two- thirds out of the hundred 
belong to himself as a legacy, and what remains of 
the hundred he must return. The second slave has 
to return two hundred and thirty-three dirhems and 
one-third. 

" Suppose that a man, in his illness, emancipates two 
slaves, the price of one of them being three hundred 
dirhems, and that of the other five hundred dirhems ; 
the one for three hundred dirhems dies, leaving a 
daughter; then the master dies, leaving a daughter 
likewise ; and the slave leaves property to the amount 
of four hundred dirhems. With how much must every 
one ransom himself?"* The computation is this: Call 



* Let A. be the first slave ; his original cost a ; the pro- 
perty he dies possessed of u ; and let B. be the second slave ; 
and his cost b. 



( 147 ) 

the legacy to the first slave, whose price is three hun- 
dred dirhems, thing. His ransom is three hundred 
dirhems less thing. The legacy to the second slave of 
a price of five hundred dirhems is one thing and two- 
thirds, and his ransom five hundred dirhems less one 
thing and two- thirds {viz. his price being one and two- 
thirds times the price of the first slave, whose ransom 
was thing, he must pay one thing and two-thirds for 

Let X be that which the master gives to A. in emanci- 
pating him. 

A.'s ransom is a—x; and his property, minus his ransom, 
is u— a-{-x, 

A.'s daughter receives J [x—a+x], and the master's heirs 
receive I [x—a-{-x] 

Hence the master receives altogether from A., 
a—xi-^ [*— «+^] = 5 [x + a^x.] 

B.'s ransom is b > — x 

a 

The master's heirs receive from A. and B. together 
^[cf{-a + 2b] — — [a + 2h]x; and this is to be made equal 
to twice the amount of the legacies to A. and B., that is, 

i[x^a + 2b]—^/^a + 2b]x=2—-x 

The master's heirs receive from A., — ^^-^q^' — =293^ 
A.'s daughter receives [a+ij ^—i" =8oox 4^3%^o = io6§ 
The legacy to B. is b ^ Z,, =i88§; his ransom is 

, 4a 4-46 - a 



The master's heirs receive from A. and B. together 

5a+66 ^^ 



^[« + *]"S-'-604- 



I 



( 14.8 ) 

his ransom). Now the slave for three hundred dirhems 
dies, and leaves four hundred dirhems. Out of this 
his ransom is paid, namely, three hundred dirhems 
less thing ; and in the hands of his heirs remain one 
hundred dirhems plus thing: his daughter receives the 
moiety of this, namely, fifty dirhems and half a thing; 
and what remains belongs to the heirs of the master, 
namely, fifty dirhems and half a thing. This is 
added to the three hundred less thing ; the sum is 
three hundred and fifty less half a thing. Add 
thereto the ransom of the other, which is five hundred 
dirhems less one thing and two-thirds ; thus, the heirs 
(106) of the master have obtained eight hundred and fifty 
dirhems less two things and one-sixth ; and this is 
twice as much as the two legacies together, which were 
two things and two- thirds. Reduce this, and you have 
eight hundred and fifty dirhems, equal to seven things 
and a half Make the equation ; one thing will be 
equal to one hundred and thirteen dirhems and one- 
third. This is the legacy to the slave, whose price is 
three hundred dirhems. The legacy to the other slave 
is one and two- thirds times as much, namely, one 
hundred and eighty-eight dirhems and eight-ninths, 
and his ransom three hundred and eleven dirhems and 
one-ninth. 

" Suppose that a man in his illness emancipates two 
slaves, the price of each of whom is three hundred dir- 
hems ; then one of them dies, leaving five hundred 
dirhems and a daughter; the master having left a son." 



( 149 ) 

Computation :* Call the legacy to each of them thing; 
the ransom of each will be three hundred less thing; 
then take the inheritance of the deceased slave, which is 
^ve hundred dirhems, and subtract his ransom, which is 
three hundred less thing ; the remainder of his inhe- 
ritance will be two hundred plus thing. Of this, one 
hundred dirhems and half a thing return to the master 
by the law of succession, so that now altogether four 
hundred dirhems less a half thing are in the hands of 
the master's heirs. Take also the ransom of the other 
slave, namely, three hundred dirhems less thing; 
then the heirs of the master obtain seven hundred dir- 



* The first slave is A. ; his cost a ; his property » ; he 
leaves a daughter. 

The second slave is R. ; his cost b. 

Then (as in page 147) J [««-a+a:] goes to the daughter; 

«_|_„_|_26 

and x=a 



sceives [a+b] 

la [a-\-a.-\-h' ]-\-Z "-h 



The daughter receives \a-\-b\ , ^-^ 



The master receives from A. ^a4-Qb 

and the master receives from A. and B. together 

2 [« + *] i^ 

But if 6 =0 x=^ [«ft+3a]=i27T\ 

The daughter receives ^ [3*^— 2a] = 163^ 

The master receives from A ^ [5^^ + 4a] =; 336^ 

The master receives from B ^ [8a— «] = 172^ 

The master receives from A and B. . . ^-^ [*+ 3a] — 509t4: 

If 5=0, 
The daughter receives ^ [3*— 2fl] 
The master | [«+«J, as in page 142. 



( 150 ) 

hems less one thing and a half, and this is twice as 
much as the sum of the two legacies of both, namely 
(107) two things, consequently as much as four things. Re- 
move from this the one thing and a half: you find 
seven hundred dirhems, equal to five things and a half. 
Make the equation. One thing will be one hundred 
and twenty-seven dirhems and three-elevenths. 

" Suppose that a man in his illness emancipate a 
slave, whose price is three hundred dirhems, but who 
has already paid off to his master two hundred dirhems, 
which the latter has spent ; then the slave dies before 
the death of the master, leaving a daughter and three 
hundred dirhems."* Computation : Take the property 
left by the slave, namely, the three hundred, and add 
thereto the two hundred, which the master has spent ; 
this together makes five hundred dirhems. Subtract 
from this the ransom, which is three hundred less thing 

* The slave A. dies before his master, and leaves a 
daughter. His cost is a, of which he has redeemed d, which 
the master has spent ; and he leaves property oc. 
Then the daughter receives . . \ [oc-\-d—a-\-x] 
The master receives altogether J [cc-\-a+a—x] 
The master's heirs receive. ... J [ci^d-\-a—x] 

And ^[u'-a + a—a:] = 2x .*. a: = ^[ci—a-{-a] 
Hence the daughter receives ^ [^ei-^-ii a— 2a]=z 140 

The master's heirs J [201— id + 2a] = 160 

The master receives, in toto, \ [2a + 3« + 2a] = 360 
If the slave had not advanced, or the master had not spent «, 
the daughter would have received ^[sct-\-^d-2a] = i8o 
and the master would have received J [2c6-f2^-f 2a] = 320. 



( 151 ) 

(since his legacy is thing) ; there remain two hundred 
dirhems plus thing. The daughter receives the moiety 
of this, namely, one hundred dirhems plus half a thing; 
the other moiety, according to the laws of inheritance, 
returns to the heirs of the master, being likewise one 
hundred dirhems and half a thing. Of the three hun- 
dred dirhems less thing there remain only one hundred 
dirhems less thing for the heirs of the master, since 
two hundred are spent already. After the deduction 
of these two hundred which are spent, there remain 
with the heirs two hundred dirhems less half thing, and 
this is equal to the legacy of the slave taken twice; or 
the moiety of it, one hundred less one-fourth of thing, 
is equal to the legacy of the slave, which is thing. Re- 
move from this the one-fourth of thing ; then you have 
one hundred dirhems, equal to one thing and one- 
fourth. One thing is four-fifths of it, namely, eighty 
dirhems. This is the legacy ; and the ransom is two 
hundred and twenty dirhems. Add the inheritance of the 
slave, which is three hundred, to two hundred, which (108) 
are spent by the master. The sum is five hundred 
dirhems. The master has received the ransom of two 
hundred and twenty dirhems ; and the moiety of the 
remaining two hundred and eighty, namely, one hun- 
dred and forty, is for the daughter. Take these from 
the inheritance of the slave, which is three hundred ; 
there remain for the heirs one hundred and sixty dir- 
hems, and this is twice as much as the legacy of the 
slave, which was thing. 



( 152 ) 

" Suppose that a man in his illness emancipates a 
slave, whose price is three hundred dirhems, but who 
has already advanced to the master five hundred dir- 
hems; then the slave dies before the death of his mas- 
ter, and leaves one thousand dirhems and a daughter. 
The master has two hundred dirhems debts."* Com- 
putation : Take the inheritance of the slave, which is 
one thousand dirhems, and the five hundred, which the 
master has spent. The ransom from this is three hun- 
dred less thing. There remain therefore twelve hundred 
phis thing. The moiety of this belongs to the daughter : 
it is six hundred dirhems plus half a thing. Subtract 
it from the property left by the slave, which was one 



* A.'s price is a ; he has advanced to his master a ; he 
leaves property u. He dies before his master, and leaves a 
daughter. 

The master's debts are ^ ; a: is vi'hat A. receives, in being 
emancipated ; a — x is the ransom ; J ^ci-]-d — a-\-a:] is what 
the daughter receives. 

Then et—^[u-^d—a-\-x] is what remains to the master ; 
and «— i [cc-l-d—a-^x]^^ is what remains to him, after 
paying his debts ; and this is to be made equal to 2x, 

Whence x=^[u'i-a—d—2fA] 
Hence the daughter receives ^[3«:d— 2a-|-2a— ^] = 640 



The mother receives, 1 i r . ^ . i /- 

inclusive of the debt) i[2«+2a-2a+f.] = 36o 

The master receives, 1 i r . r ^ t /:• 

'exclusive of the debt I i[2u + [2a-2a-4^] = ibo 

If the mode given in page 142 had been followed, it 
would have given x — l[ci-{-a + a— 2ft,] 
and the daughter's portion-i [3^— 2a + 3rt- it4]-740. 



( 153 ) 

thousand dirhems: there remain four hundred dirhenis 
less half thing. Subtract herefrom the debts of the 
master, namely, two hundred dirhems ; there remain 
two hundred dirhems less half thing, which are equal to 
the legacy taken twice, which is thing ; or equal to two 
things. Reduce this, by means of the half thing. Then 
you have two hundred dirhems, equal to two things 
and a half. Make the equation. You find one thing, 
equal to eighty dirhems ; this is the legacy. Add now 
the property left by the slave to the sum which he has (109) 
advanced to the master : this is fifteen hundred dir- 
hems. Subtract the ransom, which is two hundred 
and twenty dirhems ; there remain twelve hundred and 
eighty dirhems, of which the daughter receives the 
moiety, namely, six hundred and forty dirhems. Sub- 
tract this from the inheritance of the slave, which is 
one thousand dirhems : there remain three hundred 
and sixty dirhems. Subtract from this the debts of the 
master, namely, two hundred dirhems ; there remain 
then one hundred and sixty dirhems for the heirs of the 
master, and this is twice as much as the legacy of the 
slave, which was thing. 

" Suppose that a man on his sick-bed emancipates a 
slave, whose price is five hundred dirhems, but who has 
already paid off to him six hundred dirhems. The mas- 
ter has spent this sum, and has moreover three hun- 
dred dirhems of debts. Now the slave dies, leaving his 
mother and his master, and property to the amount of 
seventeen hundred and fifty dirhems, with two hundred 

X 



( 154 ) 

dirhems debts." Computation:* Take the property left 
by the slave, namely, seventeen hundred and fifty dir- 
hems, and add to it what he has advanced to the mas- 
ter, namely, six hundred dirhems; the sum is two 
thousand three hundred and fifty dirhems. Subtract 
from this the debts, which are two hundred dirhems, 
and the ransom, which is five hundred dirhems less 
thing, since the legacy is thing; there remain then 
sixteen hundred and fifty dirhems plus thing. The 
mother receives herefrom one-third, namely, five hun- 
dred and fifty plus one-third of thing. Subtract now 
this and the debts, which are two hundred dirhems, 
from the actual inheritance of the slave, which is 
seventeen hundred and fifty ; there remain one thou- 
(110) sand dirhems less one -third of thing. Subtract from 
this the debts of the master, namely, three hundred 



* A. dies before his master, and leaves a mother. His 
price was a; he has redeemed ^, which the master has 
spent. The property he leaves is «. He owes debts g. 
The master owes debts f^, 

^ [u-^a—a-\-x — 6] is the mother's, 
fit— ^[«d-f o — a-fj—g]-— 8 is the master's. 
a — ^ [ec -\- a ^ a + a: '- 1] — i — fA — 2x =:the master's, after 
paying his debts. 

Hence xz=.\ [2u + a — ^— 2g — 3jtt]=30O 

Mother's =zi\ [^u — 2a-\-2a—^i—^] — 650 

Master's, without ^t , ... —^ [4«-|-2fl— 2^— 4g— 6jei]z26oo 

Mother's, with ^ r=-iJ-[4<«+2a— 2^ — 4€+|i4] = 900 

A. receives, inclusive of i =z\ [3*— 2a-|-2^-|-4g— /t*] — 850. 



( 155 ) 

dirhems ; there remain seven hundred dirhems less 
one-third of thing. This is twice as much as the legacy 
of the slave, which is thing. Take the moiety : then 
three hundred and fifty less one-sixth of thing are 
equal to one thing. Reduce this, by means of the 
one-sixth of thing ; then you have three hundred and 
fifty, equal to one thing and one-sixth. One thing 
will then be equal to six-sevenths of the three hundred 
and fifty, namely, three hundred dirhems ; this is the 
legacy. Add now the property left by the slave to 
what the master has spent already; the sum is two 
thousand three hundred and fifty dirhems. Subtract 
herefrom the debts, namely, two hundred dirhems, and 
subtract also the ratisom, which is as much as the price 
of the slave less the legacy, that is, two hundred dir- 
hems; there remain nineteen hundred and fifty dirhems. 
The mother receives one- third of this, namely, six 
hundred and fifty dirhems. Subtract this and the 
debts, which are two hundred dirhems, from the pro- 
perty actually left by the slave, which was seventeen 
hundred and fifty dirhems; there remain nine hun- 
dred dirhems. Subtract from this the debts of the 
master, which are three hundred dirhems; there re- 
main six hundred dirhems, which is twice as much as 
the legacy. 

" Suppose that some one in his illness emancipates a 
slave, whose price is three hundred dirhems: then the 
slave dies, leaving a daughter and three hundred dir- 
hems ; then the daughter dies, leaving her husband and 



( 156 ) 

three hundred dirhems; then the master ^lies." Com- 
putation:^ Take the property left by the slave, which is 
three hundred dirhems, and subtract the ransom, which 
(111) is three hundred less thing ; there remains thing, one 
half of which belongs to the daughter, while the other 
half returns to the master. Add the portion of the 
daughter, which is half one thing, to her inheritance, 
which is three hundred ; the sum is three hundred dir- 
hems plus half a thing. The husband receives the moiety 
of this ; the other moiety returns to the master, namely 
one hundred and fifty dirhems plus one-fourth of thing. 
All that the master has received is therefore four hun- 
dred and fifty less one-fourth of thing ; and this is twice 
as much as the legacy ; or the moiet^ of it is as much as 



* A. is emancipated by his master, and then dies, leaving 
a daughter, who dies, leaving a husband. Then the master 
dies. 

A.'s prices a; his property a. What he receives from 
the master =^. 

The daughter's property = ^ 

A.'s ransom = a -a:. The daughter inherits J [cj— a + or], 
and ^ [« — a + o;] goes to the master. 

J P+2 ['*'~^+^]] goes to the daughter's husband 
and -J [^+i [^-«+a;]] to the master. 
Hence, according to the author, we are to make 
a— a; + i[«— a+a?]+iP+J[«~a+^]] = 2x 
.% a: — -^ [3ot-|-a -j- 2^] - 200 
Daughter's share =^[6e^— .40-}-^] zz 100 

Husband's =i [S'* - 2a + 5^] = 200 

Master's =:^ [2^ + 6a + 4^] - 400. 



( 15'r ) 

the legacy itself, namely, two hundred and twenty -five 
dirhems less one- eighth thing are equal to thing. Re- 
duce this by means of one-eighth of thing, which you 
add to thing; then you have two hundred and twenty- 
five dirhems, equal to one thing and one-eighth. Make 
the equation: one thing is as much as eight-ninths 
of two hundred and twenty-five, namely, two hundred 
dirhems. 

" Suppose that some one in his illness emancipates a 
slave, of the price of three hundred dirhems ; the slave 
dies, leaving five hundred dirhems and a daughter, and 
bequeathing one-third of his property ; then the daugh- 
ter dies, leaving her mother, and bequeathing one- 
third of her property, and leaving three hundred dir- 
hems." Computation:^ Subtract from the property left 

* A. is emancipated, and dies, leaving a daughter, and 
bequeathing one-third of his property to a stranger. 

The daughter dies, leaving a mother, and bequeathing 
one-third of her property to a stranger. 

A.*s price is a ; his property is a 

The daughter's property is ^. 

A.'s ransom is a—x\ u — a-^x is his property, clear of 
ransom. 

^ [tfs— a + a] goes to the stranger; and the like amount to 
A.'s daughter, and to the master. 

3 [3^+«— fl+^] is the property left by the daughter. 

^[35-j-<* — a+a:] is the bequest of the daughter to a 
stranger. 

f [2i^-{-u—a-\-x] is the residue, of which ^d, 
viz, ^\ [35-l-<«— a+j:] is the mother's, 
and 2T [3^ + * — « -}-^] is the master's ; 



( 158 ) 

by the slave his ransom, which is three hundred dir- 
hems less thing; there remain two hundred dirhems 
plus thing. He has bequeathed one-third of his pro- 
perty, that is, sixty-six dirhems and two-thirds plus 
one-third of thing. According to the law of succession, 
(112) sixty-six dirhems and two-thirds and one-third of thing 
belong to the master, and as much to the daughter. 
Add this to the property left by her, which is three 
hundred dirhems : the sum is three hundred and sixty- 
six dirhems and two-thirds and one-third of thing. 
She has bequeathed one-third of her property, that is, 
one hundred and twenty-two dirhems and two-ninths 
and one-ninth of thing ; and there remain two hundred 
and forty-four dirhems and four-ninths and two-ninths 
of thing. The mother receives one-third of this, 
namely, eighty- one dirhems and four-ninths and one- 
third of one-ninth of a dirhem plus two-thirds of one- 
ninth of thing. The remainder returns to the master ; 
it is a hundred and sixty- two dirhems and eight-ninths 
and two-thirds of one-ninth of a dirhem plus one-ninth 
and one-third of one-ninth of thing, as his share of the 
heritage. 

Hence, according to the author, we are to make 

a— a:+J [at— a-f j;]-{-^[3^4-<*-a-f-x] = 2a; 

Therefore ^=-h [i 3<* + 14« + 1 2^J = 2 I0y5_ 

The daughter's share. . --^^ [2701 — 18a 4-4^] = 136^4- 
The daughter's bequest = ^^ [gM — 6a-\- 24^] = 145^ 
The mother's share .... =/^ [3<«— 2a-f-8^] — 97^^ 
The master's =^2^ [i3ct-t-i4rt-hi25] = 420|^. 



( 159 ) 

Thus the master's heirs have obtained five hundred 
and twenty-nine dirhems and seventeen twenty-sevenths 
of a dirhem less four-ninths and one-third of one-ninth 
of thing ; and this is twice as much as the legacy, which 
is thing. Halve it : You have two hundred and sixty- 
four dirhems and twenty-two twenty-sevenths of a dir- 
hem, less seven twenty-sevenths of thing. Reduce it by (113) 
means of the seven twenty- sevenths which you add to 
the one thing. This gives one hundred and sixty-four 
dirhems and twenty-two twenty-sevenths, equal to one 
thing and seven twenty-sevenths of thing. Make the 
equation, and adjust it to one single thing, by sub- 
tracting from it as much as seven thirty-fourths of the 
same. Then one thing is equal to two hundred and 
ten dirhems and five-seventeenths; and this is the 
legacy. 

*' Suppose that a man in his illness emancipates a 
slave, whose price is one hundred dirhems, and makes 
to some one a present* of a slave-girl, whose price is 
five hundred dirhems, her dowry being one hundred 
dirhems, and the receiver cohabits with her." Abu 
Hanifah says : The emancipation is the more impor- 
tant act, and must first be attended to. 

Computation :* Take the price of the girl, which is 



* The price of the slave-girl being a ; and what she 
receives on being emancipated x, her ransom is a—x. 
If her dowry is ec, he that receives her, takes u, + x. 



( 160 ) 

five hundred dirhems ; and remember that the price of 
the slave is one hundred dirhems. Call the legacy of 
the donee thing. The emancipation of the slave, whose 
price is one hundred dirhems, has already taken place. 
He has bequeathed one thing to the donee. Add the 
dowry, which is one hundred dirhems less one-fifth 
thing. Then in the hands of the heirs are six hundred 
dirhems less one thing and one-fifth of thing. This is 
twice as much as one hundred dirhems and thing ; the 
moiety of it is equal to the legacy of the two, namely, 
three hundred less three-fifths of thing. Reduce this 
by removing the three-fifths of thing from three hun- 
dred, and add the same to one thing. This gives three 
hundred dirhems, equal to one thing and three-fifths and 
one hundred dirhems. Subtract now from three hun- 



Hence, according to the author, we arc to make 

a—x—Q. [_cc-\-x~\ ; whence x— ~ 

3 
And her ransom is §[« + «] 

But if a male slave be at the same time emancipated by 

the master, the donee must pay the ransom of that slave. 

If his price was b, b — x is his ransom. 

Hence, according to the author, we are to make the sum 

of the two ransoms, viz. a—X'{-b — j;=^2[«+x] 

.-. a+6-2^=[3+^] X ... x = a "-±i=|-^=i25 
The donee pays ransom, in respect of the slave-girl {a -x) - 375 
and he pays ransom for the male slave b—~ x — 75. 



( 161 ) 

dred the one hundred, on account of the other one 
hundred. There remain two hundred dirhems, equal 
to one thing and three-fifths. Make the equation with 
this. One thing will be five-eighths of what you have ; (114) 
take therefore five-eighths of two hundred. It is one 
hundred and twenty-five. This is thing; it is the 
legacy to the person to whom he had presented the 
girl. 

" Suppose that a man emancipates a slave of a price 
of one hundi-ed dirhems, and makes to some person a 
present of a slave girl of the price of five hundred dir- 
hems, her dowry being one hundred dirhems; the donee 
cohabits with her, and the donor bequeaths to some 
other person one-third of his property." According to 
the decision of Abu Hanifah, no more than one-third 
can be taken from the first owner of the slave-girl ; and 
this one-third is to be divided into two equal parts be- 
tween the legatee and the donee. Computation:^ Take 
the price of the girl, which is five hundred dirhems. 
The legacy out of this is thing; so that the heirs obtain 
five hundred dirhems less thing ; and the dbwTy is one 
hundred less one-fifth of thing; consequently they 



♦ The same notation being used as in the last example, 
the equation for determining or, according to the author, is 
to be 



7 * r T 

a—x-^0—- X — X— 2 [<:« + 2j:| 



( 162 ) 

obtain six hundred dirhems less one thing and one-fifth 
of thing. He bequeaths to some person one third of 
his capital, which is as much as the legacy of the person 
■who has received the girl, namely, thing. Conse- 
quently there remain for the heirs six hundred less two 
things and one-fifth, and this is twice as much as both 
their legacies taken together, namely, the price of the 
slave pluB the two things bequeathed as legacies. 
Halve it, and it will by itself be equal to these lega- 
cies : it is then three hundred less one and one-tenth 
of thing. Reduce this by means of the one and one- 
tenth of thing. Then you have three hundred, equal 
to three things and one-tenth, plus one hundred dir- 
hems. Remove one hundred on occount of (the 
opposite) one hundred; there remain two hundred, 
equal to three things and one-tenth. Make now the 
reduction. One thing will be as much as thirty-one 
(1 15) parts of the sum of dirhems which you have; and just 
so much will be the legacy out of the two hundred ; it 
is sixty-four dirhems and sixteen thirty-one parts. 

" Suppose that some one emancipates a slave girl of 
the price of one hundred dirhems, and makes to some 
person a present of a slave girl, which is five hundred 
dirhems worth ; the receiver cohabits with her, and her 
dowry is one hundred dirhems ; the donor bequeaths 
to some other person as much as one-fourth of his 
capital.'* Abu Hanifah says : The master of the girl 
cannot be required to give up more than one-third, and 
the legatee, who is to receive one-fourth, must give up 



( 163 ) 

one-fourth. Computation :* The price of the girl is 
five hundred dirhems. The legacy out of this is thing; 
there remain five hundred dirhems less thing. The 
dowry is one hundred dirhems less one-fifth of thing; 
thus the heirs obtain six hundred dirhems less one and 
one-fifth of thing. Subtract now the legacy of the 
person to whom one-fourth has been bequeathed, 
namely, three-fourths of thing; for if one-third is thing 
then one-fourth is as much as three-fourths of the same. 
There remain then six hundred dirhems less one 
thing and thirty-eight fortieths. This is equal to the 
legacy taken twice. The moiety of it is equal to the 
legacies by themselves, namely, three hundred dirhems 
less thirty-nine fortieths of thing. Reduce this by 
means of the latter fraction. Then you have three hun- (116) 
dred dirhems, equal to one hundred dirhems and two 
things and twenty-nine fortieths. Remove one hundred 
on account of the other one hundred. There remain 
two hundred dirhems, equal to two things and twenty- 
nine-fortieths. Make the equation. You will then 
find one thing to be equal to seventy-three dirhems 
and forty-three one-hundred-and-ninths dirhems. 



* The same notation being used as in the two former 
examples, the equation for determining x, according to the 
author, is 

a—x + b— x—^x =2{cc+i^x] 
Whence a: = ^-y^ [a-{.b^2cc]=r3jW 



( 164 ) 

On return of the Dowry, 

" A MAN, in the illness before his death, makes to 
some one a present of a slave girl, besides whom he 
has no property. Then he dies. The slave girl is 
worth three hundred dirhems, and her dowry is one 
hundred dirhems. The man to whom she has been 
presented, cohabits with her." Computation:* Call 
the legacy of the person to whom the girl is pre- 
sented, thing. Subtract this from the donation : there 
remain three hundred less thing. One-third of this 
difference returns to the donor on account of dowry 
(since the dowry is one-third of the price) : this is 
one hundred dirhems less one-third of thing. The 
donor's heirs obtain, therefore, four hundred less 
one and one- third of thing, which is equal to twice 
the legacy, which is thing, or to two things. Trans- 
pose the one and one-third thing from the four hun- 
dred, and add it to the two things ; then you have four 
hundred, equal to three things and one-third. One 
thing is, therefore, equal to three- tenths of it, or to one 
hundred and twenty dirhems, and this is the legacy. 



* Let a be the slave-girl's price — u, her dowry. 
Then, according to the author, we are to make 

a 

Therefore x-=- — — - [a -f- u] =-f^ x 400 r^ 1 20 
The donee is to receive the girl's dowry, worth 400, for 280. 



( 165 ) 

" Or, suppose that he, in his illness, has made a pre- 
sent of the slave girl, her price being three hundred, 
her dowry one hundred dirhems ; and the donor dies, 
after having cohabited with her." Computation :* Call 
the legacy thing : the remainder is three hundred less ^ 
thing. The donor having cohabited with her, the 
dowry remains with him, which is one- third of the 
legacy, since the dowry is one-third of the price, or one- 
third of thing. Thus the donor's heirs obtain three (117) 
hundred less one and one-third of thing, and this is 
twice as much as the legacy, which is thing, or equal 
to two things. Remove the one and one-third of 
thing, and add the same to the two things. Then you 
have three hundred, equal to three things and one- 
third. One thing is, therefore, three- tenths of it, 
namely ninety dirhems. This is the legacy. 

If the case be the same, and both the donor and 
donee have cohabited with her ; then the Computation 



* If the donor has cohabited with the slave-girl, the 
donor's heirs are to retain the dowry, but must allow the 

donee, in addition to the legacy a:, the further sum of - x ; 

The ransom is then a—x -- x, which according to the 
author is to be made equal to 2x. 

Whence x=: — — =qo 
The donee is to receive the girl, worth 300, for 210. 



( 166 ) 

is this:* Call the legacy thing; the deduction is three 
hundred dirhems less thing. The donor has ceded the 
dowry to the donee by (the donee's) having cohabited 
with her : this amounts to one-third of thing : and 
the donee cedes one-third of the deduction, which is 
one hundred less one-third of thing. Thus, the donor's 
heirs obtain four hundred less one and two-thirds of 
thing, which is twice as much as the legacy. Reduce 
this, by separating the one and two-thirds of thing 
from four hundred, and add them to the two things. 
Then you have four hundred things, equal to three 
things and two-thirds. One thing of these is three- 
elevenths of four hundred ; namely, one hundred and 

* If the donor has previously cohabited with the slave- 
girl, it appears from the last example, that the donee is 

entitled to ransom her for a—x — x. 

a 

If the donee cohabits with the slave-girl, it appears from 
the last example but one, that he is entitled to redeem the 

dowrv, <«, for a — - a? 

The redemption of the girl and dowry is 

a — X X-\-X X, 

which, according to the author, is to be made equal to 2x. 

rr>i • 0-1- 2a 

Ihat is a-{-c(, x=2x 

a 

Whence x= — ^ — xFa-f ^1 = 1094^ 
The donee is to receive the girl and dowry, worth 400, 
for 29915 . 



( 167 ) 

nine dirhems and one-eleventh. This is the legacy; 
The deduction is one hundred and ninety dirhems and 
ten-elevenths. According to Abu Hanifah, you call 
the thing a legacy, and what is obtained on account of 
the dowry is likewise a legacy. 

If the case be the same, but that the donor, having 
cohabited with her, has bequeathed one-third of his (118) 
capital, then Abu Hanifah says, that the one-third is 
halved between the donee and the legatee. Computa- 
tion : * Call the legacy of the person to whom the slave- 
girl has been given, thing. After the deduction of it, 
there remain three hundred, less thing. Then take the 
dowry, which is one- third of thing; so that the donor 
retains three hundred less one and one-third of thing : 
the donee's legacy being, according to Abu Hanifah, 
one and one- third of thing; according to other 
lawyers, only thing. The legatee, to whom one-third 
is bequeathed, receives as much as the legacy of the 
donee, namely, one and one-third of thing. The 
donor thus retains three hundred, less two things and 



* The second case is here solved in a different way. 

a ^ a ^ 

.*. X=i' 



This being halved between the legatee and donee becomes 
The donee receives the girl, worth 300, for 262|. 



( 168 ) 

two- thirds — equal to twice the two legacies, which 
are two things and two-thirds. The moiety of this, 
namely, one hundred and fifty less one and one- third 
of thing, must, therefore, be equal to the two legacies. 
Reduce it, by removing one and one- third of thing, 
and adding the same to the two legacies (things). 
Then you find one hundred and fifty, equal to four 
things. One thing is one-fourth of this, namely, 
thirty-seven and a half. 

If the case be, that both the receiver and the donor 
have cohabited with her, and the latter has disposed of 
one-third of his capital by way of legacy; then the 
computation,* according to Abu Hanifah, is, that you 
call the legacy thing. After the deduction of it, there 
remain three hundred less thing. Then the dowry 
is taken, which is one hundred less one-third of thing ; 
so that there are four hundred dirhems less one and 
one- third of thing. The sum returned from the dowry 
is one- third of thing; and the legatee, who is to receive 
one-third, obtains as much as the legacy of the first, 
namely, thing and one -third of thing. Thus there 



* According to the author's rule, which is purely arbi- 
trary, 

n "- a -• 

Whence x=za -—-, — —48 

The donee will have to redeem the girl and dowry, 
worth 400, for 352. 



{ lfi9 ) 
remain four hundred dirhems less three things, equal 
to twice the legacy, namely, two things and two-thirds. (119) 
Reduce this, by means of the three things, and you find 
four hundred, equal to eight things and one-third. 
Make the equation with this ; one thing will be forty- 
eight dirhems. 

" Suppose that a man on his sick-bed makes to ano- 
ther a present of a slave-girl, worth three hundred dir- 
hems, her dowry being one hundred dirhems; the 
donee cohabits with her, and afterwards, being also on 
his sick-bed, makes a present of her to the donor, 
and the latter cohabits with her. How much does he 
acquire by her, and how much is deducted?"* Com- 



* We have here the only instance in the treatise of a 
simple equation, involving two unknown quantities. For 
what the donee receives is one unknown quantity ; and what 
the donor receives back again from the donee, called by the 
author *' part of thing," is the other unknown quantity. 

Let what the donee receives = a;, and what the donor 
receives =:^. 

Then, retaining the same notation as before, according to 
the author, the donee receives, on the whole 

and the donor receives, on the whole 

Whence x=^\ ^^^^aa-^^ [3a'H3««-2«^] ^ 102 






z 



But 



( 170 ) 

putation : Take the price, which is three hundred dir- 
hems ; the legacy from this is thing ; there remain witli 
the donor's heirs three hundred less thing; and the 
donee obtains thing. Now the donee gives to the 
donor part of thing : consequently, there remains only 
thing less part of thing for the donee. He returns to the 
donor one hundred less one- third of thing ; but takes the 
dowry, which is one-third of thing, less one -third of 
part of thing. Thus he obtains one and two-thirds 
thing less one hundred dirhems and less one and one- 
third of part of thing. This is twice as much as part 
of thing ; and the moiety of it is as much as part of 
thing, namely, five-sixths of thing less fifty dirhems 
and less two- thirds of part of thing. Reduce this by 
removing two-thirds of part of thing and fifty dirhems. 
Then you have five-sixths of thing, equal to one and 
two-thirds of part of thing plus fifty dirhems. Reduce 
this to one single part of thing, in order to know^ what 
the amount of it is. You effect this by taking three-fifths 
(120) of what you have. Then one part of thing plus thirty 
dirhems is equal to half a thing ; and one-half thing 
less thirty dirhems is equal to part of thing, which is 
the legacy returning from the donee to the donor. 
Keep this in memory. 

Then return to what has remained with the donor ; 



But the reasons for reducing the question to these two 
equations are not given by the author, and seem to depend 
on the dicta of the sages of the Arabian law. 



( 171 ) 

this was three hundred less thing : hereto is now added 
the part of thing, or one-half thing less thirty dir- 
hems. Thus he obtains two hundred and seventy less 
half one thing. He further takes the dowry, which is 
one hundred dirhems less one-third thing, but has to 
return a dowry, which is one-third of what remains of 
thing after the subtraction of part of thing, namely, 
one-sixth of thing and ten dirhems. Thus he retains 
three hundred and sixty less thing, which is twice as 
much as thing and the dowry, which he has returned. 
Halve it : then one hundred and eighty less one-half 
thing are equal to thing and that dowry. Reduce this, 
by removing one-half thing and adding it to the thing 
and the dowry : you find one hundred and eighty dir- 
hems, equal to one thing and a half plus the dowry 
which he has returned, and which is one-sixth thing 
and ten dirhems. Remove these ten dirhems; there 
remain one hundred and seventy dirhems, equal to 
one and two- thirds things. Reduce this, in order 
to ascertain what the amount of one thing is, by taking 
three-fifths of what you have; you find that one hun- 
dred and two are equal to thing, which is the legacy 
from the donor to the donee: and the legacy from 
the donee to the donor is the moiety of this, less 
thirty dirhems, namely, twenty-one. 



( 172 ) 

On Surrender in Illness. 
(121) " Suppose that a man, on his sick-bed, deliver to 
some one thirty dirhems in a measure of victuals, worth 
ten dirhems; he afterwards dies in his illness ; then the 
receiver returns the measure and returns besides ten 
dirhems to the heirs of the deceased." Computation : 
He returns the measure, the value of which is ten dir- 
hems, and places to the account of the deceased twenty 
dirhems ; and the legacy out of the sum so placed is 
thing; thus the heirs obtain twenty less thing, and the 
measure. All this together is thirty dirhems less thing, 
equal to two things, or equal to twice the legacy. 
Reduce it by separating the thing from the thirty, and 
adding it to the two things. Then, thirty are equal to 
three things. Consequently, one thing must be one- 
third of it, namely, ten, and this is the sum which he 
obtains out of what he places to the account of the 
deceased. 

" Suppose that some one on his sick-bed delivere 
to a person twenty dirhems in a measure worth fifty 
dirhems ; he then repeals it while still on his sick bed, 
and dies after this. The receiver must, in this case, 
return four-ninths of the measure, and eleven dirhems 
and one-ninth."* Computation : You know that the 

* Let a be the gift of money ; and the value of the mea- 
sure m xa. 

It appears from the context that the donee is to pay the 
heirs f mff. 



( 173 ) 

price of the measure is two and a half times as much as 
the sum which the donor has given the donee in money; 
and whenever the donee returns anything from the 
money capital, he returns from the measure as much 
as two and a half times that amount. Take now from 
the measure as much as corresponds to one thing, that 
is, two things and a half, and add this to what remains 
from the twenty, namely, twenty less thing. Thus the 
heirs of the deceased obtain twenty dirhems and one (122) 
thing and a half. The moiety of this is the legacy, 
namely, ten dirhems and three-fourths of thing; and 
this is one-third of the capital, namely, sixteen dirhems 
and two-thirds. Remove now ten dirhems on account 
of the opposite ten ; there remain six dirhems and two- 
thirds, equal to three-fourths of thing. Complete the 
thing, by adding to it as much as one-third of the 
same: and add to the six dirhems and two-thirds 



It is arbitrary how he shall apportion this sum between the 
money capital and the measure. 

If he pays on the money capital p. a 

and on the measure . • q,ma 

we have the equation p. a-i-q. ma=^ ma 
or p -\-q m =f m 

The author assumes jp=—. q 

Whence y=-|, andjt?=f, and therefore the donee pays 
on the money capital. ... ^ a=ii^ 
and on the measure ^ ma =22^ 

Total 33*. 



( 1^4 ) 

likewise one-third of the same, namely, two dirhems 
and two-ninths; this yields eight dirhems and eight- 
ninths, equal to thing. Observe now how much the 
eight dirhems and eight-ninths are of the money 
capital, which is twenty dirhems. You will find them 
to be four-ninths of the same. Take now four- ninths 
of the measure and also five-ninths of twenty. The 
value of four-ninths of the measure is twenty-two dir- 
hems and two-ninths ; and the five-ninths of the twenty 
are eleven dirhems and one-ninth. Thus the heirs 
obtain thirty-three dirhems and one- third, which is as 
much as two-thirds of the fifty dirhems. — God is the 
Most Wise ! 



N O T 'E S, 



Page 1, line 2-5. 

The neglected state of the manuscript, in which most 
diacritical points are wanting, makes me very doubtful 
whether I have correctly understood the author's meaning 
in several passages of his preface. 

In the introductory lines, I have considered the words 

amplification of what might briefly have been expressed 
by l^lfcib j^iJt ' through the performance of which.'* I 
conceive the author to mean, that God has prescribed to 
man certain duties, ^ l^^ ^^lill ^^^ u^J^^ '^ *^' f^^ 
iX^Vs^l, and that by performing these (&c. ijoJ^\ to-tljb) 
we express our thankfulness (jx1j\ *«jI ?-^) &c. 

Since my translation was made, I have had the ad- 
vantage of consulting Mr. Shakespear about this pas- 
sage. He prefers to read «Ju , t.j,-c>-^:uJ , and ^^y in- 
stead of «Ji3 , K^^.^y^ , and ^y , and proposes to tran- 
slate as follows ; Praise to God for his favours in that 
which is proper for him from among his laudable deeds, 
which in the performance of what he has rendered indis- 



( 176 ) 

pensible from (or by reason of) thein on (the part of) 
whoever of his creatures worships him, gives the name of 
thanksgiving, and secures the increase, and preserves from 
deterioration." 

The construction here assumed is evidently easier than 
that adopted by myself, in as far as the relative pronoun ^-iJl 
representing irJw«l.sr* , is made the subject of the three sub- 
sequent verbs *_aj , &c., whilst my translation presumes a 
transition from the third person (as in wJbi yb u , and in ^^ 
irJux>) to the first (as in *JiJi &c.). 

A marginal note in the manuscript explains the words 
jt^^ ^J^ ^^y>3 by^l ^ ^Us^U ^^j3j ^^Aiu Jxl " The 
meaning may be : we preserve from change him who en- 
joys it," (viz, the divine bounty, taking <U>.U? for c-^%^-L> 
<dJl f^. The change here spoken of is the forfeiture 
of the divine mercy by bad actions ; for God does not 
change the mercy which he bestows on men, as long as they 
do not change that which is within themselves." J <dl\ ^^b 
j»g..,.,c»b U U;-*> 15*^" |*y L5^ V^^ ^^^^ ^;?*^ ^--^« {Cor any 
Sur, VIII. V. 55, ed. Hinck.). 

Page 1, line 7. 
J--j^l ^ ^^ ^j^ L5^ ^®® Coraw, Sur. v. v. 22. 
ed. Hinck. 

Page 1, line 14, 15. 
I am particularly doubtful whether I have correctly read 
and translated the words of the text from IjL.:;^-!^ to ^J>^y 
Instead oi j>^ \i\j:ij>-\ I should have preferred ljL.^1 



( ITT ) 

yi^^ benefitting others," if the verb ,^;*-.>"\ could be con- 
strued with the preposition J . 

Page 2 , line 1 . 
To the words J^f-j iJ^J * marginal note is given in the 
manuscript, which is too much mutilated to be here tran- 
scribed, but which mentions the names of several authors 
who first wrote on certain branches of science, and con- 
cludes with asserting, that the author of the present treatise 
was the first that ever composed a book on Algebra. 

Page 2, line 4. 
An interlinear note in the manuscript explains <^ix-j JJ 
by i3j:Ji^ f*^* 

Page 2, line 10. 

Mohammed gives no definition of the science which he 
intends to treat of, nor does he explain the words j^ Jebr^ 
and ^liU mokabalah^ by which he designates certain ope- 
rations peculiar to the solution of equations, and which, 
combined, he repeatedly employs as an expression for this 
entire branch of mathematics. As the former of these words 
has, under various shapes, been introduced into the several 
languages of Europe, and is now universally used as the 
designation of an important division of mathematical science, 
I shall here subjoin a few remarks on its original sense, and 
on its use in Arabic mathematical works. 

The \erh jf^jabar of which the substantive J->.^'e6r is 
derived, properly signifies to restore something broken, 



2 A 



i 



( ITS ) 

especially to cure a fractured bone. It is thus used in the 
following passage from Motanabbi (p. 143, 144, ed. 
Calculi,) 

iijd\ >^ \ 4>^ Jb i^\ ^j K Lj^\ \ 4«i aL—J J>jl! ^ I) 

^— jW l::^! Ulic jjj*4t€. ^^ V^ '^^^ ^^**^ C^^^^T^ ^ 
O thou on whoml rely in whatever I hope, with whom 
I seeic refuge from all that 1 dread ; whose bounteous hand 
seems to me like the sea, as thy gifts are like its pearls : pity 
the youthfulness of one, whose prime has been wasted by 
the hand of adversity, and whose bloom has been stifled in 
the prison. Men will not heal a bone which thou hast 
broken, nor will they break one which thou hast healed." 

Tfence the Spanish and Portuguese expression algebrista 
for a person who heals fractures, or sets right a dislocated 
limb. 

In mathematical language, the verb^^-^ means, to make 
perfect, or to complete any quantity that is incomplete or 
liable to a diminution ; i. e, when applied to equations, to 
transpose negative quantities to the opposite side by chang- 
ing their signs. The negative quantity thus removed is 
construed with the particle C-^ : thus, if a:^— 6=^23 shall be 
changed into:i:2=z29, the direction is IfcJJj d:uJu (jy^j^^ 
j^^yL^ltj d^\ ^ i. e. literally " Restore the square from 
(the deficiency occasioned to it by) the six, and add these 
to the twenty-three." 



( 179 ) 

The \evbj^ is not likewise used, when in an equation 
an integer is substituted for a fractional power of the un- 
known quantity : the proper expression for this is either 
the second or fourth conjugation of J-*j , or the second 
of "J . 

The word ^\ax mokabalah is a noun of action of the 
verb J-J to be in front of a thing, which in the third 
conjugation is used in a reciprocal sense of two objects 
being opposite one another or standing face to face; and 
in the transitive sense of putting two things face to face, of 
confronting or comparing two things with one another. 

In mathematical language it is employed to express 
the comparison between positive and negative terms in a 
compound quantity, and the reduction subsequent to such 
comparison. Thus loo+ioo:— ioa;4-ar2 is reduced to 100+ a:^ 
<U liblJi ^ Jju after we have made a comparison." 

When applied to equations, it signifies, to take away 
such quantities as are the same and equal on both sides. 
Thus the direction for reducing x^-\-x=x'^-\-4. to x = 4. will 
be expressed by JjvJJ . 

In either application the verb requires the preposition 
c--> before a pronoun implying the entire equation or com- 
pound quantity, within which the comparison and subse- 
quent reduction is to take place. 

The verb Jjli is not likewise used, when the reduction of 
an equation is to be performed by means of a division : the 
proper term for this operation being 3j , 



( 180 ) 

The mathematical application of the substantives 
and <UjliU will appear from the following extracts. 

1. A marginal note on one of the first leaves of the 
Oxford manuscript lays down the following distinction : 



[Uj] j^Ul^j-^:*- A^kc J:>>- ^-^L-^M Ij^ jltf Uli l^p-^^ 

la)! ^lill! CI-JjU Jjj ^^^ J] l^^ ^;-,lir»-iIl 

'' Jeftr is the restoration of anything defective by means 
of what is complete of another kind. Mokabalah^ a noun 
of action of the third conjugation, is the facing a thing : 
whence it is applied to one praying, who turns his face to- 
wards the kihlah. In this branch of calculation, the method 
commonly employed is the restoring of something defective 
in its deficiency, and the adding of an amount equal to this 
restoration to the other side, so as to make the completion 
(on the one side) and this addition (on the other side) to face 
(or to balance) one another. As this method is frequently 
resorted to, it has been named^e^r and mokabalah (or Res- 
toring and Balancing), since here every thing is made com- 
plete if it is deficient, and the opposite sides are made io 
balance one another Mathematicians also take 



( 181 ) 

the word mokdbalah in the sense of the removal of equal 
quantities (from both sides of an equation)." 

According to the first part of this gloss, in reducing 
^— 5«=:iOrt to .a:r=i5«, the substitution of x in place of 
x—^a would afford an instance of Jebr or restoration, and 
the corresponding addition of 5a to 10«, would be an 
example of /woArafta^aA or balancing. From the following 
extracts it will be seen, that mokdbalah is more generally 
taken in the sense stated last by the gloss. 

2. IIaji Khalfa, in his bibliographical work (MS. of the 
British Museum, fol. 167, recto^.) gives the following ex- 
planation: ^j\x\} (HLksA ^ ^JaA} to jji 'ij\)Jj^\ ^^-Jt^^ 

^\)\ \AU\ Zj}^\ ^j^^ :^jIcJ ^j^^i] ^^\ ^ s.\::^:^\j 
J jU::]J ^^^K y ^ kJ^^=>^ ^J^ Jebr is the adding to one 
side what is negative on the other side of an equation, 
owing to a subtraction, so as to equalize them. Mokdbalah 
is the removal of what is positive from either sum, so as 
to make them equal." 

A little farther on H aji Khalfa gives further illustration 
of this by an example : ^j\ J J^. li*-i» 'i\ 'ijts. UjJi ^ U^ 

JU ^\j}^^ ^laJUj^ ajl^ '^l^ ^^^i Jj^ ^^ ,^;>l^m 
bjLj^\j^ Sft,} Jl*4l ^ J^^\ is,^\jj ^Ac ^ ^.^^uJLl 
^^^oMJ i^\ iLbUlli X»-/4o- j^ ^S"^ ■^\i^^ ^J^ tj^ 



* This manuscript is apparently only an abridgement of Haji Khalfa's 
work. 



( 1H2 ) 

<Ui l^y^ ^^ <LljUi,[j ^^1 Jx " For instance if 

we say: Ten less one thing equal to four things;' then 
jebr is the removal of the subtraction, which is performed 
by adding to the minuend an amount equal to the sub- 
trahend: hereby the ten are made complete, that which 
was defective in them being restored. An amount equal 
to the subtrahend is then added to the other side of the 
equation : as in the above instance, after the ten have been 
made complete, one thing must be added to the four things, 
which thus become five things. Mokabalah consists in 
withdrawing the same amount from quantities of the same 
kind on both sides of the equation ; or as others say, it is 
the balancing of certain things against others, so as to 
equalize them. Thus, in the above example, the ten are 
balanced against the five with a view to equalize them. 
This science has therefore been called by the name of 
these two rules, namely, the rule o? jebr or restoration, 
and of mokabalah or reduction, on account of the fre- 
quent use that is made of them." 

3. The following is an extract from a treatise by Abu 
Abdallah al-Hosain ben Ahmed,* entided, A^JJill 



• I have not been able to find any information about this writer. The 
copy of the work to which I refer is comprized in the same volume with 
Mohammed ben Musa's work in the Bodleian library. It boars no date. 



( 183 ) 

^\5\} jj^\ Jya\ ^ Li\^\ or " A complete introduction 
to the elements of alsrebra." 



On the original meaning of the words Jebr and 
mokabalah. This species of calculation is called jebr 
(or completion) because the question is first brought to 
an equation ...... And as, after the equation has been 

formed, the practice leads in most instances to equalize 
something defective with what is not defective, that 
defective quantity must be completed where it is defec- 
tive ; and an addition of the same amount must be made 
to what is equalized to it. As this operation is frequently 
employed (in this kind of calculation), it has been called 
jebr : such is the original meaning of this word, and 
such the reason why it has been applied to this kind of 
calculation. Mokabalah is the removal of equal magni- 
tudes on both sides (of the equation)." 

4. In the Kholaset al Hisdb, a compendium of arith- 
metic and geometry by Baha-eddin Mohammed ben al 
HosAiN (died a.h. 1031, i.e. 1375 a.d.) the Arabic 



( 184 ) 

text of which, together with a Persian commentary by 
RosHAN Ali, was printed at Calcutta* (1812. 8vo.) the 
following explanation is given (pp. 334. 335.) (^l^\j 
^j^\ ytj y^'j] ^:: ui3 J Ji.« ^\j)^^ J^^ ^\:J!^}\ jj 
iLLUU^^ Ufw< LiLJ ^t-i^yi ti ^,f^\ «LjU:i11 (jJ[:^'^\j 

The side (of the equation) on which something is to be 
subtracted, is made complete, and as much is added to 
the other side : this is Jebr ; again those cognate quan- 
tities which are equal on both sides are removed, and this 
is mokabalahJ'^ The examples which soon follow, and 
the solution of which Baha-eddin shows at full length, 
afford ample illustration of these definitions. In page 338, 
1500— i-»=^ is reduced to 1500 = 1^0;; this he says is 
effected by jebr. In page 341, *^x=^x^- + ^x is reduced 
to i2,x=x^^ and this he states to be the result of both 
jebr and mokabalah. 

The Persians have borrowed the words jebr and mokd- 
balahy together with the greater part of their mathema- 
tical terminology, from the Arabs. The following extract 
from a short treatise on Algebra in Persian verse, by 
Mohammed Nadjm-eddin Kuan, appended to the Cal- 
cutta edition of the Kholdset al Hisab, will serve as an 
illustration of this remark. 



* A full account of this work by Mr. Strachey will be found in the 
twelfth volume of the Asiatic Researches, and in Hhtton's Tracts on 
mathematical and philosophical subjects, vol. ir. pp, 179-193. See also 
Mutton's Mathematical Dictionary, art. Algebra, 



I 



( 185 ) 

Complete the side in which the expression ilia (less, 
minus) occurs, and add as much to the other side, O 
learned man : this is in correct language called jehr. 
In making the equation mark this : it may happen that 
some terms are cognate and equal on each side, without 
distinction ; these you must on both sides remove, and this 
you call moJcabalah.^^ 

With the knowledge of Algebra, its Arabic name was 
introduced into Europe. Leonardo Bonacci of Pisa, 
when beginning to treat of it in the third part of his 
treatise of arithmetic, says : Incipit pars tertia de solutione 
quarundam qucestionum secundum modum Algebrce et Al' 
mucabalce^ scilicet oppositionis et restaurationis. That 
the sense of the Arabic terms is here given in the inverted 
order, has been remarked by Cossali. The definitions 

of Jebr and mokabalah given by another early Italian 

2b 



( )«6 ) 

writer, Lucas Paciolus, or Lucas de Burgo, are thus 
reported by Cossali : // cojnmune oggetto deW operar 
loro ^ recare la equazione alia sua maggior unita- Gli 
uffizj loro per questo commune intento sono contrarj : 
quello delV Algebra e di restorare li extremi del diminuti ; 
€ quello di Almucabala di levare da li extremi i superjlui, 
Intende Fra Luca per extremi i membri delV equazione. 

Since the commencement of the sixteenth century, the 
word mokabalah does no longer appear in the title of 
Algebraic works. Hieronymus Cardan's Latin treatise, 
first published in 1545, is inscribed : Artis magncB sive de 
regulis algebraicis liber unus, A work by John Scheu- 
BELius, printed at Paris in 1552, is entitled : Algebrm com- 
pendiosa facilisque description qua depromuntur magna 
Arithmetices miracula, (See Hutton's Tracts, &c. ii.pp. 
241-243.) Pelletier's Algebra appeared at Paris in 
1558, under the title: De occulta parte numerorum quam 
Algebram vacant, libri duo, (Hutton, 1. c. p. 245. Mon- 
TUCLA, hist, des math. i. p. 613.) A Portuguese treatise, 
by Pedro Nunez or Nonius, printed at Amberez in 
1567, is entitled : Libra de Algebra y Arithmetica y Geo- 
metria. (Montucla, 1. c. p. 615.) 

In Feizi*s Persian translation of the Lilavati (written 
in 1587, printed for the first time at Calcutta in 1827, 8vo.) 
I do not recollect ever to have met with the word^^-^ ; but 
^JjviU is several times used in the same sense as in the above 
Persian extract. 



( IST ) 

Page 3, line 3, seqq. 

In the formation of the numerals, the thousand is not, 
like the ten and the hundred, multiplied by the units only, 
but likewise by any number of a higher order, such as 
tens and hundreds : there being no special words in Arabic 
(as is the case in Sanscrit) for ten-thousand, hundred- 
thousand, &c. 

From this passage, and another on page 10, it would 
appear that our author uses the word jJLc, plur, j^jAi^, 
knot or tie, as a general expression for all numerals of a 
higher order than that of the units. Baron S. de Sacy, 
in his Arabic Grammar, (vol. i. § 741) when explaining 
the terms of Arabic grammar relative to numerals, trans- 
lates J^ifi by noeuds^ and remarks : Ce sont les noms des 
dixaines^ depute vingt jusqu'o, quatre-vingt-dix. 

Page 3, line 9-11. 

The forms of algebraic expression employed by Leo- 
nardo are thus reported by Cossali (Origine, &c. deW 
Algebra, i. p. 1.) • Tre consider azioni distingue Leonardo 
nel numero : una assoluta, o semplice, ed e quella del 
numero in se stesso ; le altre due relative, e sono quelle 
di radice e di quadrato, Nominando il quadrato sog' 
giugne QUI videlicet census dicitur, ed il nome di 
censo ^ quello di cui in seguito si serve. That Leonardo 
seems to have chosen the expression census on account of 
its acceptation, which is correspondent to that of the 



( 188 ) 

Arabic JU, has already been remarked by Mr. Cole- 
BROOKE (Algebra, (fee. Dissertation, p. liv.) 

Paciolo, who wrote in Italian, used the words numero, 
cosa, and censo ; and this notation was retained by Tar- 
TAGLIA. From the term cosa for the unknown number, 
exactly corresponding in its acceptation to the Arabic ^^^ 
thing, are derived the expressions Ars cossica and the 
German die Coss, both ancient names of the science of 
Algebra. Cardan's Latin terminology is numerusy qua- 
dratum^ and re*, for the latter also positio or quantitas 
ignota. 

Page 3, line 17. 

1 have added from conjecture the words IjJtf Jjk^j'jjjcvj 
which are not in the manuscript. There occur several 
instances of such omissions in the work. 

The order in which our author treats of the simple 
equations is, 1st. x'^=px; 2d. x'^=n', 3d. px = n. Leo- 
nardo had them in the same order. (See Cossali, 1. c. 
p. 2.) In the Kholaset al Hisub the arrangement is, 1st. 
n=zpx', 2d, px=x- ; 3d. n=x^ . 

Page 5, line 9. 
In the Lilavati, the rule for the solution of the case 
cx'^-{-bx = a is expressed in the following stanza. 



( 189 ) 

i. e. rendered literally into Latin : 

Per mult ip lie at am radicem diminutce \yel\ auctce quantitatis 

Manifestce^ addiice ad dimidiatimultiplicatoris quadratum 
Radix, dimidiato multiplicatore addito \_vef\ subtracto, 

In quadratum ducta — est interrogantis desiderata 
quantitas. 

The same is afterwards explained in prose : ^TT 

Tjfti: f^"^gwr %?rt%ri; gfoi^ ^ ^ 

^"^^rrff f^^ ^5? rT?3f q-jff TTI^: 

^^ln>, Cl i. e. A quantity, increased or diminished 
by its square-root multiplied by some number, is given. 
Then add the square of half the multiplier of the root to 
the given number: and extract the square-root of the 
sum. Add half the multiplier, if the difference were 
given; or subtract it, if the sum were so. The square of 
the result will be the quantity sought." (Mr. Colebrooke's 
translation.) 

Feizi's Persian translation of this passage runs thus: 



( 190 ) 

With the above Sanskrit stanza from the Lilavati some 
readers will perhaps be interested to compare the following 
Latin verses, which Montucla (i. p. 590) quotes from 
Lucas Paciolus : 

Si res et census numero cocequantur, a rebus 
Dimidio sumpto, censum producere debes^ 
Adder eque numero, cujus a radice totiens 
Tolle semis rerum^ census latusque redibit. 

Page 6, line 16. 

<L-*4o- U^ j^ jcj-il! u. o > d: . ^ J Such instances of the 
common instead of the apocopate future, after the impe- 
rative, are too frequent in this work, than that they could 
be ascribed to a mere mistake of the copyist : I have 
accordingly given them as I found them in the manuscript. 



( 191 ) 

Page 7, line 1. 
JjeU CJ^^j ] The same structure occurs page 21, 
line 15. 

Page 8 J line 11. 

i^jj*A\ i::^] aj^j Hadji Khalfa, in his article on 
Algebra, quotes the following observation from Ibn Khal- 
DUN. ^ (*:t^^^ ^^^ L/^. ij^ ^^ ^^ (j!^*^^ fj^} J^ 

Ibn Khaldun remarks : A report has reached us, that 
some great scholars of the east have increased the number 
of cases beyond six, and have brought them to upwards of 
twenty, producing their accurate solutions together with 
geometrical demonstrations." 

Page 8, lime 17. 

See Leonardo's geometrical illustration of the three 
cases involving an affected square, as reported by Cossali 
(i. p. 2.), and hence by Hutton (Tracts, &c., ii. p. 198.) 

Cardan, in the introduction of his Ars magna^ distinctly 
refers to the demonstrations of the three cases given by our 
author, and distinguishes them from others which are his 
own. At etiam demonstrationes, prceter tres Mahometis 
et duas Lodovici (Lewis Ferrari, Cardan's pupil), 
omnes nostrce sunt. — In another passage (page 20) he 
blames our author for having given the demonstration of 
only one solution of the case cx'^-{-az=.bx. Nee admireris, 



( 192 ) 

says he, hanc secundam demonstraiionem aliter quam a 
Mahumete explicatam^ nam ille immutata Jigura magis ex 
re ostendit, sed tamem obscurius^ nee nisi unam partem 
eamque pluribus. 

Page 17, line 11-13. 

The words from (j LjtX-s l^lj to -^JuJl ^^-Jw.•^ are writ- 
ten twice over in the manuscript. 

Page 19, line 12. 

f^\ jl 1*^5^^*^ LT^J'^^ -' ^^^Q root of a rational or ir- 
rational number." In the Kholaset al Hisdb, p. 128. 137. 
369, the expression (jjai^ (lit. audible) is used instead of 
aA*k< , which stands in a more distinct opposition to ^ 
(lit. inaudible, surd). Baha-eddin applies the same ex- 
pressions also to fractions, calling ^jt^^ those for which 
there are peculiar expressions in Arabic, e. g. ui-^ one- 
third, and j^^\ those which must be expressed periphras- 
tically by means of the word ^Jp^ a part, e. g. ^y^' *^^ 
^ JLc^ L.^A^ ^jA3 three twenty-fifths. See Kholaset al 
Hisab, p. 150. 

Page 19, line 15. 

The manuscript has JW tli3wi ^^J^ . The context 
requires the insertion of j^ after ^^ 5 which I have 
added from conjecture. 

Page 20, line 15. 17. 
i^oA^\ <-,>w3.» U ] " What is proportionate to the unit," 



( 193 ) 

/. e. the qtiotient. This expression will be explained by 
Baiia-eddin's definition of division (Kholaset al IJisaby 
p. 105). j4.j-Jil,\ L^ Jc^yi ^\ <kx^ JS£. c-Jis U^\ 
<l1c /•^•Jili ^\ Division is the finding a number which 
bears the same proportion to the unit, as the dividend bears 
to the divisor." 

Pflg^e21, line 17. 

^j^ ] The MS. hasjj^ . 

Page 24, line 6. 

f^t-^xscr "j 'ijya l^ IjJkC^ J An attempt at constructing a figure 
to illustrate the case of [loo-j-j;'^— 20:c] +[504-1 oo;- sx^] 
has been made on the margin of the manuscript. 

Page 30, line 10. 
v:>w-£i to j^ J A marginal note in the manuscript 
defines this in the following manner. l^jt^\ a-j1 15-^. 

lie means to say : divide the ten in any manner you like, 
taking four of wheat and six of barky, or four of barley 
and six of wheat, or three of wheat and seven of barley, 
or vice versa, or in any other way : for the solution will 
hold good in all these cases. (Note from Al MozaihaJVs 
Commentary).''^ 

Page 42, line 8. 

The manuscript has a marginal note to this passage, 
2 c 



( 194 ) 

from which it appears that the inconvenience attending the 
solution of this problem has already been felt by Arabic 
readers of the work. 

Page 45, line 16. 

This instance from Mohammed's work is quoted by 
Cardan (Ars Magna, p. 22, edit. Basil.) As the passage 
is of some interest in ascertaining the identity of the present 
work with that considered as Mohammed's production by 
the early propagators of Algebra in Europe, I will here 
insert part of it. Nunc autem, says Cardan, suhjungemus 
aliquas qucestiones, duas ex Mauumete, reliquas nostras. 
Then follows Qucestio I, Est numerus a cujus quadrato 
si abjeceris ^ et 4 ipsius quadratic atque insuper 4, rc- 
siduum autem in se duxeris^Jiet productum cequale qua^ 
drato illius numeri et etiam 12. Pones itaque quadratum 
numeri incogniti quern qiiceris esse 1 rem, abjice J e^ 4 
ejus, es insuper A, Jiet t2 ^ei m: 4, duo in se, fit x^j- 
quadrati p: 16 m: 33 rebus, et hoc est cequali uni rei 
et 12; abjice similia, Jiet 1 res cequalis -f^ quadrati 
p : 4 m : 3j rebus, &c. 

The problem of the Qucestio II. is in the following terms, 
Fuerunt duo duces quorum unusquisque divisit militibus 
suis aureos 48. Porro unus ex his habuit milites duos 
plus altera, el illi qui milites habuit duos minus contigit 
ut aureos quatuor plus singulis militibus daret ; quceritur 
quot unicuique milites fuerint. In the present copy of 
Mohammed's algebra, no such instance occurs. Yet Car- 



( 195 ) 

DAN distinctly intimates that he derived it from our author, 
by introducing the problem which immediately follows 
it, with the words : Nunc autem proponamus qucestiones 
nostras. 

Page 46, line 18. 
The manuscript has the following marginal note to this 
passage : jo-lj' ^ ^„j^ j c-^^*^b J-«ju ^LJ^. ^JJ^ 

jSa ^Js> \Ji^j^ ^^j^ ^^ l;^*^ ^J1^ u/^ J^^^ 

jWj jw 4-i^ j*^!^ ^^j'^ L?^, ij^j^ u^- tJ^^ 

^J Ui Jl^l J^ This instance may also be solved 
by means of a cube. The computation then is, that you take 
the square, and remove one-third from it; there remain 
two-thirds of a square. Multiply this by three roots ; you 
find two cubes equal to one square. Extracting twice 
the square-root of this, it will be two roots equal to a 
dirhem. Accordingly one root is one-half, and the square 
one-fourth.* If you remove one-third of this, there re- 
mains one-sixth, and if you multiply this by three roots, 
that is by one dirhem and a half, it amounts to one-fourth 
of a dirhem, which is the square as he had stated." 



[x- - ^ r-] X 3a; = a^^ 

2x= 1 



( 196 ) 

Page 50, line 2. 

I am uncertain whether my translation of the definition 
which Mohammed gives of mensuration be correct. Though 
the diacritical points are partly wanting in the manuscript, 
there can, I believe, be no doubt as to the reading of tTie 
passage. 

Page 51, line 12. 

I have simply translated the words <t-s4X:^i ^jitA by 
"geometricians," though from the manner in which Mo- 
hammed here uses that expression it would appear that he 
took it in a more specific sense. 

FiRUZABADi (Kamus, p. 814, ed. Calcutt.) says that 
the word handasah ((LsJC^l) is originally Persian, and 
that it signifies the deternrnning by measurement where 
canals for water shall be dug." 

The Persians themselves assign yet another meaning to 
the word <!UjJcJ& hindisah, as they pronounce it : they use it 
in the sense of decimal notation of numerals.* 

It is a fact well known, and admitted by the Arabs 

^j^^ss>- J ^ Jusr^ \ yf"^ «J^^.y c:^UK 

"Hindisah is used in the sense of measurement and size ; the same word 
is also applied to the signs which are written instead of the words (for 
numbers) as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10." Burhani Kati. 



( 197 ) 

themselves, that the decimal notation is a discovery for 
which they are indebted to the Hindus.* At what time the 
communication took place, has, I believe, never yet been 
ascertained. But it seems natural to suppose that it was at 
the same period, when, after the accession of the Abbaside 
dynasty to the caliphat, a most lively interest for mathe- 
matical and astronomical science first arose among the 
Arabs. Not only the most important foreign works on 
these sciences were then translated into Arabic, but learned 
foreigners even lived at the court of Bagdad, and held 
conspicuous situations in those scientific establishments 
which the noble ardour of the caliphs had called forth. 
History has transmitted to us the names of several dis- 
tinguished scholars, neither Arabs by birth nor Moham- 
medans by their profession, who were thus attached to the 
court of Almansur and Almamun ; and we know from 



* It is almost unncessary to adduce further evidence in support of this 
remark. Baha-eddin, after a few preliminary remarks on numbers, says 

'ij^^L\,\ hi^\ /♦l^^^ *y^\ -tU^ y *-w?j *XJj " Learned Hindus 
have invented the well known nine figures for them." {Kholdset aUHisdh^ 
p. 16.) In a treatise on arithmetic, entitled Jlc ^1 ^I^^ .^^ 
/_ »\...gj \ which forms part of Sir W. Ouseley's most valuable col- 
lection of Oriental manuscripts, the nine figures are simply called 
A'lVj.^11 /Jl^^U See, on the subject generally. Professor von Boh- 
len's work, Das alte Indien, (Kbnigsberg, 1830. 1831. 8.) vol. ii. 
p. 224, and Alexander von Humboldt's most interesting disserta- 
tion : Ueber die bei verschiedenen Viilkern iiblichen Si/steme von ZaM- 
zeichen, &c. (Berlin, 1829. 4>.) page 24. 



( 19S ) 

good authority, that Hindu mathematicians and astronomers 
were among their number. 

If we presume that the Arabic word handasah might, 
as the Persian hindisah^ be taken in the sense of decimal 
notation, the passage now before us will appear in an entirely 
new light. The iUaJc^t J*^^, to whom our author ascribes 
two particular formulas for finding the circumference of 
a circle from its diameter, will then appear to be the 
Hindu Mathematicians who had brought the decimal nota- 
tion with them ; — and the ^^ (^H^^ «-J^^> ^^ whom the 
second and most accurate of these methods is attributed, will 
be the Astronomers among these Hindu Mathematicians. 

This conjecture is singularly supported by the curious 
fact, that the two methods here ascribed by Mohammed 
to the ^-oJc^^l Jjbl actually do occur in ancient Sanskrit 
mathematical works. The first formula, ^ = v^iOfl?2j occurs 
in the Vijaganita (Colebrooke's translation, p. 308, 309.); 
the second, ;?= ^^^^^ ^ is reducible to -7^^ , the pro- 
portion given in the following stanza of Bhaskara's Lila- 
vati : 

" When the diameter of a circle is multiplied by three 



( 199 ) 

thousand nine hundred and twenty-seven, and divided by 
twelve hundred and fifty, the quotient is the near circum- 
ference : or multiplied by twenty-two and divided by seven, 
it is the gross circumference adapted to practice."* (Cole- 
brooke's translation, page 87. See Feizi's Persian trans- 
lation, p. 126, 127.) 

1 he comcidence of ^^^^^ with -7^^ is so striking, 
and the formula is at the same time so accurate, that it 
seems extremely improbable that the Arabs should by 
mere accident have discovered the same proportion as the 
Hindus : particularly if we bear in mind, that the Arabs 
themselves do not seem to have troubled themselves much 
about finding an exact method. f 



* The Sanskrit original of this passage affords an instance of the 
figurative method of *the Hindus of expressing numbers by the names of 
objects of which a certain number is known : the expressions for the 
units and the lower ranks of numbers always preceding those for the 
higher ones. ^ (lunar mansion) stands for 27 ; H^^ (treasure of 

Kuvera) for 9 ; and 3f t5"(sacred fire) for 3 : therefore ^^T^^?^. 

1^=3927. Again, ^(cypher) is 0; ^ |0| (arrow of Kamadeva) 

stands for 5 ; M^ (the sun in the several months of the year), for 



12 



: therefore <G|C| |U|H^ = 1250. For further examples, see As, 

Res. vol. XII. p. 281, ed. Calc, and the title-pages or conclusions of 
several of the Sanskrit works printed at Calcutta ; — e. g. the Sutras of 
Panini and the Siddhantakaumudi. 

t This would appear from the very manner in which our author 
introduces the several methods; but still more from the following 
marginal note of the manuscript to tlje present passage : \,^^JsCi ys 



( 200 ) 

Page 57, line 5-8. 
The words between brackets are not in the manuscript; 
I have supplied the apparent hiatus from conjecture. 
Page 61, line 4. 
A triangle of the same proportion is used to illustrate 
this case in the Lilavaii (Feizi's Persian transl, p. 121. 
Colebrooke's transl. of the Lilavati, p. 71. and of the 
Vijaganita, p. 203.) 

Page 65, line 12-14. 
The words between brackets are in the manuscript writ- 
ten on the marg-in. I think that the context warrants me 
sufficiently for having received them into the text. 
Page 66, line 5. 
The words between brackets are not in the text, I give 
them merely from my own conjecture. 

^1 Wj^t^ Axj ^j tljOi &Ji>. ^Ic Ss>-\ tJib ^j ijit^^ ^ 

^-ir^' jj^ J\jJ^l 2^j^ ui to ^^;-^\j <dll ^1 A^lxJ ^ ^;jc>- ^,^ 

approximation, not the exact truth itself: nobody can ascertain the 
exact truth of this, and find the real circumference, except the Omnis- 
cient : for the line is not straight so that its exact length might be found. 
This is called an approximation, in the same manner as it is said of the 
square-roots of irrational numbers that they are an approximation, and 
not the exact truth : for God alone knows what the exact root is. The 
best method here given is, that you multiply the diameter by three and 
one-seventh : for it is the easiest and quickest. God knows best !" 



( 201 ) 
Page 71, line 8, 9. 

Tlie author says, that the capital must be divided into 
219320 parts: this I considered faulty, and altered it in 
my translation into 964080, to make it agree with the com- 
putation furnished in the note. But having recently had 
an opportunity of re-examining the Oxford manuscript, 
I perceive from the copious marginal notes appended to 
this passage, that even among the Arabian readers con- 
siderable variety of opinion must have existed as to the 
common denominator, by means of which the several 
shares of the capital in this case may be expressed. 

One says : Uj ei-Jj ^J^j ^j ^-^ c>^ J^ J^^ 

^r-^. iU<j C>y^^ ^'^^^^3 ^^^ ^J^ <--^^ CT* f^ cT*^ 
Find a number, one-sixth of which may be divided into 
fourths, and one-fourth of which may be divided into 
thirds; and what thus comes forth let be divisible by hun- 
dred and ninety-five. This you caimot accomplish with 
any numberless than twenty-four. Multiply twenty-four by 
one hundred and ninety-five : you obtain four thousand six 
hundred and eighty, and this will answer the purpose." 
Another :* ^j.«..>^ fs . ^ ^ j LU J^ssT ^^jl^l <^p-j ^J,^ 



* The numbers in this and in part of the following scholium are in 
the MS. expressed by figures, which are never used in the text of 
the work. 

2d 



( 202 ) 

ii l^^li y (^,wk>- ^ J CPI ^'Jc^-j ^^Ij eJ^l ybj ^ill 

iT7 ^j\ ^.^^u\j vnr ^^^^\ ^^^u j " ac- 

cording to another method, you may take one hundred 
and fifty-six for the one-sixth of the capital. Multiply 
this by six; you find nine hundred and thirty-six. Taking 
from this the share of the son, which is one-third and one- 
fourth, you find it five hundred and forty-six. This is not 
divisible by five : therefore multiply the whole number of 
parts by five ; it will then be four thousand six hundred 
and eighty. Of this the mother receives four hundred and 
twenty-five, the husband seven hundred and eighty, the 
son two hundred and eighty-eight (twelve hundred and 
eighty-eight ?), the legatee, who is to receive the two- 
fifths, fourteen hundred and ninety-two, and the legatee 
to whom the one-fourth is bequeathed, six hundred and 
ninety-five." 

Another : X>UiJj j cJ^l ^*-J ^ ^, j^] [^j] Jj 

fj^j^ l^Ju*s Ujci-kj fjti^j^j tjt^y^ U^^ tt/!^ ( ^ -N**^*^ ' 
J-tf^ L^j^'^ C^ ^J^.J^ (jr?!? W^ j^^ c-i-JiJij j*^ 



( 203 ) 

c^j ^ ^iii ij^i li [MS. ] * ^r] Wv ^,/.. Tr 

As-) iUjlj {^j^^ J <bU2JJ J L-2II ' According to another 
method, the number of parts is nine thousand three hun- 
dred and sixty. The computation then is, that you divide 
the property left into twelve shares; of these the mother 
receives two, the husband three, and the sjon seven. This 
(number of parts) you multiply by twenty, since two- 
fifths and one-fourth are required by the statement. Thus 
you find two hundred and forty. Take the sixth of this, 
namely forty, for the mother. One-third out of this she 
must give up. Now, forty is not divisible by three. You 
accordingly multiply the whole number of parts by three, 
which makes them seven hundred and twenty. The one- 
sixth of this for the mother is one hundred and twenty. 
One-third of this, namely forty, goes to the legatees, and 
should be divided by thirteen ; but as this is impossible, 
you multiply the whole number of parts by thirteen, which 
makes them nine thousand three hundred and sixty, as we 
said above. Of this the mother receives eight hundred 
and fifty? the son two thousand five hundred and seventy- 
six, the husband one thousand five hundred and sixty, the 
legatee to whom the two-fifths are bequeathed, two thou- 
sand nine hundred and eighty- four, and the legatee who 
is to receive one-fourth, one thousand three hundred and 
ninety.'* 



( 206 ) 

there remains nine of it, and this is the deduction from the 
completement. , Subtracting it from the completement, 
which is thirteen, there remains four, and this is the legacy. 
as the author has said," 

Page 98, line 8. 
The word l^ll^ which I have omitted in my translation 
of this and of two following passages, is in the manuscript 
explained by the following scholium : ^ l^ <U jL-::.^ l^*X« 

^ILJ! J j^\ J jJJ\j JW J ^.^^^\j ^\ J jj-^s!^ 

Adequate, «. e, corresponding to her beauty, her age, her 
family, her fortune, her country, the state of the times, .... 
and her virginity." (Part of the gloss is to me illegible.) 
The dowry varies according to any difference in all the 
circumstances referred to by the scholium. See Hamil- 
ton's Hedaya, vol. i. page 148. 

Page 113, line 7. 
The manuscript has the following marginal note (?). 

^\A\ 4 'i;^\ ^lcji\ 4 l^ " The OA:r ofa slave girl 
corresponds to the adequate dowry of a free-born woman ; 
it is a sum of money on payment of which one of dis- 
tinguished qualities corresponding to her would be mar- 
ried." See Hamilton's Hedaya, vol. ii. page 71. 

I am very doubtful whether I have well understood the 
words in which our author quotes Abu Hanifah's opinion. 

Abu Hanifah al No'man ben Thabet is well known 



1 



( 207 ) 

as an old Mohammedan lawyer of high authority. lie was 
bom at Kufa, A.H. 80 (A.D. 690), and died A.H. 150 
(A.D. 767). Ebn Khallikan has given a full account 
of his life, and relates some interesting anecdotes of him 
which bear testimony to the integrity and independence 
of his character. 

Page 113, line 16. 
The marginal notes on this chapter of the manuscript 
give an account of what the computation of the cases here 
related would be according to the precepts of different 
Arabian lawyers, e. g. Shafei, Abu Yussuf, &c. The 
following extract of a note on the second case will be 
sufficient as a specimen : i<*)J^^^ ^^ t/*^^ ^-'*l^^ ' 

^^ L5^ ^^^^ (JJ^ ^} ^^3 ^'^3 ^ \"".* t-j^^yi /^ 
Ji4>.sr< Ai£ J ^^^llJl [^^X< jtfU ^^^^ '^^^ J^^ ^ J ^^ 
<Upj ill J <U^ '-r^ ^ t-^yi ^l?j J«^ (*) u-:^?^^ ^i;^ 

j^^ L::-Jxa lH J.»«jw Ju-i:>- ^^1 Jji j^Lcj l^ls*- j^^ ^i]/Jt 
i!i LUiJJ ^jji^ '-^'V.V J^ liii(*) yjj '— fl-'j:^ (^^ «--^«X« 
^jl c^JJl ^:a Jt^ ^^ ^-^ Jj^. ^^ JUj j l*-^ 
X--^«ci. ^,<^t Jjui ci-^lij ^-^j^ \^^ ^} ^j >^V 

* These names are very indistinctly written in the manuscript. 



( 208 ) 

l^bjl AiJj ij, The solution of this questlan given by 
the Khowarezmian is according to the school of Abu 
YussuF Wazfar, and one of the methods of Shafei's 
followers. Abu Hanifah calls the sum which the donor has 
to pay on account of having cohabited with the slave-girl 
likewise a legacy ; thus, according to him, the legacy is 
one and one-third of thing : this is another method of 
Shafei's school. According to Mohammed ben al 
J.MSH, the donor has nothing to pay on account of having 
cohabited with the slave grirl :* and this is aarain a method 
adopted by the school of Shafei. After this method, 
one-third of the donation is really paid, whilst two-thirds 
become extinct : and there is no return, as the heritage has 
remained unchanged. According to Abu Hanifah, you 
proceed in the same manner as after the precepts of Abu 
YussuF Wazfar. Thus the heirs obtain three hundred 
less one and one-third of thing, which is equal to two things 
and two-thirds : for what he (the donor) has to pay on 
behalf of the dowry, is likewise a legacy. Completing and 
reducing this, one thing is equal to seventy-five dirhems : 
this is one-fourth for the slave-girl ; one-fourth of the 
donation is actually paid, and three-fourths become ex- 
tinct.*' 



* I doubt whether this is the meaning of the original, the words from 
JkA^S^ till it^lL being very indistinctly written in the MS. 



.LcUUi 






^-- 


> 


.k-J ^JS.^ 


^^^, JWj 


c;J^'. 


\^ rp 


i-jiasTj 


J=^j 


1 rc 


A-^^4 


>^'. 


!P 


^■j^i 


i>cj 


11 n 


^. 


^ 


/ re 




t^ -*v'^~ 


A n 


>« 


■ji^ 


^ Pr 


4«J_y i(L-*i- 


io^_,^^U.^ 


ic ic 


tuijj 


J^.i 


ic vr 


eJJ, 


'^^ 


1 vr 


rf' ^' 


r^" 


1^ — 


J^i^cr* '!)=r-cP^iir* 


j.jjij^ecr' 


Ic vc 


cs~^ 


u-^ 


ir Ai 


*«j^l 


SmjS Lw^iil 


f- AV 


^j 


■ >. 


r V 


j*j 


J* 


ir — 


&lj 


*iJJ 


V 11 


JU ^ U«- j^jlj j^U j^ 


J^^ 


n ir 


Jiscli 


AS.' 


|v 11^ 


S^ 


^ 


M 11 


W»j 


C^, 


p 1 ♦♦ 


IL^Sl 


^^^\ 


1 - 


liaij 




ti ~ 


IJuc 


Jufi 


V 1 T 


LL 




n iM 
n tn 


ciJ.'^ 


hL', 


tr Mr 


.^li 


'^. 


IP tn 



irr 

y&^ i^^\ ^ \^suu JtAJ *L<-i> u» a .A* J ^(^_5**j Uji^J 
h^ i^cflfs^ ^^Iaj ^^ ^Jii AJfe^ J bJJj Ujb^J ^^ ^ 

ybj [^ ^jf^\j hJ^\ ^ jjj i^ iuL jjjj ^^1 

cL-Ji <LJ\^^ f^\ji^ ^^ (JJ^ (^^'^ U— j'j i^Ujb^J 

^J\ L!J3<i J^ Uibjj ^^y^ ^j JH^ c^lj ^j^ 
cLj1 <u-«c*. j^'j <)lcLj"1 Axjjl ^1 ^ j^ l^LJi 

ciJjj Ujb;u> j^yijj Hj <^^1 *-^*V3 s^^r^^ rt^*^ t*^\? 



ir 



* u^^ 4 (^^ ^^ 



(*\^ ^ ^ d Ujbjj ^^ au^> 4 -k; (^^ ^^^ 

^. 3 J^^ ^in'. *^^ Hr* 4 ^^^ (*^ C^b"^ i^ *^-5^' 
''^ J >^' 'ir:'. J^ ^M (*>^b^ ir-^ '•^^ ^J3 ^ 
^ L^jSU UjbjJ ^,j^ -^V^^^ '^ e;/^ (^b^ Ir^ 

yb^ ^^^i> J^^^ s-^ j^ UjbjJ ^/ij uiCli lKj ^j 
^^ ^^^1 ^ ^cJJ^ ^^b ^^ j^^ L^^\ ^ 

* i^bU^U ^^J^ Uybj >jbl^J 
^ 4 ^^jo ybj Uibj^ ^^^ «>:; J^ {^--^ l;^ 

^ Jj '^\ lL-1 JW ijw«|^ ^^ J^^ iJ ^ \SUI) J ^^j^ 

^jjjb ^1 ^ i3^. c5J31 Jxs^i ^d*^ Jiw^j <Lk^^! 
ybj ^.^\ ti;-* l/^ ^ L5^ ^^ ^^-^ tirAr^ 

R 



-^^^ cJ^ Jjc^, Ufc^j ^jjj ^ ^^^\ (^^ ^^Li 

C^jJl ^^1 (^ JjOO ^^ ^ ^^^ c-o^ ^^/J 
^/-J UjbjJ ^^ ill >^| j_ft^ ^;^^ *^^J1 ^^ ^( 

jAj ^1 jci^^j *^-i, (_a^ ^ e;.jV-j uW.^^ ^^, 4 
u^Jj ^(^-^ ^r^^ c:;>>^J ^*;>Uii5 2rJo 4 J-^^^ Ai>l^J 

^.^ L)^^ J^^^'J ^{^S^^ L^ ^*^3 ^LS*^ ^"^^j 



hUxij\ ^f^ s.\.^\ ^i^ ^^ j^^ -"lS^ iJ^^ U^ 

^jLs C^s> JjUii ^^-i Li^j -\^^ ^W* J^^>. 

* u&;t> -r^-'Vj^j ^^ J*^. '^^V^ "^^r*^^ 

l^:;^ ^ 4 ^jV t>r;^ ^-H^-? ^-^ ^^ ^^ 

I>UiJj l^::^ JU^ ^\ ^uUi j^k:a:^^ j^^ V-* J^- (*^ 
^JJ ^^-^3 4 L5*^ ^^^ *-^^^ ^ ^^^ ^ f^^^ 

ji^3 "-J^ u^. ji^3 ^J^ ^> J^ ^lT ^^ ^ST* 
j^ ,^ lL<33 jt^"^ "^ u^. J^ ji^3 ^J^ 
^^U^^ as Jc;.U ^^ ^j ^yd '^5^ u^ Jl ^^'^ 



^\sux> U^IwJ «J^1 ^V^^ L5^^ Jy 1^^ ^"^^ cJij 
<U^ ,^^iL-i /^^ <jjj y&j ^1 J, J -c^ ^ LUiSj 

J>X*J ^^ ,^_^ J ^i;!*^ ^ «^,Uii5 ifjj J ^A.>J 

L5^ ^^Jj "^(^^ *-^ J ''l5'^ '— ^^ ^r^^ ^^ 

^^^\j -^U-i»l ^^1 Jj^. ^^^---^♦^j iiuUjlaJ ^^;-:u^j!l 

He iLaJ J (j^^ J <^»**' y^j <^j u^ J j^ 

Jr^\ JcS-lj -i,^ ^^ LUilj ^^jiif-i lL-i» ^-^jll Jjtsr 

-^ *Jb;J LUjOjI i^tX> ^j jLa3 ^^c-^ lIJlj ^*i. ijto 
j^^\ iL^lj -^^^ LiJj ^1 J,j -t^ <jJj J ^^ 

,^jiLJ J^^ (Jjjj llJi Jjill i^j J2^ cUbJlj «^ 
ul^i J ^j^\ J^« J'^^. -^^1 ^ ji^ (*^J^ ^.^^J^ 



^^^\ Jx 5Jjj ^^ cU5 J ^-^^ Ll^i ^li 
^ ^^Ij ^^ lUjj ^\^\ ^ Jj^t^. ^.Uii5 ^^ 

^^ ^^U ^j jjfilftljill lIJj iiJ L— !U>^1 /*jij J ^(^_5i^ 
^ ^U*J^1 C-^yi Xjjj t^Jol <i J^ -^(^ *-^ 

k^J^ U^ Ui^^ J^ ^^J3 ^^ J^-^ -"lt^ 
dx»-Jj ^U ^j LUx^j^ ^ 1-^^ jLs- »\>-l jjA3 -'Jjs*"' 

^ 1-t^ yLc Jo-1 j^ "^i^f"^ i^^^j 1^^^^ J k}"^ 
^j x^^jLW JxrsT ^su:.s>^ ^^\ JjJ lij ^ j^"^ 



n 



^^ cr?^:*^J '^ J^. ^J^^ u/^ ^, iSi^ ^^ er* 



<d JU ^ <C?j^ (^^ ^S ^J^ Jsfj^ S^^J J^J 

jtoj i|^^ ciJj ^^ ^J £>lo (J^ij i/^Jill «J-Jj 

^,^li ^^ll-i, iiOi J ^^ ^ j^l l^^\ h,< tl<3 J^ 

^/-i ^^^1 Jj^ ^Jjj ^^ cj^j ^^ ^.Utlj^l 

H5 L!J3i ^r^ ^^j ^(^ tjjjj ^Li.1 HS Jjou ^U«j\ 



!U 



U&;J ^>^j ^j^ ^j iil^^ j^ ^J^ ijiri!^^ cT* 

Jjj^ *-r^j J rt^^ ^,^ W^^^ ^jW" (j^ u^ 
bybj ^ Mj^ji' ^^y f^*^ iJU-— 4^ W^*^ ^i;V 

jcA--i -^^c-2i uL^li ^ <^*^J^^J {»^^ <ijU-w«C>- *^jV^ 

^^J^ jJ:. *Jb^J LU:;<«j £j;jll ^^kX>3 <4j^ ^Lf^ U-^*>^ 

JtXjo uj3i t^{lav5 iLj-^jH ib.0 uJ33j -^Lj-^ (^ ^-^ 
j^^ il^ill i^ji^ L!i3J j^^ ^^-i» ^^ l^jcf t^j^- 



IIP 



ajUj! L.A^ CJ^j ^ ^^^ J^ CJisi JjU- 
l;aA^j <u./K£^ j ^U yb^ 4ir?^U ^\^\ L.^ jc^l^i 

kjW (J^J s^ J (^'^ ^.^ ^"^^^ ^liJ ^'V^ j^^ ^^ 
c^\ ^ j^\i hj\A\ ^^^^ i^j^^ "i &j\ &ju^ 

«U-J JxfisT ^ ^^^ * t:;:f«-^ W^ CJ^\ ^if^ 
ti jUy ^^^ L!j3i jj^ L^^t AJb;J aJUwid- hj^\ 

Ji^ ^j «tJU viJdJ J^ ls*^j1? -^lT^ U--*>^ J -'^-- 
^^-r>-U ^j^ ^r^^ ■'jc^ j*^ LUsIj ybj (*^k^? J*^. 

^Ll\ Hj Jj^, LUiJj ^/J ^^ ^j -^^5^ cKlJ 
^^^ ^yuJ X,.U: ^> ^>^ (^j^ ^>j -^Lr* -^-^ 

uj3i ^^ ^^li <U-JjUj ^^^^ jL^j ^Ul H5 J^. 



itr 

*j&^j ,^.^ J^. *^^^y^ s.^^\ ^^^ <u^ \-^ 

^ Uj>. jLa Lt^ ^ s\js^\ X-.^d- J ^5;J ^/^ J 

^ J LU ljb;Ai: ^ *J^;t^ iJU..-^«o- If^ ^.jV <-^^ 

j^^o-Up Lap* J*^j (^^ ^.^ *V^' *^r^j 4?^ S^ 
^jd LU ^1 ^Jj ^J^. ^ ^y^y^ ^^^3 fD^ 

^j,^ LUc^ L)jj^\ i^dj\ 4 J^ -^ls^ U"^ j^ 

^jj U-^ JJ^. j^j-^ ^.UiiS lL^J ^/J -^^1 ^^ 
ibU LU^\ ^j;^ ^JbU j^jj X»Ujj ^^ ^U=5.1 
j:^ ^^U^l iiJjj U-^ J^. (^j^ W> L5*^^ ^W: 

Q 



Cl^-tfjl iXJJj ^^^ et^j ^ib;J ^J^j U^J U^J ^J 

U--J J Uji;J L;j.y-^^ u^ J (^"^ ^.^ y^j W^ ci-J^ 
^*{;^J t^i^j^^ ^j^j J^y^ L5^.^ •'lT' t^-? C^J^ 

ijjjj Ajb^t) «^^ tj^j pl-ji '^.j^3 ^'♦^^ i^y^ J 

-*^-i» J--J J (»^i^ «--J' bJjj cLjI ^U*j UajJ (i)^-J 
t/^J ti J-^^ <^i-^i»- <)j^ aj Ij|^ ^^ ^^ ciJj^ 

j^Ulj Ujb^J U^J ^J^J U^\^ *--^^ t-jL=:i -^(^^ 
^ ^jj ^ Vj:>. ^^j ^ ^ Vy=T ^3r^ 3 

^^ lL^j ^^ -^^1 W^ ^>:'j -^1/r^^ '^^k 
^ ^ ^-trr i::^,j^3 (j^^ 3 ^^ e>^-? ^-'^-^ 



JuJJj ^sLaJ L;:^iJJ s^ ^Jhi:^ ^^ ji^ h\^ h}xj\ 
J\ ^^J^ fa-go"-' j<&j c:^iJ! <Lii>- c- t? ^«,.g T" . ^ Ai^j 

ju«Jl A> ti ^ t^?^ J^ -''(^ trL^-? U,^"*^ J *V.^ 

-"(^ cT^ ^"^ vr=r^ ^-^ ^^'^*^« ^L5^ ^ ^ ^'^ 
^J^ (j:Lr^J L.^j ^^U ^/^' -c^l J^ i^jjj 

Jc^^^l ^^Jl3 Lli^lJj JjUi J^J^ ^^ IL^ J^. 
yU fa^J*.) Cl-w>tfj|^ 1^1 (JI^^'j Lll^iJ! Cl-^'lo J <5JU 

Ujb;J c;>^j ^ ^3 *^^* tjJ^ ^*tfjl jij ^^ J 



ijij — ♦cs- _5 LUiij tl^ J L, p .. ^ :^ j -t^^ ^<i)j Ju;J| i[^ j Sd^ 
L^y^ -^LS^ (^»X^ Lli3 j^^^li U--i» Jjxj J^^ (j^X^jJ: 
*i,S^^ ^f^ ^\^ 4^*>w>j 11^ Ja.^^ ^^^---^.♦cLj LUllJ 

^,UJI ^j jy,l l!J^1^2w-j1 Uj jw^I I^' ^--4^i L^^\ 
^s^S^ C/^.^^ l!J3j ^< Jj-*-^ UJ^;J ^^--.Arv. J aJUiJj J 
1::jU Xj-tfjll ^;-i. «^^:»^l <uJ> ^j LU-JI Jj-jO* J >ib^J 
<%^ Ujb^J (^ j >^^. < ^^ j^ii^J ^Ux-J _j t-iJI tJu*i /»i^;fc) 
<UUls Uj&;J jj**-./*^ j -«Jb^J iJUcuj tjJ^I l^i jj^ 



aA,J c^l (^^ »X.^i lij ^ <Uilji Ujb;J i^^j^j 
^^\ ^^J l1J3 J ^^ LS-^^^ ^J'^ L#-*J I;Uiij ^c^j 



bU ^J Ju^^ L5^J ^^'^ cr?--'*^ ^ ^,U*^ J Ull 
j^Ull ui3ji AJb;J LUc-^ y^J jjjll Jjs*J' t/iJ^j t^i^ 

M& -^^j Ujb^J jjj--.4^j iLU::-->j t«iJl j^.5^ ^^ 

*ibjj L-aJ^ ^Ju-i ^j-w4c;^j LUjcwjj u-cJl ^j ^'^yr^^ 



S^'VJ 4 L5^ ^":i^^ L5^J ^^ ^y er* A^ UjbjJ 

^1 j^\ L^j h^ lL<1Jj l^j ci^^j ^.^ H'^^ 

^jd uJJl *X-xll I^' JxasT j^l iLoUw AJ^;J bU ^J 

^^ ^'^rr^i^ "^lS^ U-fitOJ J ttSijii LUcU-a y^j Jt:«M <uj^ 
jf£. »fi>jj h\^j\ ^Ju^ aJ^J <— 2-11 ^j Ju^\ i^* 

^jJ bU J;^J jjji^l ^,jJ Lli3 J ^^ ^_^* -^^^ c-ioi. 
^jll ill^ Jjjtj ^j<-i. (.^fi*a3 ^ *ib;t> l:^.^ l5*^ 

<0 Jjliii Unoj j (^.Ar* J-^ (*^J^ L5^.^ U^^ '^^5^ 



^Li^i ^u«.4a> (J<^. f*^'^ <uW«^ .A*^a»i -i^^ ^■Qoi^) J 
<U-M*jj ajl« Jc»-l^l -^L<^^ rr*^ '^ J^^ -^Lj-^ ^»o..fl.> J 

j^to^J 1:l)U j^^iifrji jf(^ *^r^ (j^ ^i<^ ji^ ijUiJu 
jtJbjii i>U ybj d^l^b <— fl^l Ju^JI <JJ,j Jl ^-^J.^ 

tt' v^^^ ^^^ l^i-a:j (jrPlr* •V*^^ ^*^ J'J^. tl^^ j 
j^^ lL^j j^ ^^ ^j ju*Jl aL^j Jjoo ^^ 

•'^5^^ ""ls^ t^jj ^ J^. f^j'^ ^> uj^ ^^ 

^j 4^^1 a;'^" ^^U Ujb;J (jJt/^j i^W.^ alfUJ^j 



1-1 

^Lil iU-M-o Jax^, Ufcjj ^^^;^.--,4^j Julc (Jl;* fjjLa 

h\j^ Ll< Jaxj A=^1j1! -'(^^ ti)J^ 'V c^^ liajj 
i^:;^ t^jjl Juxll L^j uLi^li^ Jtibjii lI-Jjj Ujb,J -1^ 

tUiij Jiwoj l!J3j Ji^ ^^1 jux!l L*tfj J Ajb^j ^UsJj 
^V J^ t^ •'lT'^ ^^^ ^" ^ ^^ ''^T' 

^^ lL^J ^^li ^Ul i:*^;^ Ui3ij C^^( ^5^ 



4fcU^ t/JJl Cl-^Ui ♦Ajt^ X>U — 4r?- j^1\ <UJ ^ *Jb;c> 

^ ^J^ U^ Ss>-\^ J^ j^yu-j *^ 4 mUd^ ^\^J^ 
<ti>U«j J U-Jj j»Jfe;J iJUiJj i^:;^ t/JJ\ J^l L-tf J (J«sr 
^^i:^ v/jj\ s^\ L^j J«?^ J ^(^ ji^ iJUlJi 

^^lK-..4k£>. '^^U-J J •*L5'^ Li^"^ -? ^"*^ (*^'^ ^^U— .<»:>>. 
X^J Jl« ^tX^ ^^^ -4^^ ^^ J -^^ _^ *Jb;J 

Ci/ J . (i»^t> LUiJj <*:i^ cJ«>i^ Ll^Ui <)w^1j Ji^ J 
ui3 J j^ u-g*tfivl1 J^,^ ^ A^t^ ^.U ^'i^^j t/'-VJ ti i^^^Vrr^ 

i^to Jlr ^wvJ 4 >r^'^ -^Li-^ L5^J -^LS^ ^ 

P 



I*r 



cl^J ^J^ ^^ ^^ ^Di ^ jj, UJ>*J ^.^^ 
<^\ l^ jJ, LUIj jui Utojj ^^^ J^l ^^j 

x^ ^Ul ^*^ ^^UibJl ^^ ^ILA tl^o^ ^^ 

^^;^A J^. X>Uilj ^^ ^^^1 J^ iJiJi ja^- 
or* ^Ifr' ^^*-^" *^^ ^ ^J^"^ ^^ f\^\ ^ji 



* Ju*U ^^l^ 



X4J LiJJ «^!Ui JuJ! ci^U J l^fl^fu-li ituJ ^*Jij 

^^U ^j iJ^SUO) U^ vl^i vjJjj >J^;t^ X>U «-J.l 
UfwO JO-lj Jfl AJb;J <J^J Ujb^J liij^j ^j fy*^ 

^j^ ^\ Ji (jJjj Ujb,J CTf^J ^ 4 ^^ L5^ 

UJ ^ju-jj L*tf^ *Ji^4i l5^-? Uji;t> ^^^:^j ^ X),lil 



iUdlj ^^1 lijjf ^^\ 4 J^ <-^^ J^ (Jj^-? 

^ >^J:y^ U-oJj ^J^^t^ J*^. ^.UiiS (jj^ li^rf^:?^^ 
U3J^ ^.^ *V-i^ L^J '^^ '-^ ^J^ ^-^5^ U^' 

Lli3J ^^ Juj«ll L^j Jxfl^ ^^1 ui3«i u--^ ^^ 

ui3j iiT* i**^^^ -^iS^ J f^j^ h^ 'V*^^ 'H^j s^'Vj si 
l5**j1^ -^Li^^ ^'^ (j,^*-**^* ^5^.J (^^b"^ ^;^-^ybj ^^^^1 

^ ^^;t^2^^ -*^ ^j ^j'^ ^j^ ^j) <-^^ *^'V 
J^J -'l5^ ^Lji ijujl^. l^jJ^^;;! ^l^V lL^j 



IT 

^^ JU Uj ^^^\ Lri. Ji^ /jJJ C^l^ ^^1 ^ 

r 

J>-; j:ii:l jJ i^J^j * ^^ ij^ ^ J JuJI 

^yc-j. J It:^ J^l ^J J-^ u^ ^^ ^ "^J 
*V. 4 J^ ^lT* >r^ ^'^^ y^J 'S^ cr* L5^ ^ 



ay^ V>j^ ^Si\ 4 j^, J ^^ jJ^ ^5;^ ^y^ J 

^biSj >Ji>l;J i^ yj>^ j^ U c.iaj 4^5^V «-l^^ 
^jbj Jtibj d^i ^^ ^jl^ ^^ ciJj ^ ^liiS J Ua,J 

iU! iujl Ui3ij CiT^^^l L^l^ J*^. ^lT* '/'^-^ 

\fi)jd i^ ^ \Aibjii JLs. ao-i J ^U ^f^ L!i3i ^^U 
Jj^. oc^yi -c^^lj s^^ ji^j slJ;>\ L^AsS^ Jaxj, 
^jj ^ Uj9u ^ iu^ ^ -^l;^! hjt^^ Uj&jJ ^^^ 






i^y.1 L>^' J <^5^ (j-Ui alU cJ^ JsrJ ^^\j 

nr* LT^' (^ ^J^ C:^.>*-**^' c/t^ f^b'^ ^ y^J ^;f< 

j^l ^1 ILA /y.1 Jc^U il ^U^ U« lU^I J \Ljl 
iLi. U>1 (jJill c^^s9-U Jtvoi d^ cJi^\ C;.^U 

jjjx^jj i«-..«cS- ^jJ-Jl ^^ c^J^J (i \^J^ ^^ UjLa} J 

ulOi ^y-^li -^Li*! ^jl Jjuo uJ3ij liaJ J i^ ^1 

lL-1 jlL& Jc»-1 Jjju U^ er?*^v '^^ e;/^ ^^^ 
j^ Ay>\ £ijj \^J y;-c ^*-^ Jj^. Jtf-yi Xj^li 

AJbljc> IjLs. \^ j^j ^ji^ X>U ^ WxP U^ 
<jJl^ c:^-tfjlj ^]j^ ^/-t '^'^Z J ^jj^' Jt*^ <.::^'to ^ 



^1 

bJj J Ujfc^ J ^ot^*-^ J *^ ^^ (irA*^^ ij^ *^j -^j^ 
yr> tl^j ^^ j^ljll -i^^ls f-^ ^3 ^j^ i}'^, 

i^j^t iJaxJ J\ l153J ^^U* l^U cUiJ Lli^^jlj j^!;<^ 
J*2^ <U«^ <Uk« y ^^«*-J 15^. J ^;f* mMji^ ijLs:' 

aJ^IjJ y;^ l^j uJ3i ^ ^joa;^ ^^ J j^Jblj^ ^^jLfi 

^y^ ^jjjl ijjj c/<^J 4 J^ ^^ ^^ ^^^ '^^^ 

L5^ ^JJ^ ^^^ ^_/h3 ^^;:**-::Jl jf^\3 J^Ji» LL^ij 
jtj^^ Ltf^J ^riA?** J*^. ^^'^ (^;:?*-^* UJ^ i^j:?-^^ 
Jaj^ Ujb;J ^yjj J ibU ybj ij[^\ h^ Ll^i ^ ^^\i 



1A 



Jj^. ^Ijll j^lj Ue er?^^ i^tj ^.UcJ-j lill 

^J J^S< k^^^ ^J^ ^^ji^ J ^W* ti)/:> t*i-Wl 
* w;Lr-^ J <u-jj i>Ux-»,^j c-aJl ^jJJ j^^^ y 



* Jjjjl 



l^lo ^^j \>^ «ij JU ^j AJfc^J hVa ^Js. djyo jjo^ ii 

^ y ^. U 1>W fj^ i^Ji J^ ^Uai ^jjJl C^U 
jfkC. Uj^j tj^*-^ (,.5*?^ t— nIj ^^ llj-i» ^V^j J*?^ 

j^^AmJ ^^^ LiJj J AJb^J C-Jj _j j*Jfc];t> ^ yij yU 
«jJjj *Jb;c> ciJjj .(^1;^ ^ y^j c-iLaill <LSl^-r« ui3j 

^LS^ J^J ^U' ^J ^ >^J lT' L5^ ^' r^^*^ ^-^':^ 



»» 



4«.^4^ fcX^l^l /»^V3 A^ [j"^^^ ^ /^' <U-M-J Ji^, ^U 



^^^ U ^^yv-* ^' ^l^SLs^-lj ^]j-o\ «-->:t-^ cft^t iU^ 



^ \j\ uJol JJ3j (^Liii XA^l J^ lU^I ^^ 
*^j J^ or* '-iHr cT^vJ^ J liTt^^ i^r* 'iHr ^-^ *^*-^' 

l^ 4jUU U^ t'l^ ^^ (J*^, (*^ ^^^ ^^ J f*^^ 
hoj\y t^ h*^ J'^M ^^J^ J^ ^**^ L5*t^ (♦V^^ ^^ 
iujlj <lU)U9 <lL: *X>JJ f^\ ybj tlflU *^ Afri ^L-al 
Xojl Jj^. Jto ujU^ (jjLi 1^ y:^ ^x-J j^ Aje>'\ 



o 



11 

^-^ J (^J J^ U-'^^ '^-^ lJ^ ^/J U^ ^\ 
U^ f^\jt^ h\^\ ^ jJU U^ yu: jyj Jjco >e 

^--^ Jj^. cJ^j U^ jL^ ^:>.\ Ji^ ^ ujIoJ J 

J^ L5*^j!^ ^^. U--*^J M ti^'j <^*U Js^j 
L5^- ^ t/-^ ^' t:^^^ c-,'^:?'^ 4jw-JuJlj jj-^l LUiiu 



u-^ jJli JU (JJJ j^^ iLU^I j^ LiJdJl ^^ 
ill (jU^ C^x^ j_5%^ t:;-"*^ ^^ L^'^^J '^'^ ^ JW 

^U^ lLnx^ Lf^W* '^i^ ^' (*^ t— a«3J ybj ^l:Ki::w-j^l <l1c 

1^5>- ^i^T^V-ij ^-^ e;^^ J^^ lT^ "^ *i^ tJ^ 

Jjjcj U^j ti;:^^-' J J^ U^ ^t^ u;:tr-^J *^.^ t:r° 



^,1^ Sjc^ u*^. u^ LL^«^ i^y^ l^jJ f»V^^ iiT* 



ci^ J A^j J to Wj ,^^il^-i ujic< 1^ U ^;wJu*5 ^1 J 

j^ ^ Li^-? ^'^^ Ci^^ W-* J*^ ^ J*^. f^-* 
Wj J U^ jL£. Jk^l Jjoo JU bij ^^^u^ U^ ^^ ^ 
<— ^^ ^jLi l^A-a) /»l^! j^_^ jjjJ ^ y^j uj3to jmc;5. 

^^Ujj <U-**j:i- JW J^^ W^ ^;l..ft ix^ J*^, J^ 

^j^ic ^ ^^Ij ^*?^ ;j;^' ^j^^j iLA=>- A^lj 

*^ JW ^^ i^^iuj to ^j lUO ill Jl c-^sM^ JUJ 
Itto j^^ U^ ^ic iiJj ^l^li v,:ucj c^^jmu:.) iUSoll 



^je 



j^i3 ijjj^\ uiOi juu Lj\ 4-.wa:j j^p^i a"iiu ^:^j 

JUi j^yLJ e^l S^^ U^ ill <u.i^ jJl J Jl 

^' J*^. (*e' ^^ l^rT er?^ ur* ^t)^ j^ ^ 
^ l/;^ U^ yLc HS ^^ aUt jJli U^ y;^ 
^\ ax^ Jj.^^ JU ^^ 1^ ^^j-:;*- ^ 1^ y^ 

jm ^/J ^ H3 ^^ ^1;5^1 iojlj Ue ^^i ^jA 

j^^^^ U j^^w^ci- <?-Ujcj ^/=^^ (♦^l u '>.*>av,< JW ^j 

lUOj Is* a\^\ Juu JW ^ ^ U ^1 jj U^ ill 
h!^ ^ ^ ^U>.1 aiiij J ^^-.ff- jJli U^ ^ HS 



^r 



^ iLU^l 4— >l> 



<L^ (Jib* J ^)Ji\ Xj-^jll j<JJ>j c:^ '^r'rr^ W** ^^ 
^^yuJ LJbi! ^jl\ ^j /♦^l S-^:t^ (iT'*^ ^^ ^J ^) 

^ ^ji^jJLs- ^^ Uj^ jLs. Jo-1 lL^^v< i^yL-i *f*jl &li^ 
LS^ JojJ jjl ^j Lli3to J^ A^^ ^/L& Jax> Jto 
^jCs l^ 1^ jLs:> Jo-1 ^ -^1/s-l ^x-J A^^l ^^1 

JW ^^ yu£ 4X5-1 ^Jl Jto-li j*^ ^^ Uj^r y^ 



^^U^l hoj\^ JU ^ U^ er?*L>^-? er^.^ cir* W- 

Uto,Jj S->:^^ ^^*Uo-l ^J^J y^ (^^ *^i^ (i^^ L^y^ 
^^^^ L5^ t:>J^ y^J J^^ t:r*^ ^^ cr* J-^^ (*^ (^'^ 

Lmoj^I XjUsJI J^ iTfcjjj ^_^ Ui L!J3<i j^^ \f*aj\ 

A\s>A ht^ jj ^j^j(^ ^ <-r^:?*^ ^^Uc^l <iSi5j U^ail L3l^ 

j^^ iK!3j ui3U J:;^j j*^j^ ^»;^ Vy>- ^^ 4^^= 

^J ^:Jii^}^i ^^' iL/*^ JWj i:;-2-.*j iji^^3 '^S^ 
^x^j iJUuj ^^Ij t:^j*-«^*^ ^^* J k^J^ Lt^\ 



^^-Ufi^l <U^1 jf£. 4.-/40. iJif-J ^-'^!**^ (j^Uc>-l ^J^3 

(JjGto Lgii-^-i Ujb;J^ *^^ ^^ cL^J «-jj JjJ^ c, -^ . i*^* 
HS J >^ ^j CJ^\ J^ clClfi i^^iui ^ y&j jm ^^ 
jjli,! j^ L5*fr^' <»»^.r'^'^ (^Uci-l ^-Jj J jk>.lj »j> u>it> jj^ 

3 ^;^U:i.l H2.J lL^ J ^ft^^ [V*^^ ^V J<^J 

U^ ^^1^ fL***^ l;^^ ^^ L5^j L^irt jtfLc l&jjj 

Smm^**n> ^j^U^l ^ J L^t ^U^ J*^. (*^ ^y ii^j 
^li JU ^j JlS'' ^sXs?^ f»l^l 'T^^ ^ C^^l 



*^,jJi ^j ^j^\ J ^Ju*j^l L-^hj UaJirt iXiJ\ 
^J^ ^ \AJbji^ J 4-r^ liT* ^^Jsr J^ iv-* iiT* ^tf^ 

^..^^•Jl J^J-U j^J ^^ l.t^ ^^ ^XjMrf ^ Uj5»- yiC 

^^ Jll^ u/:ti y^ ^*f-J >JbjJ>!\ ^ U^ yl^ <U-w» 
<U J-4a:l3 Ir^^^ i^jJ^ Tl>^ ^ ^^^j\ ^^ J:^ ^Jt^j 



J^ J^^ L5*^j!^ crt^^^ C^ ^' ^' J^ 

j^ tuUiU ijjjll cl^i jW"^ JW j--.^ 4j^ ^^J 

^ ^;— >-l <^-jil IJJb (i ^j U^U.-* *Jb5;j31 -^J^ U^ 
iuj^ Jill J £uj []<d*59-l3] <Uf*J*j JU ^j j^b* ^jl 
u»^y<3> j-^ ik«j |Jif;t^ W^ ^J^ i^r* C?^ (ji/^ J 
^_;-,^*»£»- ^\i ^^^^■Ma3 ^ iu-^ ^Jha^ U&;J j;J! J 



tX^l S"^-^ J^. uWj^ L5^j1? iiTt^ *^;^ *— ^' l;^ 

^1 LiJJ jJuJ Lmo} <Uo jJU JU <J^ Is; (jJiJI 
*j;j uiJj jJu-i UJ^;J cJ^^j ^-^**^ ti^-? W^ ^^ 

Ui^l ^j-« U-^*^ L5*^ ^^^ ^J^' (if* <--^^ L5^« 

*jb;j j^_^ ^! JU ciJj ^j^ *^'l ^ ^jA) A^! iU-.4j>- 

^Slc Jj-3 S-^:*^ ^M-liX-ol <U-^*^ ^^ AJfe^J L5^J Uji;J 

JL.i^ il! J ^jd ,^ i Ujijj ^\ JU ^ U^ ^-^ 
(^^ U"« tli3j jt^la V^j^ ^"^^^l J^. ^-r'tt^ ^A«j\ 
^Lr ,^ U^ jLs- Ix*^ ^^ L^^^ i^i-ii ^^^^ i^jj 

tj ^Ju*5\ L-4«ci.j L^l ^jl Ji^X^^ Jto ^ jts^ 

N 



AA 



^3 {^<^j S-r-*2J c g ,.^ - ..» CJ^3 j^\} La3l <Lm4c^ 
^^-IfcX**)! <U-.^«d- lSx^ Uv^ LaJill ,^J^ IbJjj Ji^J cb .1 
^j Ujb^Jj ^-^r**^ J-J^j Loil L-Asi- JjoO* JU 

j^^lj La3^! ^ jjjj ^^1 y,j u!J3U J^ ^j ^b,l 

^ J^. J^ tliU^ ;^,yLi [^^.y^^ Ji^ ^^j\ HsJIj 

^*f-» JW u/-^ ^/:-fr Ajfe;jJt J ijts. c,,.^^! J«rl* 

l^J ^j^l ^^ ^^1 C^J;1 Jj ^ U^ ^[^j 

^.wo; ^ ^\ cJ^\ lilj lL^ JuJ LLitl\ c^ 

Aib^Job U>^ ci^'J <*>-^:**^ (^Jui ill *^ ybj (_>x^ v« 
j^^ilS jix uj3i iXj u. ^.yg) u-i-33 iJ! (jUf-j lLsxx) (^yL-i 
V,.^wa? t^jLaJ ^! «lfi <U-»-j ijj^ ^r-^ <U^*^ ^j Jill 



AV 



^Ic iJUl ^^1 ^Ij ^jJb \ss^J jLs. J^^^ ^ 
j^yLi ^^^) &\jj\ ^iljj ^j^ j^j \^\ <UJ^\ L*aiill 
Ja5»-U j^ J ^J^ ■^]}^^ j^ ^*-^ (j^ ^^j^ 3 liT?^^ (»*-^^ 

* JW ^J^ J^l /^j'-^^ e;/:?' ?r^^-? 

aAJo-I k-j^-j*^ Ji-r. J^ ^5-^Jl^ ^i;:^^ L-^^ CSj ^ 

ju^ Ujj jL), u. jJi jj c,.^ eJ^ in LiJiii 

U< jJl jj Ujj ^Ij «_^ ^ ill cJ^i bdj tK«^ 

fj^S^ ill ijjill ^ *^1 'k^ ^ ^ fi^ <U^j l!JC«^ 
tl^ ^^1j ^1 UftjJ jJi jJ j»^t^ ^j ill J s-^ 
cb,l Hj J U&, J il\ J w--^ uji-2J ill (jJ^l c-ffi^ 

^IjwI Lj^ ^f^ jm ^/U CJ^'^ Ja jjJ ^jd 

JjoO AJb;J C.b;l ^j Ui^jJ illj L,>^ ^ t-fl-oi ill JU 



A1 






^ <>< Jet/i ^5-'Jb cT!^ ^J^ ^j ^^ J=-j 



^J ^j tl^ ^^. U ^j j^ J L-a3 ill lUj 
tlXx^ ^..5%^ Ujb;J l4>1 ^JIJ J L> i %.n'> *j^ ill ciJj 

c-.^ ^y ^ ^1 Jlil ^jybj JU ciJj gV,l iilS 

lK^ ^/J Jilt ^ ^ tl^j jcr-i Ufc,j illj 
^Vjl ^ ill JU ^^ ^ ^^1 ^ t^Jr^ ^ j^l 
iiibj tl^JS ^;-£»-U Loil iujl Jjljo Ujy,j i!!j 4^^ 

L5^ OT* ^tf^ .y*^ '^^^ UJ^ f^J"^. J S- •^j'*^-'' f y 
e, ^*>^'> cy diU J L^l icj! J<^. JU ^J^ jJL& 

aA;JJ1j LoJill j^JlC iSJjJ ^\ jJfcj i^U JJiXi Uib;Jj 

Jjwu JU tl5U^ (j^^ ^^ ^"t/^ >r*^ '^^^^ cT* ^"V^ 
ij^ '-t^r ^ ^' cr* L/iJj^ J U^( " 



AO 



ill J JW yl* uJ*^ ill viJj ;^^iuJ ^1 W^J L-,.>^i 

c:u^ liii ^^Ir ijJ^l J*^^ 2?Ul:u-il ^^JJI y^J ^^yu> 
^xijlj L-*^ li\ 1^^^ ^V *^^-* i^ ^}^\ jts- t— P,.a> 

^Lfj <L-^4o- v^j l;,^^ Lma)^!! ^^^ ^jAsiiJ* U «-^ ill 

^' <J^ iiT^ (^Jurf J jju lL$>x^ i^jf^ U.y***'J *V.^ 

<Xi^ ^^^ Uj Lli3;> Jl^^ \^\ h^ J*^. WjJ ■j.».»;»ffl.> 
• * •• • U 



Al^ 



^j Lli^ L5*^ ^^*^'' *^^ L5^ tJ^ t^J ^l^ *^^ 

^ ,^yu> u j^ l!J3J Jojji C;--:?*^ j^jiij ^\ ^j W.^ 
1^ ^^^ ^ \s-j>- ^j^,jts. J iii-j LliOj ^^jLi (jJiJl 

*^*J JW ^j^ (^ U l!J3j j^ t^ J (.«,«-«a} ^ 1^ 

ul^li (j^lji ^^.Jwj J ^j ^ t^j^l? «J^! <U>^ cl/io^l 
^*f^ J*^. J^ liT* ^li^ iji'j^ tiT* ^-^/r ^r-^ '^^^ 

^W J*^?. J^ tl^*^ (jj^ ^^j^ j^ *^*^ c^ "^li^^ 



^1 v»,^ Ji< J^ ^j^^ ^, ^ <^* J^ 



Ar 



j^yuJ lS/>-\ '•^^^ «-«r^***^ uJs)(i ^^r< i^^iib' J c,»^;*^' ^j,.a£>-j 
jJx JJJj *J c,.»,*>a> fj...4^j (Jft^r^ ^^ ti^ (^r-'*^ J ti^ 

^^ j^iib* J ^^^tt*^ ^^ Vj^-jLs. L--*>rs- ^ U^ ^Ix ^j|^ 
1-^^ jjjj^P-c J iU^ ij'W' J^^ t=r*^ ^^Ju*i (^c^> tlflJ 

l-^jss- ^^Ix <L-^«^ t^ ^-0^ T"^ ^^.J^ J \^^\ «LjUj Ja*j» 



j,^^ Ji/^j (<*tf^\j yi:>- (Jx LsJ^jsS] l::^!^ ^^li 



J LmaJ iJi JU LJJj iJ^tji U^ ^"^^ L5^^ J^ **^-^ 



Af 



^^^^---♦tfS-j iU«J ^^ !^ d^,.j^ 3 ^J W^ ^W 
^U^lj U^ t:;:t*-^j ^-.♦d^j ^.^j^ Li>y)l -•l^ 



=^ J^ d^) ^3^3 cT^^^ cT^^ ^J U^ 

^--.4^ (j-Jwj ca^;> ^\ ^J j^jlj '^ tl^ii "^ 
JU ei3 JcLl:i5 eUiJl ^ l^ b>^l ^j^ ^l^ JW 

L-ai 1i\ cjJj ,^-4d. J lii* ^'^ ^^ S-^ U-*^ 



At 

^^U51j X>UiiS uJ3i n^ ^jL-^i^Mj 4j:r-^*^ J i--4^ J 

Alls l^Jx iXj uj)/2lcj Jo-lj i>U i^^jiifSi U^ ^-x-Jj 
:%J vl^i cJ*-tf ^^Mj c:^! 



AJx ul;«*. i i U fc}jj ^-»*-*a} 1^1 Jto L»4f^ L5*fr?* l:;t^'H^ 
^^^x-i <-,>- S M<i ? jll-ti <U^* ^] l^..M»»rv J ^^jM-^ks-M %-^j ^j 

^^yLi J 111 ^^Ufi>^l ^vilj l!J3 j ,Jx J^ S^'-*^ j\JLe\ 
j\jLc\ A«-Jj ^ j ;>*^A > i!l JU ^;w«^ jllLcl iuJj illo 

ax«Jj ^; «^*,«i a . v> lI^j j^U ImsjI <u^ J*^, l -;-^« 

A«*-J' J JU U.^ UJ^ U^ill 1^^ UjJj C-.>^.*.>aJ jUlcI 

M 



A> 



^^J U^ U^^J ^I^^J k^ ^ JXJ ^\ Afu^ ^ 

* tl^lj ujU-^i j^^'^lj c:^iJl ^„^ 
cT* L5^" ^ u-'*^^ ti;^ 7^^-? <^::-^l ^--t?^ J^**^ 

aillj <l1c j^ S^^ ^ ^^}^ ^..j^ ^ ^-^ j^ 
JfcVxJ t-T-'^rr'^ ^^ ^-^^ U^>;"*^ cT* ^t^^ j^ '^^ ^^ J^ 

iU^ 0^^. <*-^^ UT*^ <U-mJi jJ-s IfcJjj u. ^**r^ ■j»< 
A^i*;^ v_ i-rr' ^^ '-i;^ L**!.j^ tiT* "i^^ t'^-ii fcX>-'j Lmo}' 

La)\ iU-J Jjou J to tliC*^ ti)^ ^-^J=r (ir?*t;^j '^^^ 



v1 

^Lc t>jj L-^w^ ^ l-^>- (jrr^ (J-"* ^'^r^ ,r*^ ^^**^ ^^ 

\ibj^\i Utfj* <^*-M-j (Jt3^, t.-.'waj jj^ '-i^ t:;:;^ (iT* '"t/?" 
cLmsI iii-j ^j^^ L^^\ hi^\ ^^Js. U&L>jj 1-t^ y;*£^ ix-ij 

^ \^\:>- jj-^ jj^ 1-^^ _y^ iU--J J Lrfi)! <U^ J^. 

CSx^ to J^ (^_^ i^^,yi ^ ^^j ui3to JJ*^ <-r^r^^ 
t.l,>U^ j^yLs U^ (iT?*-^* 3 ^J^ c;-* ^-^ ^^ '^^ 
X)U (j^ 1-^^ ^^--J J «U^*j L^j] ^:?^V J'>*:'. J^ 

^j^^ UJ*t^J uW.^ Jhi,:^ ijy^J ^W*j ^.^ j-^j 



VA 

u-^j i!U ^/J JW ^ Jx ui3J jj ^ ^^^ 
L^\ 'ixij\ Jj^j s->:r^ (j--*v>. ^ L«a3 11 JU (JUlJ 
J[«jjj^l v-5^ ^^Jj <-r^ C/-*^j ^->:r-3^ JW ^^t^U 
La>1 JL/to. Jax> JU ciJJ u-^^j 1^0 (^/tti L^iJl 
j^l ^j doA^ JU jjl lL^j jjjli u^^^i jj-«*»-L J 

^^.CLc ^li-i jjA) jj5»>. ybj tU^ t—C-a) tliU^j U^ ^./aiuj' 
J j^UjI <U-**) J La3\ <UJj1 Jd«j JU Lliiuj ^;-^ 

I J ^i^ h^ JUlj ^^j ^«^- JUl J«^li 

<l1c J|J *>^1j l^-^^o- X«-^i^ jjJdll ^^ ^Ju-i JujUj 
Ix^ 'L>cyi\ ^J^ ^jJ' (j^ *U5a«jl t^jJl Jc»-ljll 
^ y^-? J^^ L5^ W^ ^ ^ <jJl!l ^j^ j^^^ 

J^ c;::^ ^J ^ U^j ^^ i:)/i^ U^ blV^ J 

* <^UJ jjj! 

I'l JU \xa^ 15%-' .^^ ^-r^.t'^ *'-*'* rj"^ J^^ Li***** 



vv 



c-^^--i:3 ^^ ^t fc-!-^ -^j j^j ci^J^l (^ j^yLj to wjj 
JlVl ^ ^^L JjJ u^^--33 gl^jl iiij i!\ JU ^ij ^J^3 

iJl JU ^J^ ^-^JT y^ ^5^^ cT* ^0"^ J^ '^' c-l^ 
^. Ll>3i ^^f^li - W^l ^xijl Jj^, ^-^ ^j^ '^"-^ 
Jo-^ clii^o i^y^ l-MA)iJ\ il*jji!\ ^^^ l&Jjj *-r^»^ f-Vj^ 
Jisijj L^l X'tJ;! J-^. JU i:/* ^r*^ 15^^ ^ l-f^s>-^;^Lc 
^J^^ \J^ "^-j^ ^^ -^^ tl^U J-^ S-r^ ^j^ 

^jj,,^*«o- J Ax^ JW J ^,1^ Ss>-\ {^ ^**n\\\ J*f-li ^U 
^JLJ jt^ ^\s>'\ c^-^-^l l!J3J ^ip* jJLc- ^x-J ciJiJIj 

^ U^^iU£ Jo-l ^\ J^ ^^^ 

^1 1-^-^3 JiUJ Jr^J ^__^j\j ,j-:j IvJ ^y ^ 
Lj^ ^11 eJJ ^yuJ U-^ ^:.« ^^Ij bJJ ls> (jJ^l 



* UJ./^ J '^^ J ^.Uii; ^jjJl u--vMaJ J ^j;ti<-^ J 
S-rr^ Jl^ J=.J ^j\^ i?]^^^j ^^,^jX=5.\ t^J^j- ^\i 



to cJ^ L::-^rLt J^ U^. ^^ ^\ ^ l^ji^\ ^Jij* ^^! 

^_^ iJ! JU ,^^ L-^ ^^ Vi JU j^ ^ j^_^^ U 
LiJj J JU ^UjI HS J jjt> <)K Jl^li 4-!^ (J^^ ^^ J 

<C'Uj^ Hj JW ^ ^^li L^j ^\aj\ L^Asi^j tUj 
j^UjI if--^*^j tUj AlU JjjO* Jli,l ^^UjI <L^*^ jJLji 

d^j\ (jjjj J CL^U Jaj-; V.W^ cT* ^^ *^-? 4-? 
^j) j^j iUj Jc»-1 u-^ J^ J^ ^^^-^ ert^ 
ijjj ^ ^ Ul V^jll e^l ^^ ^-^-^^ ^:^ L5*tV. ^ 



v^ 



j^^jiL-i u^*«Jj hL yj>j "r-Jy"^^ c--^;;'^^ J^j {^jit^J^J 

^Ju-i *^1 «^ Jill iH^'*^ {j^ ^-•*^^1j W-* y~*^ ^ 
jm j^ ^^^iL> to ^-i-^ J ^*-J' j^_j:»i>:3w-sl J a^I <i;il5 ^1 JU 
lI^Jj U;.1xj ^^l h^ ««.-J ill ^^Ixj JU W-J ^ 

ilU uj3j 4jj^ /*^ j^ l-^/*- (jtr^ t:;-* ^l/T ^r*^ '*^*^' 
^^15 ^ \sj>' jLfi- Ix^ J ^\ h\j ill \jLji J U-J J 

l^bvo j-iLc iisi^l ^lii ^Jp A^ j^ l.tj5*- ..Ix. <LuJ J 

JU ^\ ui3J j^ ^ ^^ Uj£- ^^' ^ Ujs- ^ 
^oU ^^ '"ij^ ^r*^ <u.-j* 4iUjj j^ f^^^akJ jjl *J^J t\>-lj 
^^Uj J U^ jts. d^ JtXxj JU j^^iuJ j:]j5wl JUu^* J 






^Lj\ Lj\^ iH:^ i^j^ JW ^rr*^ t*^'-? uW-' ^j^\ 
JU cLjI ^U5 J'>*.-> W"*" .y-^ <L-4»:i- e^y^ iryr*^ 

?r-^j JW i;/*i^j -^jj^' S-r*^ J^ iji^^Ji iji 
^^j j^lyL«!lj ^^1 A.V JW ^ ^. c^^^j Ue 

U^ yLc ^ Jjoo j^j JU ^^ 1^ ^^jl ^^ 1-^ 



vr ■ 

^j ti^hl! ^^ j^ to ciJj ^iJ^ J ^^1 S^-rt*^^ f W^ 

tliOJ J^i u^-*a3 j^ i.^^ ^^ f .*^^ L5*^ ^^^ J^ t*-^' 
S^r^ L^V^ ^'^ J^« jA^*^ ^U5 V:,<»^i JW j^ ^ 

*^^V cr^' -^Ir^^ ^^ tl^^ J ^-r^ t^ i^5^ j 
•^Ir^^ ^^^ W^^ ^"^ J<^' J^< ^^ ^Uj ,^,^Li 

^j^^l L^j^\j U^ ^^ A2l5j j^l:iit<' JW J ^^j-.*.^- J 
:iJ • U^ ^ ^ ^,/» rv J <^::««i ^l (JkJ jjji*j j 

^j^ ijj^l^ j^l^' j»Ju ul^ j (j^U^i Jill ^-^^ ^-<5i^ 



vf 

Lli3ij (^r-^cu Vf-*^j tr^ V-^ u^v. ^y 'i^^ 

4^*^ ti?^ ^Ir*^ y*^ k;:*^-^ ''^ *^ UJ^'-? ^L-4c5- 
^J^ e;^^ J^^ Ir^ tJi3J ^ a3 ^,5^^ [ji:^j^^ ^^--*^ 

JV. Jrv^ ^5*^Jl^ l:i:^j ^^ ^j U Ci>- ^U 
'•-^^ jl hi^r*"^ c:^ S^*^ J^ ^^ ^"^ '^^^^"^ '— ^"s*^ 

i[::u*a J LU-li Uj^^ai t/^l ^1 Afvo c^li j! *§^j 

1j^^ J ^Uilj Lli3ii ^jLi ^j^ J k^j LUilj ^ 
d^\ c--^-^ Ji^ J^ ^jlj ^^ H5 (JJ/ ijli 

^^ ^ JLL|^1 ^V-» ^ Ji tKli c^-W" '-^^ 
u5>l o^ j.^*^ t::-^l^ jij i^J^^J^J ^^^-? '^^ ^-^ 



vt 

4JUj lLsIJJ UJ^J ^Uj ^l^rj^ ^^.s*****^ CT* J . 

/♦l^ ^si^ CL^i^J\ j^U (^yi5j LjUj <LtfU- ^ j^\ 
UJt ^^ ^Uc-Jj uiil j^U ^ d^'^ l^-^:^^*^ l4>^t 

ajojI <-^ J <-^t« j5>-j v.W^ lt* j^^ ^^ 4-? 

Ji^ I'l ^j^\ dcA «--w^ J^ J^ ^^3 t\j<\^ ^^ 

^^jjl ^1 JJO- c:^U ^-s ^r^l J^j ^j^ ^\ t\y^ 
HS XJj^I jJx l>P j^l «-r^:?*^ f^-^l H3 <u aj ^^ 
h,j^s^ L!J3i j^^ L-tfjll ^^ HjjJbj ^^'1 <-r^ ^y*jl 

^1 c^^ Ji^ ^U; ^,-^jl^ l:;:oj ,^1 lL^' ^li 



;=?: |J J c;^~^'l ^r*^ l,ff~*^'l ;:;^« j^\ Ji 
t_,o-U <uj <_>;-4; i5_,jll ^-.*s>- jjii jjU- yjJjfjU 
^Mj ^^1 ^Hlj j^;i ^,^ y^ ^1 ^jLi uLS3 

*^1 Ail; ^uX) ci ^^. {J^ ^J^^ J^ J^ ^}^ '^2--a>. 

^t< j^^ ''i^f" ri-iC <U*-J igJ^J l^^y>a'> jj^ <tl*a>- J ?-{/' 

U^ jLs. ix«j J U^ l&jki d ^^ '^^ ^yj uy^ J 



tU-w-o ul^ J ^ ^^ U^' jts. ^^\ jj^ ljbjc>>l:x5 X^j^l 

u ji^ Ujj ti ^jui^ ^\ ^j^^ ff\ to!j ^ u^ 

* j^Uil y ^^l^ U j-rr^; '^^l^ yi>J tbcV^, ^ ^jSsT 

JLj U ,^y^.j ui*iy <LjJuJj (JJj <Uj^ j^^^ ilU d^^ 

^ j*^ * IJ^J^J c>W.^ tli3jj ^;J^ ^^ l»--«:V. 

cr° ^J^^ ^f^ ti^^ ^"^ cr° ^JJ^J * Dj^ 



1A 



^ J^b tli3ji ^jll ^ (jJill ^^ jSil Ij;j1^ 4j^ 

l^!j \^\^ ^^^j '^^yj (j:^Vo il^\ ui3t) Jli^ 
(♦W (^ J^ 4-^^^ c/-^ *-^W^ ^' '^'^ o^-* ^ 



IV 
J^l j^ ^t/T^J z^^;*-^ ^J^ 1^^ ^^}=^ j^ <^^^^ ^ 



* bUjl\ j^« ^p^! c->b 



*ljil <U;.£^lj ifU-lj <)uj^lj <u\ Ci^'j Cl^U J^ 
>Jy jjl Ci3j ^j-*l»J ^jli iSJlo %--^ (J^T^ L5*^-J^-? -? 

ix^^JJ Cj^«>ll jjlj <uLj1 ^Uj c:^-:t*^ <U-J> c:-'^JyJ JU> 
ciJjj ^^j WjJ ^"^y 3 c:-^^ i^^\ Jl5 j^l* 



I 



11 

^^ ^J^ ''^^ u-'*^^ ^U&,j ^^ jJU j^;-**^ 

Li^J^ O^l ^^1 jJj-*3' j^* ^j^-i, j^U^-l ^j^j aA1;J 

^ [^^ ^^^.wic;. ^^i ^j jjUi ^^\ J^.y>j] -^^5^ 

^^ ^Jjj <Uiij J^ iULc Jujj ^ jJDj -^1^^ J^ 
X>-^4^ jj^Li (jjjj ^jUjb;^3 yij (UiJj Ji^ uJ^lj ^sisJl 

jj\i ^^^-^1 J^>-1 ^Xs. L>J 1^1^ J ^^IXj L^ AJ^];t> ^;.l^ 

^^Ujb^J^j 'S?*^^ ^o'^^ '^ Jj-*^ ^-r^^J ir*^ ki?5^ ^^1^1 
^^-i> ^^«Urv.l ^J^J z^);'^ ^^ L5^^ ^L5^ U"**^J 
*Jj)|^J dx-J" j^y^ Ujb^ti ill Jlj <Dil Ujb;J ^j^:^ J 

^^ ^^ ^^r.^ LtMij ^^ tr-^J l^b^ ^ liJ^^ 



16 



* V.W^ M^ 



* C^.^'j U1^^ 4 ^'^ tiT* ^V 



jo-1 |J-s IjJ aaIjJ ^^ J L*c AJi];J ^^ cli/ J 
«jjjj eJj^ *ib|;^ iil5 ybj ^U eJdJ ,^j\ ^!i l^ 

l^ J I^J «jJj J >%Jb5; J Mj ^^1 J^ <-r>:r^ ii,r?^^^ 

UlSj j,j&);J H5 Jjoo ^^ UU i^^iuJ -tj^ 4jJiJ ^^ 
oujjj <U*a3 J^ <uU *VJ-^1 ^\^^ J^' u^ -^^^^^^ 

aJ^IjJ ^^j \:«^ *ii>];J ^;1^ CSpj ^^\ CJj (^U 
^jjj iXjU {.^r-^ (-kji t/^jlj cr?^^^ ^^' L5^ M 



K 



IP 



i)^ ij-j tj lL-i> ^-J;*AJ jj\ Ito^jM^ci *^\j ^j^j ^i-c Ujfej 

«_-..4^ JU ^^c.At f\ AJii\ lxij\ ij^ ^ <— a-a) J 
^j^Li ,^^4ii*lt ^ihll ^,--Jo" y> ^j;:^J^J 4^^ J^^' -^W^^ 

i^3 ^i j^Ur^l ^*i;^j ?;'^^ ^J^ ^^ ^j^ '^^^ 




^j:j^ jJx l34>j U ^^-^* tli3i^ cjJjj y;*^ ^ y>j 

c^i ^j £j*^^ ^-5 ^'^ ^ ^''^ ^-^ ^'^^ ^^^ 
Jy^\ j^ uJii; CJj ^ Uji ^^*-J' ^ -^ ^5A^ 

* ajj^ i^jjij ^y^^ 




cjjil "ijLs. ^^\ 'ijLs. <uJU- ^ Hiw« ^j^ JJ ^U 

i[*bJL^ jy^ 4_J/0* ^\ L!i3i u-*^' ^.J^ cr* *-r^V 
^yLi <dL« ^ iLj y&j ^jLDlfll\ cJ^ c-.^^* ^^ yst>^ 

^j L3U5 Ujj^ '^'' (jj^j ^j^ ^5^. 'V.^ ->^> *^ 



J[:j\ [^ la-flsTj ^j\ a[^-wj U^ 'ij^Xo \^UA3 L->l:i3l 

^-x d^\ ^j l^ kosr; t^JJl j^^\ (^Juaj ti cJ^j idj 
CI^^-mO-I ^J<i U,---^' ^^ U*a3j 1^;-^^ ^^ JJJ^ 

^Uj i^^^i^t^ L, o » rr'> J ^^.1^ ^j l^x^ t^g,.A*j l^x^ l^:^^ 




c jjl ajujl ij cjj\ 'ixij\ AsuA ^^js^ t>^ Jl5 j^U 
»dA-o\ ^^--Jo «jJj jjli ^^-yi J^Xs--* Irj^-* <J^ ^^ ^ 

L.*i« <)J ^^J^ ^ u/^ ^h «J^. L^^*^ t^^' (^ C^ 1^^ 
t_»^^' ^^li l!J3j^ ui3 J ^l^ Uli iUjj^l cji^ ^jU5i!l3 

JjU\ 4 *Uj^ cJjj iL-*^ ^j jA-.-il^ ^r^^' <-^ 



It 



crt^jl; L5^ ^'^^^ ^^ ^. \^^ <^y^\j J^ ^^ yi>^ 




dp-^« Lj|j l^ (^^ jjb^ ls>-jk^ Li-Jli)l 4j-«i.^jlj 



Xi^x^ 



<U«*J 



3l>. 



u-^j 



>l^ 



C^-^J 



dllwJ 



JU 



uJO/ilc: l< Jll^ jJ-& Uj^^i^j ^-^ isiL*.* Acj l^:>-jl£v. 

* Vjj-^ ^'^j c^^^ <»^^^ i^^j ^^^-^^^ 4 




jjw3 4 ^jir^ a ^^-^ (^* ^/ ti^^ izJsj^a^ \^^ 



j^j:;«j J <U^* J <LU ^ife^ l^liw<! J, jLs- h^ ^ Ai^ssu ^ 

JLjUj J a*/. l*rj-i» ijjy^J ^^J L^J^'^J <U— J ,Ji--i 
ht^ ^ <^.U ^ [ji.j^ J '^*--' cH^ t^r?^^ J '*^**^* J 

H^l ,^, U« ^^\ ki-^ ybj X-.a::5^ .^j^i^l V^^ 
Hj ybj <diL« ti Vj»/^ ^. ^'"^^ t^^ c^r" V^^J 

j^l ^ L!J3i j<W ^j*i;^ J ^j^ J ^.^'^ ^<^ ^ 



6l 



Jlc UL? jttf JcJj c>jA«J\ ^ il^j J*^ "^^ L^.J*^**'^ 
iwksll ji^ <-r^^ j--^^ tl^J,^ ^U ^;;-::-^li ^^^tr^^ 

ij ^< ^j <U,.^ jj^ jj^ ■ L<-^ y^ sj <L-^ks)' 
« U »4> r^ > ^ ^-X%«sj <U-.^ ^-^^ {J^.j^^ ^ <U*.*»^ j»A>- 




c^j;^ UU \sSji jLs. <&Jj fc--^W" ^ir*j ^b"^ y-^ **^J^ 



CA 







bIjfjJi i^jU j^iUirt ajjUai iJ^U JbJi j^^l Uj 

^ laiL-^ ^Is irj^li ^^ jy^ U^ r>^ ^^^^ iJ^ 



^ 



\ laii-^ uJJU- lil:;£i>l ^li j^ULaJ! Ijn^l Ul 






cv 



'ijya i^^j ijji3 Jxli Cljlihii^ C.-jL-rs- ^\ ^ys^j 

1^ *^^V <^^t^^ 




fr 



in 




j^^l ^^, ^^y ^ ^^. lj^\ J^ ^<^^ t^-^" 

^^J^\ U^ :^ .Ui) 4 bjj^ Jjl^^l t^^ tiT^ /^^ 
Ijj,^^ J^!^^ ^-Lil^ ^^ JjI lil^ UfJ.*.*^j ^Ui3 4 



^ ^.^5jJ\ aU^y h\^,< ui3i Jiu^. :^ Ito;^- ^ 

I 



Cl 




jJ^U ^^j\ d:;^^^lj L3U5 \^jjbi j^U ^jJl L-.4C;. 
%-L3ji J c. ^ji <L-./4»>. f j*^' **— ^♦^ ubULi U^^i.^ j»^-!« 







CO 



l^iil Li^^l ^^ ^j ^V^^ ^5.^-:• iilU!^^ le^ ^j^ 

y^ ^\ j^j ^^V ^U^ ^yij *V.l^j UL.vd:xr;^V 
^Jiy:^ ^j^\ Jj^ ^i^^-^ ^^}jjj (^Uk^-* k^j^j 

j\ bUjJl ^'Ijj ^^1 Ju,^:!-^ ci^Uj^^ ^^ ^'<i Ui 
J^! L^j^ J Ito^" ^li V.^J>^^ '^'^' t!^^^' '^^^'^ 
^^j\ tl<li Jk<j :%: ^rr-^^ >'^ t^ ^ u^j*^^ 4 




cjil LjUj 9)31 LjUj yjis Xxj^ j^j! 4^^1j 



Ci^ 



i 



iL lass^ Uji^yj p-1 uJ-a) y^J <di^ ^\j c-j1 cJ-^u L^ 
lt*-? J (J^ ^ u^ %^ tr>^ tKSJ^j aUjU LjVj ^ji 

4 1^^ la>- e;^ l:J ^Jtr^ ^ j ^^ yt> lJ^\ ^'^\ ^^^ 
^fcCi U>^Li-4^ ^-iil^ jif^' ^' J (j::^'^ jfr^ *^^ 
<L-i3 ci l^'if J-<^ ^ ci^bil^ ^jl ^rr^' <^^ t:^^ 



:ti <0* 



^JJ^J 




it'jbjJi jxj ^ Ui iJ^i^uVo L-i^ er*^^' U-y^' <-^^ (^^ 

u^j t^y ^ L^ i^^ u^' * j^r^^ y^ j^^ 

^\^ Ui ^;^- ;«i^i ^^ ^r-^' J ui3 j i::^L^ ^li 

♦ cUij J JjL^l j-LdJl C-^ j^ ^ylj 4^jJl Ji^ 
\j\jj]\j ^% uf^U:^ U.^ U^ J*^ l3l LL^i ^lJ^^3 

1? <LLii3 ,^^ ijt^^'^ ^-'^ t^ t^ (*^ J lJ^ ^J=^ 

^^j £=1 ^ ^j d^-U)^ y^^llj Jl^^j! L^.L^. 
^^' cr^ (T/^ r' ^^ ^^^ -^"^ ^^ —^ 



^ 



tr 



1*^ 



1^^. {J^ ^ ^^ C^y^^ %l^ 'ij^Xc ^ ^xlajj J^j 

(^ Jji (jU IjIj I^ ^;j«X< c-a^ ^ Jj^\ u-d-aj J 
^1 J^ U\j ^;*Xo eJ*^ ^ Ji\ ^ J^\ 

^ Ui >^! ^^^ 7r>r=^ ^"^ 'iij (*^^ l5^ <U*JI J adi^ 
(*^ cT* i?^^^ ^^"^ ^^ (j^li ^j-^-* *— a^ ^/-o J^^ 






j^ ^^l^ Ui ^^Kic^l ^j^ ^^ ^ h^ 4 f ''^ 4 

4 ^;i3«Jl <— 2*^3 j^U ^;^J^« J^j * J^\ -^j^ t^-? 
iUjL:;^ y^J^ <^\ Ll^U J^ ^^i^ ^r^-^^ 3^ Ji"^^ ^"^ 



Ci* 



^j^\ yb ^\j\ ^ ^]^ ^ jxJ^\ y, ^\j^ i^ aijyj ^m 
tliCli ,j/-i yc-Al ybj ybllall JJo«!l j^ ^jjl ^*;i^^ 



* tb^ 4 tl^'^y* s/^^ ^^ J^^ H^^ ^^li 
^.^i ^jj ujLaJ w^'U- J^ ^^^ ^^^. tir* ^ <-^-? 



jx^\ yb ii::^ ^!^j j*-X\ ^^^^\ ^ IjLs- i^^ ^*?jV ^^-^ 

JA*1J ^Lo ij^jt!! yb ^cjj\ ycJLl jo^li ^^1 y> ^^jjl 
Ujbj i[xjj^'\ J Sjt^\ ^-y^li ^j^l j^j jj^l y& uf JJl 

^:<-i)l JJxli ,^ l^Awili ^^jl ^j/-i ^^Hall ^l:JLdt 
JAxl! ^jbj Lirt^J ^'*^***' cA5^ *^^ yfej -^1 jJb (_5Joi -fclliJl 

l»^ ^>» 4 c/i^^ J^fs^^ ^j^^ ^ ^'^^ "^^ J^^ 
^yi jjotU ^L« ybj^^ljjb ^^jJl jjotJl ^ LjU^JIj 

^^-jjU:^^ i^^iy\ ^^.J^V*^^ ^-^^ ^j^ j-^j (j^^^ 3^ fc/*^^ 
^ttt-^j (j!^*-*^ iJv^ <L3Uj ti <^j^ ^•ii>j ^;£i-i51 ci l♦J^J^l 

^^*'cJ\ c:^*i^ljt< j-^-f- Ij^j LH^^— J ^-1^ ^c::Jl ^^11 

^Jb^;-^ ^yu ^1 4 dJj>-\j^\ JUi ^UL JL ^li 

H 






jA«iij ^Uj ^^\^ j^\j j^\ ^^ j;ui i^ liftij 

i^Wl\ 2xjji\ ajjtj ^JAil\ ^ ^;^\ jj^ ^Lc jtJ\ ^ 

^^/. J JO iii i^m jIac^ii Hill ji ^- J lKij 

^^JJk«\ C-^.g ",^ ^Urs-La! jjjlt*'* U.^:^ Jcs-lj jji jjUjI l^i^ 

* <lLc e:^^/^— J t/33! JA*11 j^*^ 
j^ iL-j ^^ (Jii J-1' lil <u<< *^-_^ 4 ^"^ tJ^'"^-? 



j^V 



J\3>Pr\ Hjj c^Ju^ JW blii ^j JWj 4-ji^3 JWjis> 
U <jJj Jo-b' J iij\^\ dx;j\ JjjO* JU Jli ^JS 
^jLi j^jcss-^l ^xjji\ tuL: t3^;5 ^^ic^ y-^ ^'J\ J^<3 ,<iu 

\j^ il! Jtc jjus.^ JU, jj^ Ij^ 1^;:^^'^ iJ^ ^5^ 
jjcj- (j-^jJl ^^ jl^j JU jj^ ^.^ jllJ ^J^J^ J^. 

i!l ^Uj *Jb);J dX)j\ At^ J IjjvP- i!l iJlt^J'^ u^ J^ 

dxJ^^j \1U ^^Ls ^ Jjliii 5^1 J^ )^1 !^U Ja*.;j^3c?-1 lx)j\ 

h\j ^^ JU: f^ ^j^ J'^\ ^j ^u j-^. j^^jj 

ybj Idjj Ujb^J Jjx) jjes^li AJb5;J ^*^^ JaxJ j\j^\ 

^ .*jbjj cl^i J*^j ^jd JWj JWj^ 



J^ UjbjJ Jj^, ^i,jj J^ ^^.JU ji^ ^^A-:^ JU 

^^ (i?^ (^b"^ ^ ^/r^i ^ 4 l^**^^ C-^U jjc-j 
^''^ 4 *^^1^ J'M^ cJ-a:i *ib1jJ &x«j j4Xx> Ij jc»- J 

U c-fl-dw jA>j adl^ 4 *^=^ <-:i-^^ ciJJl jjcs^l i^juaj <)^ 

(Jjil <uLii3 L^si^ ^\ij <Lilj 4 <^^ JU Jli* ,jU 

4 ^^^t^ c-J^li c- g ^ '» J <U-M-> j3^ ^^ilj 4 i— ft<^ ^ 
c— s.rr't J <U-uj 4 fUji *^j|^ qLjI <UJj1 (JJ^ (jrr^ 

i!w*^ ^^yLi t—sLai J <U-M-» 4 ^ J ^-^ ^-y^^ u, g .a. > j 

4 ^^r^* JU Jli ^l* * L.A^ ^isL^ (jir^ J 
Jli ^10 Jj^l JW Jb^\ L.^ ^f^ iij\d^\ HS 
jjus.^ JuiJjj Jjili JW Ji^ li,!^ i^j^ 4 *^> J^* 

i^j 4 JUl L^j^ ^ iUilj ^5ai:'• JU jU (^li 
v^y lil LL5i'l UJ^ ^^1\ JW ^^v Jj^^ JW j^^^ 



Pc 



^^-.^ l^Awsj J^ L-**^l ^^ Jjj Jii^ J^^ ^y^^li 

\LJii <^j^ ^ <u»Ljli t^jd L-.g»^} i^^Jill c-^LtfU j;«^i 
j^J ^^^^^ ^^-i» ^, o ^A 'j -tj--i> ujLaJ ^U lL-2> J«^ 

jj^**4Jfc^u> JfcXXJ -t^^ tli^ (J,^^ <UsciU -^^c-ii t— ff«^'» Jii^. 

H< ^jl j^t JWj ^W J^l y^J 

<j (jj3i .^-y^l J j^ 4^4x-j ^jLi M^ ^d^\ ^^juJl 



jLs. ^\^ \jS:>- J^ ^\ ^^ jl j^lt\ ^p.^.vj ^j^ 

(JL^Ij jj^ Jci-ll3 ^J^„J^-^ J <L-«40- J iJUo-*: jj^ l.ij5>- 

^,J^3 <u.4o- j^ 1^ ^Lc a^slij Ufc;t> ^^ jo-1 yj)j 

-^i;j ijojij ^jj <dj jjj*j t/jji ^-^^m jiy y>j 



^^\ (ULiii iL/ici. j-Li <UiJJ J a;:^^ JU JlJJ jjU 



lT^^ J^ ^i}=r ^ji^U^ ui^' cr^ir* ■'lT' t:r* >r^ ^' 
y^c a!:;.-* *ib|;jJl «^^^lj *J^5^JJ1 i^Vj^l? -«';^ ^ j^ 

<u^|^ <)jU ^ l.£j:>- uXr^J <L^A^ lIXvc J-rf2£^i jj^lj 

Uaj J ^ ^tj ^^ ^jbj J^.:i!l JD.1 J^^ jjcw cUj^ 

(^b^ a[xj^1 i^yLj jL£. k^ ^ jL,s- ^\ ^Ij &j JjUi 

1-^^ ^«-C <U^« ^ <Uw»,rs- <J, CS*-^ U) %--«^»5j- L^jou^ jj! 
^U (^^^^ {j-.J^'*^ J '^—./♦cs- ^ 1-^J5>- yic <Ix^" J i.»*A.:>~ 



^Jf^J^ jJU UbjjyLi a*l1j \jj^ Jsxi^ ^\^ ^ \s.j^ 

jLs^ j^\j^ js>- ^;^s^ c-i^ L5^ J^ er* ^ >'T" ^"^ 
*'^^-' ^^ uJ3jj dU^li J to ^j^Juj ^—i^ Jjx> Ujb;J 

oVj U^J wk-J i^ (V**-^ ujLy J j^jJ Jljj ^li 
ti ^^^^1 L^j^\i (iJfAt--* 'J^l^^ C-->UU -c^_^ J tX>.J^ 

i&-a}j Ujfe;j J^, ijAfr^ J iji^^ u.^ ^\,s^^3 A^^yi 

aj JjUi ^jJ ^j\ ^ Jjco ^^j J^ J>i2J ''^ft^ 

Ci^^^j -^J^J ^j\j <U^jj *^ C^fi: JU Jlii ^_^li . 

iU^Ui Uj&jJ jLj^ ^:^\ i^^Vjj J^^ '^^ ^^ 4 LS^ ^ 
^ ^|js-\ L^aJ- JLJ ^ijj <^ fjjf^ ^ ^^ tl6^ 
j^yuii *j&);J ^j\ \^ J)}^ ^^ cr* ^"L^ >r^ L5^' 



jjbj cLiOU (J/i-^U *^5^t) ^juJ J JU cLJi <ujj^ Jj^' j^j^\ 

t--ytflj ilU (j^^ f-{;j j^;^'! ci 9^^'^^ i!*^ji3l ^-;/^' e^^ 

^-^l J Ujjj ^^y^ (j;j^. ti^ (^^ 4 (^l"^ '^^**^' 
Uj^.j \^*J:> jLs- Ars-1 (j^j^ ti^-? c^"^^ 4 J^'^^^ ^Swwi^M 

^J^ «^J^ ^/-i s-^ ^j 4 "^LS^ *-^ ^j^ J^ 

liVo j^ jjbj U-^y--c ^\ J*^x) Jll,l5 L--i» Ja*^^ Jto 

^j^jd ^ ajuj 4 ^^^j ^ ^■>^' J^ J^ (i)^ 

C--y3J J JU (jw«tX-a cJ*a3 jj^^ -^^^ ^J 4 ^L5^ 
ti^ 4 ^J^J J*^ L5^ ^/^ ''lT' ^-^ 4 C,t."t^J^ 

y-^ L5^''' CT* ^1/?" 7^ "^^^-^ U>^J^J J^ ^^^^ uJ-uJ 



h\j Jj^, J\yo\ ^j\ ^/j^^Jcwl a*j^^ J l^j^ c-^^- 

Ji^ ^^^^ U&;J ^^^-i-c ^ JoJJ JU Jli ^li 
Ja*j Ujb;J ^jy^ J JU Jjiii' ^^ iULfti ^i^^Lc ^1 
d:x-o ^^* yi^ J l^^\j j1 Jc>-^1 cji^-ii l;3c»- yic ^^1 

^^ U jj^ J^>-J j^^jJ^ ^^_^^l^i IflwO j^^Awli ^^^-iJj J 

JU jA-j-l Luj\ ^ 4 J^ ^^^^4 r^b^ ^' ^^ 

^ 4 ^i;j ^3 % jj^ -^j^ j^ 4 r^b^ ^ ^b 

^);J ^*_»' ^];J iidj ^! 4 j,^!;^ ^ ^1; L;>b'^ ^J^ 



n 

Jill LtM J^ ^^\ ^j\ d ^^-V JU Jljj jjli 



JW Jb^cl Hj J^^v t"^^ 4 ''^r^' J^ <J^ C:^^ 

jIc JW «jJj 4 jJ.Jl Lii^y Ul l1<3\ <uUi jyj\ 

* iU^' ^J&j ^j^ ^ J^ ^'^ Jy^ J^^ 
C-.yiiJ jjl <uLiii Ujb^»t) (Art^jl? ^J^ ^'^Vjj J^^ ^J*:^^ 

ilU Jjee i!U^ ^'1 ^,/J jlj^l h\j dj^\ ^J 
JUj ^U JW ^Li: ^;yt\ ^^ jJli Ujb;j ^^^^^ iixjjl^ 

* JW j^j ^f^.J^ ^^^ W-^ 
^;ljc»-l L,^*^ 4 !;^<^^ ^j^ <-r>^* J^ J^' u^ 

CSi\ iUUi Ujb;J ^t^ J ^ ^-Vjj JW ^J^ ^jf^ 

JW e;:'./-^^ CL,*^ ^/l^' ^j^ t^j ^^ e;r:^^ J^. 
^^j i!::««j Jajo ilU ^^ LjUj (^^iit^rji ^^Uj ^lU 



Ta 



JW i^li ^J^ ^^ Jd^ iU«4^ li)j JU Jli ^\i 
\.ijs^ jLs. ^j\ J J^li jS^ «-*«: ^-i^j ]^ j^ Jj^xj aj^ 

*— ^J l"^ J*^/. J'^^ jii^ ^'^ J^ 4 J*^ tt^ 

>^=>'^^ JWi J-t**** i— 2*^ J ItAswlj ^(^5>- j^^^ jdC^ «--w-a 

t^-^ (^^-*«J J iia»*j J ajU ^^-« '-«j=>- (j:?^ Uv^. '''— '♦^^ ^ J 

4j^Uo-! Axij\ J^c a;-v*^ jAjjI Hj JU Jljj ^li 
a^j Ji^ i.^,^3^ (Aij\ ^ ^^^ j^.-^j ^ aLo'uiJi ^^3^ 

fULjii ^^yt^ Ui^^ <^ll^l ^j^ ti '^{;«*^* JU JU j^lj 



rv 

U j^ ilU ill jl j^^ ^-^ l;j^ >r^^^ 4 ^"-^^^ <-r^^ 

^/**^ ci W^J <Lwi^ <*^^i/^ Lf*^J ti^-? fU--*^ A^oJul 
ill >^U1 ^i: y^J C-^j^l JW lL^ ^> ,J^ W 

^/J uJ^ J Ujb,^ e;--«^J t^trr^'l ^5^ J^l ^Jj JUV 
^ .».M^,^ r^ j j^:Ji Jaxj jtX^ t-J^j l;*^ UJLr*^ CSx^ 



n 

Ss^\ji\ ^^t!i\i ^j^^j^ Ja*j (^^j ^\ji>\ h^ ^J^ 
*^>-\j J^ aLJ! cIjj Ajfe^j ^ ^^ ^ ^ ^Ijs^l <xxj^\ 

\J3r^ 3 ^^ l!J3j^ l;^-^?^ J^J cr:*../*-^^ v:;:?^ ^ 

* 1^ 

^i!|^ IL-i* ^^W *^<^\ J*f-^ /»^i^ ujLaJ A-Jill C-jUU 

a;i«itfli ^^--i» ujLa3 Jj^^ aJ&;J ^j -t,^ lJ^ -^j^ 

<j UjbJ^^l ^^:^j^ e;:r*^ lf2..4wJ> ^^ Jli jjli 
t-^wrfiJ j^\ tuLii Vl^li <^-iJ 4 ^^^ /»-**aJ|^ ^/^ 
IL-i lil i;/^ c-^* J -^U^l ^>^ ^^/::i ^>^ 4 ^ 



re 

^j ljb;3c>- jcs-' ^j^ i^y^^A^j L}^ ti;!? u"*^^ i^ti^ ^l^^ 

jjj*«,4cv ybj j'jc»-^M t— a*^ jj-o l^iiili i_i^j U^*^!? <U*J' 
* jj^j.4— iiJ' <Jo-l Jj^J Jc»-1j i^^^tr* L.Jt^ ^ 

fcXs-lj J^ u:^^xj L::--.liii «^j i,'*,tjl tujci-l lL^oI^ Jj^ 

^^j\ jLl3 ^^ J iUJj\ iJI^O^ -^^^ ^J^^ ^^ 

^iiJi -'(^^ ^—2*^ J^ J^^l; J^ ^:iJl '-^i-^j ''\^' 

^l;^' ybj ^^^Jj3t t:;.^ ^< J^>^* -^^^ ^ ^^oi ^Li^l 
^LJl\ Lt^ ^J^ s-^ i^J^ ^i^^ ^ji.j^^ ^, ^ cMj 



i'l l^L^\ J^ s\^^\ L^\ Uu^ CU^ cKli ^l^ 
U-oJj ^--i» jU j:lJbi!l aL^l u^i^ cjj^l IjU 11^ 

Lll^vJi Sij e;r^--'*^ *'^ cli3^ ^/rJ <L-4^ J Ijjj^ 

<j llJj ill ^^ ^-y'^ cJ^j (i>^j-- Lli3Uj JW c>U 

^^-.'♦o-^ MiijJ ^Uw--^*^ uJ^J (^^^^ ■'W^^ <L-A»- ^1 ^,^>.«„i>jv 

^Jl Lli3i jjjli U-aJj ^^j-i-i Jj^. -'^ ^U ^1 Jl^l 
lL-i» ^^^^ ^^^ ^^j (^^ ^:^ ^^^ uj^ ^J J^ 

•*V^^ ^yC-JtH t>jj «Lll,l vl^i ^r^^ -^^^-^ lS^ Jjot) 

Jaxj JUj aJ^;^) ^^0 uliU^ ^^;^«^ -^^5*-^^ t-i-^j ^^ 
* ^^^-^wi!l i\^\ ytj Lj\aj ^Ji^ ^jj £^ y>j 



rr 



Ss^ JU J! (JjOi Jl);U s.^ ^^^ IL-i ^^jl^ ^^ 

^j,»A£>^j ^J,.,>^] i^J^ to ^---'♦'Ss- ^ <\SL^ l^..^.^.>- ^j-.,**^^ 

^3kp-l ijts. Jajo Jtoj (^jj^r-^j ^j^ ilJjf^ u^ w^^ 

^_jj^ l^h^ ^ \^j^\j Lwi£*> ^^j j^jc^^l t-jLali l^wk>> 
t-J-oi j^ a-^li Jk^lj ^j ^J^ ^^ >S>-^j f^JLj^ Jli.\ 

^^1 ^ L5^y' (^ -L.-JU :tLi»l L-^*o- ^yli iU**^ 



rr 

^^izi -*L-i»^l cJ*^ Um-Sj jLs. Jcvl Jjoo JUj 
Ui^ j^^li ^jj (j;?^ c^.^^ ^^^ ^ k{/^^ l&*^j 

'^^ t:!.'j J^^ ,J^ JW ^ ^^1 ^.r^^3 ^^^ 
* ^^^^j/iJl fcXs-l yb^ hcj\ (JLj 

^^Lfii * Ua-;^ ,-./K^^;J ui3j i_Li Ijjb ^Jji Ijjfej 
J Jet) IL-i i^,J^ ^^ cJrJ^^J '^.■'.^ ^*^^ Ll>li ^-'♦^^ 

f^ ill ^U,\ ij/^ Ll^ij U^ 1^1 ^>x 4 Vj)r^ ^ 
^^1 ^yLxli .>jj iL^ ^^ i!^ ^Uj ^u ^5^ i-^ J 



\AJi>jO ^jf,^j^ J*^. ^'rr-^ ^.J^ "^^ ^^y^ s/^^ ^'""^ cT^u^ 
Jjco J^^^jll ^^li ^-^ ^.J^ J^^' ^J'^ i^y^ l/^' 

^^Lc ill i!Uj ajU ^^ l^ 4 ^^ -^ '^J^ <-r^' 
ojjj JUj \tJ:> ^,jt^ 'i\ ^Uj '^.^ ^^ ^^ t^ 

ill ^Uj ^>£j i>U lI^J t^ c;^ ^^ ?r^ '^'^^^ 
CJ^^ IJlj U&;J ^^^^^^--..♦^j a[xjjl J^^ lL-i> [ji.J^3 ar^^ 
^;j<.„.^j <5joj^ Jjkx> ^^^Uj j^^J ^^l^j ^U l::^* Lii^lij 
y^J Jc^l^ JU Jl ^W Oc>jli IL^ ^j ^^^J Ufcj'^ 
l^j ^^^^^*«^j iL^^j^ l;^^ uli^ U c-i-^ tX^-b* (^' 

ht^ ^J^i llJi ^^ Jc^lj Ujb;J [J^..J^3 '^^^ J*^. ^^^-^ 



* ^ik^M j:u^ c^b 



^^^J^ (^ ij^^ ^^'^^ ir^ J^' J^^ J^^ u^ 

^lU il\ ^Lll 5^ ^/J IL^ ^1 ^ 4 11^ c-yli 
^ ifjjj JUb 4*^^1 ^;/;^\ ^.^li ^_/^j 1a>-1 Jajo 
j^^^^j Ia^I ^j>^xJ iLi»l ^yic u^^ (^riy^l^ *i^^\ 

l^ll^ ^i Wir^^ X-**^ ^^^^iuii j)jc>-^l i^juaj (Ji\3 ^U^ Ujb^J 
dx-Mrf c;j^ jl j^ill L^LsJ ^Js. ^Ji\ jjc&- Cl^j; u:^.*i* 
L^j^ ^j\ ^li c;^J^ L5*^ >r^^^ cT" '-'^^^ ^^^-^^ (*^ 



^illj ,j-/«wiill cJc9-l jjbj Hj j^yui (L.^ ^ ^^ j^^lsll 

* bj^ J"^ '^'^j <-3V^ 

-tj^l ^^Ij L!i3U J^" ^j^jLs^ ^1 ^ JU ^jM.juj 
ti ^ji,j^^^ l^^ji\ ^-y?|; l^ y^ ^^ ;^. y^ 1^^ 4 

^^:Jlj Ujb^J jjJUjj ^"^^^ L:;W.^ lI^*^ j^*a,^ Jlj:. ^j^\ 

^J W/^\? ^^ U^" J^*^^^ ^^P>%;?*3 ^U Jj^. |^j^»- yl.^ 

J[*ji|^ X>U^ ^S^ ^JUJ^ ^■rH^i^ LiT?^^ LT^ bbii^jj ^^i<^ 
Jsiii JWJJ^J ^.j^j^ ^j^ l1>1J (j;^^ '^ ^j 

:i; ^^^^1 Jj^' fc-^Ar« 



rA 



cL^ti;>-^£Ll ;^iii JlJ,\ y^J y;^ ^^1 SJ (.J^j dijj y^^ 

J^ UjJj LaiUl ^^1 t:;:l>^V cr^W^ '^.W^f^r^ ^^^ 
^^,;j*«.*^j LjUj Jj^ e;-'^^ h)^ U^ ijir^^3 ^^^ 

uLjC'l uliOij ^ JjIa3 ^U.^ ^>^j U&jJ .^./^j ^"^ 
^ \^j^\j L.^ ^p j^J^iSl <-i^ ^^^ ^^^ J^.^. 



rv 

^ i;^ iiu>c ^y 11^ ill ■y^y^^\j iL^ ^^^^^\ 

'^j^ tc ^^:^j^ U ^^Aj CSj\ ci-viic jjj ^^1 ijj^ ^is^ 

yi>j ii::^.w> t/jJl J 1^1 JjoO" ^L-il ijujl (^^ j:^^ ^ 
A^f\ Ixijf^ ^ ^Jj^ ^^db i^Lxll ^^^Is U-^ ill 'ijLs. 

ybj ^lij! fcX5^1^1 -^LS*^^ !/^ J"^* "^^^ «L-./*cN> uj^ 

Hi IjJu: Jjew jjjcj- *Jbj i)a*Jl 
tj^ j^ Uife^J L^j^j^ JU 4j-Ju; uJuaJ (jj^ ^^^ ^J 

^lt* tir^ ^lT* ti^ ^^ ^^-^"^-^ ^^^' "^^ ^^ ^^^ 
^ jJLi Ufcjj ^^ Jjx,, j^jj^ ^^ ^j^ ^^ 

^uo) Jjw*}' Ujfe^J^^lLc cU—J ^<iu-j /^J>^ Ujfe^J |j-ljtll 

^U^lj Lli3U J-**^^ J^^^ %-J;j -^(^^ Ct^j JU 4y-iX-5 

JU Lliv^^j-*^ ^;^^ ^^'1 j^ lLs*^ U Ji <^j^3J ^\ 

^»g.i"'.'i Ujb^J i^r^.j^J ^^yiijtP^y^ u'^^. J^^^^ ^"^^^ 



ri 



tK-ic^-^l Jjiiy:;.^! ^\y^J J:j\ '^\ ^^ SU\^ d^ 

<U«iJ j^ Vjj*^ (jrr*^^ "-^^^^ J^ V**^ 4 ir**^^ ^-^ 
«»::--w-j <L-i3 ^^ ljj|yd^^p-il\ J^^^ i^ cL-J'i d*^^ ^j^'j^ 

i!U ^j^Li ^iJ ^^ ^jl\\ i^j^^i^ U-^ ill ^^i'lj 11^ 

(•^ (J^ ?.l— Ji <U-M«>j 1^^ Ut^ ^uJi <U-w«j^ (^r^^' i^ (*^ 
cLjI ajcM^j^ (J^ J*^* <V.^ (J^^ ^^^ L5^ ir^^ ^j^ 

jii dLllsr* i! ijojl ^^1? j^^A^i-Jill Oc»-l jAj <i:i->5 lfc;j^ 

UjbOo-1 u:^v4-Jj J ^^;rr*— ^ IfUwJJ i^^ * ^l^i ^i^-i\j 
Jc-1 J*^ J ^Iroii * ^jl |%--iiJ^ ^^^ >^^^ ^ 



re 

«_jjl ^ill ti ^^^-./♦Jl Jc!-\ Ji^ <Ui3 ii (—Jj^^^l jLii 
ir-^ j^yLi li--Sj ill ^^ li ll»-i> L^jA.zJ l*--i» i!\ ^^ 

y>^ j^^-1 ti ^V-* ^^■^* (*J JV<^ '^.j^ ^'^ ^^-^ (iT^l;^ 

^j^j\ u/r^i JW ;^ Ifc^Jj J^^^l «^j^V t;:-^^ JV*^ 
j^icj-l LjUj Jjou A^y\ JllLlJ Ji^.«1 iL-*.r^ Jjsx> L"-^ 

^^ c-^y*i3-il ,^^/»«-jiJ\ je>.! jjbj (LjUj lifc^jc*- j^^^^j ^j^ y^ 

E 



rp 



^^j i^ ^1 I; j,\ U il^ij ^U JUJ jj.^ jj5 ^^u 



* ^ji 



CLX 



^ 






^U 



0^ l«»*-/«^j LU CJj\*a jliXf-l ^l^j ^^^---./isri. l^-La CUJJ 



t:i,vjtiu Uli JU ayli,! ^ ^jl^ ^j j^'^"^ ^>^ ^^ J^; 



rr 



j^ ^1 ^,jts. ^< Uyu^ lijts. 'j\ ^J*l^y< j3^ iLLc lc\j 



ir iLkiiJ Jl L-J ^ J-^j e^rriy^*^^ J^^J ^ ^-^ ^^ 

L>. j^l UJ ^^^* Sij (^ \ Lra- J^ ybj ^^^U J jc»- ksi. Jl-o 



<tlaiu ^ l:^^li t) ^ ti-i- ^ c-^ p>~ Lrs- j^^^,aiu) ,^1 






.\ u 



t^^J 



>• jjl^ J»J iari- ^^-'♦^ 



^j^\ Jj LcS^ (j^ (^ U ^ J- k>- j^t U ,.t— :X3 ^ i^ kcL 






rr 



lari> (j^ 4^ ^ J^ iJfP^S^ J^ e/* ^^^ ^^^'tll ^ ^ -Lkftj 
<LilRj |J\ ILs*^ C— > LLiu ^ TT/^ (*^ ^-'^ ^ j-^j ^-^^ 
^/♦i 'ijJLs- yb ^5 JJl 5»- 1 lass- ib*^ jJfej i^^yL*l\ la^. ji>j J 
l4>l ^^.^J*^ J^^ ^-^^ ^ J^ ^ ^^-^ LS^^ ^ ^^^ 

^ l;^ -^j ^J ^ ^J '^ ^ ^ ^-'^ *^ ^ cr* 



n 

jl ix«-J jj^:>- u-fl^ jl ^1 jl ^X-J j\^\ ^ C^*J;1 jj^ 

<dll ^U> j^l t«^*aj aJvf£.U ^^-UiilS \sib ^Ixi ^Ji U^l Jil 



^^ JW cl^i j.^=r 1-^^ Ji-« t^^' U ji^ ^jLi 

•lL^ j^j ^u*J" jj^ lJix^ ybj iiw-j j^^^ ^jc»- j^"* ^sJjj 
o^U j^l LTJiij] jjli 4j c:->^ ti-^ IacUu ix-Jjjtfw 

Ujj (j;j^ u-i^J ti lLa3 ^-^*AJ uJoli <{»-J jicj- t^jLaJ 
U^jc»- jc>-l::j Wj^ jrTrt^^ L^.^^ *^***^ S^ ^v' ^j^ ^ 

j3^ ^Js^ JU-J' jj^ *-Ju jj^ LUiij] ^^^ * j»^l 

U,j^ \xij^ ^\ ^ffi ^.j\ ^ ^v-J ^ Llioli a:>Ojl 
j^^ Ci: J;1 ^^^^ * <— a-3J J Jc^^j ^5>j J^ljll J-r-^. ^ ^ 
^ huj\ ^ CSj\i Ax^' j^ ^^ ^J^ J<^ f-^ 
^J^J Jc^l;!! c,.,-^. U U,3^ ^1^ jA^- ^j^ v:;/-^ ^^ 



ijUj * Lie d>^ j^ y^ c^.^ b'^J c;."!^.^ s^*^ 

Ulj * \jd^ (jtjiij ^1 Jl^l H5j U&;J ^;--.4^ ^ (^U 

^-^::j^1 U jio- ^;-^ Jill 4 f^ c;^^ 4 (J^' ^-r>^' 

jb^i i[ijj t'*^' ^ >''^ ^^ jw 4 (*^ ^ 4 '^^ 

jjcp- CjLaJ Jo^b* ^1 CJJ,1 ^Ij * *LJis Jb4l iJJb (J^ 
JW 4 1*^ Wj iJJ^ *— ^ 4 ^^^ S->^* U^ L5*^ <-5^ 



|A 

^ST* ^ ^ST*^ ^ ^^ ^^^ * ^'^ ^^ ^^^ ir^ ^/t?^ 
4 ^L5^J *^.lj •'^^ 1/^ ^^ 4 -^L^ ^^^ ir^ ^^ 

^ib;J ^jU ^1 JU Jybi «L:«l3 ^Li>l ^^j i^ J ^*c 

^^ Jl5 J\j Hi ^j,J LU il\ JU j^yuJ laiflj ^Lil 
eu^li ^U--l <U-*^ ill AJb^J u-a^ ii -^^<-^ c- a^j ^Jb^J 

^JbljJ lijLS' <J ^l*-i>\ iL^^ ^^Ij Jj|j ^^^ «_J^ .i^^ U-Sao} 
V\ J^ib\J^} fU-^io- Lli^lfc^ t!^**^ <J!^^ ^«^'0 ]^i^ jj^*-^«ci. 
<U-^*c>- ^j^ (f^ J^^ ?y *^J \j^^ (i^^jl? ***^* 
LaJjlj U«a3j M:t^U ;jj^ J^Jj J'^f" C»g*fl3 ii X^lj j^ j<i>"l 
l^jc>- /2;r^'VJlJ ^^^ ^!^ U*a)j j^U ^1 ^\j'^ <U-4.£>- Ci^ji 
^1 -c^ 4 ^^j ^^Ij: Jli ^li * jj^ ^1^1 iiij^ 
^^^ ^}y::j ljL£. IS >^^ J ir^J -^i^ J^ '^^ !r^ 
i!lj irjjlj j:LS>l ^^ ^^ 4 '^j^i ^Jj J^ -^Li^ 4 
jjLaiulb i^fc^Vj^^ ci^As I^l3 -tLi»l ^/Lc ^^-i) 4 ^^ 
tri^ JW jj^ Ljja:^ ^jU ^^ii 4 ^Ir*^ ^^J J^^ L5^J 
t-^J\ ^^ ^^l^ U J^j ^ ^;J a:;)U ^\ JU LL<li 



l^j ^\J^\ ijLs. x^ J ir^j k^ i;^ 4 ir^ ^"^^ 
^jLi Si\) JU x^ 4 -^L5^j ^3 -^V-^^ ir-^ ^i^ 4 
JlS j^l^ * UJj lUUj 11^ t:;:>.^-^J j^J^ k^ i-^^' 
illj X>Uj i|^l^ li ^i: Culi lL-i» ill 'ijLs- 4 llj-i ill ^^.1^ 
»JLS' 'ijts- 4 W-^ ^^1? ^-^U •^U-i'l ^;.^ i>y^ 4 ^^^ 
I:U LliOJ ^/J jo^j JUj l^j^ ^1 4 X'J:. t^ L=3\3 ^L^l 

^•^-j /^^ii ci vfJb^i^ L^j*aJ ^' *^r^J (j^ti^MJl ^^Juj J 
AJb^J 4 ^(^--^ ^'J tj^w (j*,X^ f^J'^ 4 ^'^ ^1? UjbjJ 
^^iX**J Lj4X«o ill ti Lj<\^ ill^ (^Wj ^_<iu-.3 4^0 ^Ju-J 

ill^ il^ ^^ 4 ?;^ <*i^ -^LS^J ir*^ 4 ^^ ^^ ?A^ 
ij^ ^^ 4 ^^3 '^'^ -^^^ ^T^ ?/^ 4 ^ 
LU uJsli jj^Li j^l3 JU ^1^ 4 ^---^ % ^'Mb ^^^ 

cuJi :^^5i» 4 ^ ^'' !r^ J^ L>b * ^^ ^' (^^^ 

^U JU ^(^ 4 ^-^ ^1; -^^1 ir^ J^;jS-^ 4 '^J^ 



D 



J 

11 



iSi\} ijts- ^^1 4 J^ljllj I>U ^^1 J ^^Ij |^.J1^ 

4 (^ui j^t^tj Lu ^^\ 4 '^j^^ ^^^j ^^ ^/*^ 4 

laj\j ijLs- '^jL^\ 4 ^^ u^U\ A5-yij (Lal-lj ^l.£ ^1.^1 
do\j dc>-\j ^_^U]1 4X>.^jll 4 u^^l Jc^y\j (j^W' cl^J^ 

ir^-xJl 4 0!^W^ A5-[jl!^ LU ^/Lxll 4 ^<^^ ^'^l; ^^ 

(^^^;.*:u«,it jl JJc& 1^*^ j^l^ 1j1 ^^^^ 4 W*f jil--i>irt ^-^ 

ill Lto Jyi::i X-ajlj^lJ^I ijLS' j^^j ^Ix 4 ^r-^ "^^3 *V.^ 
c-^li ^^Ix 4 ^\^3 ir^ J^ l;^ ^ -^^^ ir*^ 
irjotj ^Li»\ ^^^ '^jLs^ 4 ^^:;--'j ^l^ (i?/:*. ir^ 4 Ir-^ 
l^ 4 ^^3 ir^ J^ l;!^ * ^^^ '^j^^j Lyo j^ifj^ 



J 



^i^ji jj^ V^ jij\ '^^j^ «^^'^ ^^^ u^ 4 ^^-^ jj'^^ 

^ ^1 J^l i^ (^ ^ V^ U'^^' '-^J <^ J^ ^^^ 



If 






^ ^t L^ Jjjj J^ ^ JU L-^ ^ ^ ^^ J..aA:J 

^.--ixi JJc^Jl ii^j^l yb ^5J31 JT ^ JiU J £= ^J 
k;^ ybj i.Ju3J^ S>-\^ jjbj j^^^l U-L33 ^ Jill ^j j1 

ybj ^^ ^>=r y^ s^*^^ c^ ^ >w^ ^^"^-^ ^"^^ ^c 

to L!i3Jj ^jj-^ ^i^j Ci\ ^ ^ lS^\ Jill j'^ J^J ^^ 



jWjJc»- »ii^ ^^lij ybj :>-] Irs- ^^j1Jc»-\'^ i^Ju^j ^ c^jil 





r 3 


s 


d 






\ 




-^ r 








<i 


> 


> 


p «■ 


5> : 


^ 






\r 

I«Ul^l ^ L^ k^;^ ^£ Ijj^ J^ la^ Ji^ L«- kri^ 
Ji^ ^t, LrL^Ui ^1 ^_^ k^ Jx "^ J^ ^l*^ 

^bu c^Li\^ aL^icL £=k kr;. ^\ L! ^- ^l^ j^^ L^ >-> 
\>d.^\^ * 1 k J^-* j^ k ^ ^ ^J^ j^^ y> t^iJl ^k 

^1 kJ ^»--ji ^^ kr^ Ji^^j J^ k^ 1*^ k^ ^J^ 
jJfcj -^ J k^ £=/» kri- ^ J*^J J (♦ ^^^ J^ r-^ k!>. 
J U ^^ Ik ^^-^ Ji^ j(* >^ jUxi ^^ kri. Ji.c 
^ _ybj L-Jif ^ «>*J(* ^ '^ W>* ^^ ^ 
^ L^ Uii ^Ijt^^ L^^ ],^^ J^ Si^ c.^3j^3 

L.^^^ ^.^ U" J^ yi'j ^j ^-^ ^j ^:^ ^^ k) 



jJ-J Hj ^^ ^^jJl ^t ^A ^4 ^-iljj Ix^ Js. 

^j JuLc Uj) U J5^ cu^ Law !Jli Uii^!l JaJ^ ^\^\ 
nj^ ^^ JW ^ t/jJl ^-^t -sT^ ^-Ls jjbj <^ j^ L-^4^ 

^j^ ^*^J <u*«J JWj 



2=- 


L_> 


ro 


^ 



j^ jj ji ^ y>j ^ii^i jj^ uj^ y^ jiii jxsr 

^^« ^r=^ J^ ^^ tiJo^l ^^ ^^ ^jL ^1 L«L: Joj 











XL 




•> 


\ 


^ 




\7 





^^^1 LLoui iJ^s->'\ Sjts. Ji-0 JuLc JkJp ^1 Ijt^li JIU 
u-a^ ^j ^jj\ L.^ U^^ ^ J^ J^ ^,Ui J 5j- U-^ 

U! e^-Jui c-j^ ^ ^ Ji^ ju^y^ j^j^^l ^^1 

UUjji ^j ,l.ij £«-»^ ui3iW «_/*i. ti L-kS- ^_;-c |JatHl 



^ ^Ltf wo l^ ^j J^ ^;:r^ J u-Oo^. ) J (j^t^^ j^j ^;^ 

<UH 'T'^^^*^ t/«y^ J^ U-A^ J ^^^1 ti 4--a^ J ^U^ j^loJiil 
^jl dll^ J u-a^ J ^l:-!^ ,^^\ ^^, ,^<:xr>- ^Jbj-^^ ^r* 
L^ic jjjj ^ UJtA^J ^l-^*^ *^*:^'♦>^ i-^^ tV<^ C-.*]^ 

2^ J ^ ^Jb^ ^^^i!l ^1 j^y ^' (ITl ^ y^J L5^ 

j^^j i!x^' jjb t_cjj^ Jjwtll ^_Jx bbljjjj y^ J W:Jj^^ 

c 



lfeJ|3 U>j J ^^\ ^^ \^ J l^yili \suii^ 1a>.)^ ^^ 

— liiar; U UU 5i: Uj^a-ilj l^lJ L;:->itJ Ji^^t j^l l^ 
^Ift l^ Jjct^, ^j^ If^ c-^V J^ '^^in?** J A:sjjs-* M^^V 
Uib^J ^i^-iiSj iu-j Jj^. j^'^^^ ir*^ J JU lU Uli 

jjy u^^l JW ybj ^^1 J;ff^ trLr* ^ ^^ br^ 



^Ikx^j y^tj^l ^^^ J*^. 4^Jo?ll ^j^ UjbjJ c;jt/^j '^j 

^^ACiS^-] to ^l^ Ufe;J a'^y^ J ^^^^1^ ^"^ '^^ '^^ J^ S/^ 

^josj^ jjrly"^ J X--^*^ j^yl> l^liw* (j ^^l^ <U-4^ \:)T^ 
lxij\ ^^Ju^ JUll ^ ^\ ^^ ^\ [jij^^^ A^^j!i l^ 

^!j ajc^* JD,lj i^jjy «^JJ1 JWjic>-y&j SJJ i^^ii^ 
c:-?J,j UU * uj^Jj ^^* JWj ^J^y u?JJl JW 

LiT* ?;^ 4 *^^ Lr^^ ^^'♦^ ^UiAiJlj ^'^Vj-^V cU^. 
JW ^ ^^^ i^b'^l ^iT* Jil ui3J t^ u^ l«l^ 

jw j^ Wv (^b*^! J^ u^ ub At^=sd-^ ^iL-iu 



4 ^L5^ J^ JJ,li U^ ^U ^ :^U ^1 ,:^.^ Sij 

^J i}'^.J^\ i-.4ci.^ ju jm ^10 ,0^ j\ ai^i 

t:;^' u/^ J^-^^l i-«^ cM>r^ J '^J^ i^J^ t^ 
^Ji\ Js: lfcj;3 lu^^ ai:^ ^^ \^ J l^y^ ^^ 

i-A>. ^j l&,jc^ j^i Uj^^ U,j ^* ^yij c:;:!^^!^ 
«-fl^ JIS j! lL^ J^j * ^U^- jUlj JW jjc- j^j ^* 

lT**^ ^J'^ {jLJ^ ^ ^^ J*^. j^*^' aL^i^j JU 

2^1Jo-l L-^«ci> Jl,<( <U<^ ^ CL?J) Ul JU cil L!i3i 
tl^U J^* jjl JJ^ Uib^J jj:>yift J ^^ tli3 J iJj 
UK (-_a*j|^ <U**cli <ii*»flJ jjl ^j Ub* iJU iJ-j j<s>- 

^ Jaxj j^j^l ^^j i!U i^yli idjUj U^ LliOs^ 

Ij^^I jj^' jj--w*^^|^ <ii-Jt iJj: Ujp ^„j^^ <L-/*cJ- j^yo 
jjbj jl A5j"j(l u,a>ifl'> <U^(^^^ls <U«J^j U;tX>. jcs."' j^Ujj 

^' JUJ]^ aj"j,l t^jJI J^^ j'^y*j ^j^ L5*i*^ ^L^ici- 



JUJlj L^aS^ Jjk«j (A^yi jjcsdlj ^ji,jts. J4XJU jV^S hoj\ 

Jlj^l i^j <L3^pjU ^;-.li5»-l Ail5 l^ (j.;^ LiPV. ^"^1^ 

* iJl^l Ja*j JJcC^ 
JU uj3y Ji^ JJcrll Jj^* jj:Jl Jjj^l; JV^^l t<,ls 

\j1 JU c^l i^lix^j Uib^j i:rt^-? ^**^' J*^. ^Ij^l ^^^j 
* ^j;rjSi5j A*^' <d^ ui3i jjj jljc^l 'ijLs, Jl< aj^ c-jj) 

4— fl^ ^^^ j^^K^iiiixi ^U3^ ^j^ jc>-li5 m:*3--jj ^jl l^^^ 
*^* V*^' J^^ J^ y^J ^ L5*^ ^-^i^ jJbj jl jc>-i{l 
^1 jl Ji\j\ h\ij\ ^U^3 ^ Lli3j^ J * iuJ- JUllj 

^^iJU LL^y ^ ^^ * JU!1 iUl c^jj; U J^ 
^^U t^l ijUx^j Ujb^j (i;^jlj ^^ J<^. j^<^' ^/:^^ 



^J ujj^j ^--^ J^b *^-^ JUJl jis? ^^Ijc^l JU^ 

^j1^ ^U yb^ 1;^ y:x ^1 Jj^^ ^ JUlli Jj^l 
Jt^l *-^ ^y J^j * >^ ^r^l i;i^j c^^jlj 

^ji J5 j1 ^\y^'i\ ^J^^ cl^li^^ * ^^\ JUl^ ^l:Jt 

* JU!! aJl j^^ U Ji^ 
Jj^. JU LL^y JU3 Jj^l Jj^- ^i Jl^l Uj 
^]\y<\ L.A^ (jj^^ J * ailj ^ic>-j JUll ^ ^jc-J 
* yL^<fc;-jybj ^j^[^\ ^J,^^^ ,\>'\ji\ JUlls ^Uj Jj^* 
£u5 Ja*j JU]li j.^ ^Uj J«>*> JU <— a»aJ tli^^ii^j 
lbA>3j Jlj^l Irr^ L!i3j^j :^ A;^ ^'^J (ji:r^3 
*\!j JU ^ J*l c:^l^ (j|^ «i^2>-lj JU jjl iij) l^'Uj 
^ l^tiU Uj Jjwj* t^j^ J Ub' i!U J^' ,-:;>. l^Jic 



c:^jc>-jj * dWt ^^^j^ 4 Jr^b Jc^yij do\)\ ^ 

^;1jJ1 jjl Jo^yi Jjlsj- to JIAC^I ^^ aj laiij to J--.45»- 

J« U^ (JJ^- J y^l ^ ^ Jo-ljH ^^'<' ^^ 

t-7 ^x^- v,» J5 t>^ J«Xc^ JV*1 J JJ*^ L5^J ^3j^ ^^ L5^ 
^ jj-«^l ^ <0jfc3 Uj j|*Xc^l iif< ^'^y bcj A^ljll ^^ <iL-i3 

JjjO* Jlj^lj * ll?*^ J*^* Jlr*^ vlfljiif jJfe; Lyt^ 



<^ 4 ^j J^j ^b ^'^^ 'r-yj ^^^^5u-^ iii^j ^,jo 

* <UftJ J«j ui3i ^ jS^ ^j iULc jIj 
l^J^ A:^^ V-tb <U/1^ l^^^l ^ jU ^1 lii^ 1 j^ 

^ ^^-lill ^^pL> UJ ^d-l5»-j wi>L^l t^O.;.y] Ltfl:>- LiSILS.-* 

UjJJu AJy^ij ^^j ij-« cl^Jj ^;-4^ <UJ^X:^^lJ jV-^^ v-i?;^ 
^JCJi <);>L J-A5>-j <)L>^1 J-^-J (J^* aJJi ^ ^ lyrJ^l 



^^^-::^^ ^Ull A^jilj L]U)t ^i^^\ ti ^UL«JI ^JjJ Jj 
U jixJl ^^U ^ j%^ ,^j ^jj ^ij Lli3i ^1 ^ 

B 



A L^\Z^\ 



2sb\A^\ J Yf^\ L-^\-y-^ jc-^ 









%iA^^. 



^\ t^bs3\ 



3>b\il»i\ J jfS^\ ^->V^: 



(p 



LIBRARY USE 

- „M TO DESK FROM WHICH BORROWED 

RETURN TO dII^"^^ use 

^ book is dueon U.e ^^J"^"- *•'"*«>» 




<^^i^fro%-|i2°3 



rr»-^'*^*"ai Library 



LD 21-100m-7,'40 (6936s) 



uLiicn#iL LionnnT - u.u. bbnl 




2i9Z23 



THE UNIVERSITY OF CALIFORNIA LIBRARY