
603 
In the progression a (a +
1) (a + 2) (a + 3) ... (a + n), if x is the distance from the first or last
number, we get:
a+ (a + n)
= (a + x) + (a + n – x).

604 
In the progression a (a +
1) (a + 2) (a
+ 3) ... (a + 2n), then
a + (a + 2n)
= 2 (a + n).
In the
progression 2a
(2a +
2)(2a +
4)(2a +
6) ... (2a + 2n), then
2a
+ (2a + 2n) = 2 (2a + n).
In the progression (2a
+ 1) (2a + 3) (2a +
5) (2a +
7) ... (2a +
[2n +
1]),
then
(2a + 1) + (2a + [2n + i]) = 2 (2a +
[n + 1]).

605 
In the progression
,
if x is the distance from the first or last number, we get:
.
In the progression
,
if x is the distance from the first or last number, we get:
. 
606 
In the progression a a^{2} a^{3}
a^{4} ∙ ∙ ∙ a^{n}, then
a∙a^{n}
= 
607 
Cf. alKhuwarizmi,
Mafatih al`ulum
(Cairo, 1349/1930), p. III; alBiruni, Kitab attafhim,
ed. and tr. R. R. Wright (London, 1934), pp. 29
f. As alKhuwirizmi
explains it, a muthallathah
results from adding the numbers from one on; a
murabba'ah results from
adding every second number (or from adding up the adjacent numbers of a
muthallathah); a
mukhammasah results
from adding every third number; a musaddasah
from adding every
fourth; and so on.
Thus, the progression 1 2 3 4 5
6 7 8 9 10 11 12 13 . . . yields the
muthallathah
1 3 6 10 15 21 28
and so on. The progression 1 3 5
7 9 11 13 . . . yields the
murabba'ah 1 4 9 16 25 36 49
and so on. The progression 1 4 7 10 13 . . . yields the
mukhammasah 1 5 12 22 35 and so on. The
progression 1 5 9 13 . . .
yields the musaddasah 1 6 15
28 and so on.
Ibn Khaldun, however, proceeds in a slightly different
manner. He always adds a muthallathah
to a given progression, in order to obtain the next
higher one. Thus, he has:

1 
2 
3 
4 
5 
6 
7 
8 
10 
11 
12 
13 
Muthallathah 
1 
3 
6 
10 
15 
21 
28 




















1 
3 
6 
10 
15 
21 





Murabba'ah 
1 
4 
9 
16 
25 
36 
49 




















1 
3 
6 
10 
15 
21 





Mukhammasah 
1 
5 
12 
22 
35 
51 
70 




















1 
3 
6 
10 
15 
21 





Musaddasah 
1 
6 
15 
28 
45 
66 
91 





A perusal of the largely unpublished Arabic literature on
arithmetic will certainly provide an exact presentation of the table
that Ibn Khaldun has in mind. The theory of polygonal numbers (as well
as all the other theorems mentioned in this section) came to the Arabs
through the work of Nicomachus of Gerasa, which was translated into
Arabic and is preserved but not yet published in its Arabic form.
However, though Ibn Khaldun seems to refer to the geometrical figures of
Nicomachus, which provided the terminology for the subject, his table
would appear to be one made up of numerical pro essions. Cf. Nicomachus
Introduction to. Arithmetic
ii. 811.
M. L. D'Ooge(tr.), (University of Michigan Studies,
Humanistic Series, No. 16)
(New York, 1926), pp. 241
ff. [* Ar. translation, ed. W. Kutsch (Beirut, 1959)].

608 
Actually, a new sentence
should begin here ("By adding them up, a triangle is formed"), but the
text does not permit such a construction, and no correction is
permissible. 
609 
That is, 2
(2n + 1). 
610 
That
is, 2 (m + 1) (2n + 1).
In literal translation, the Arabic terms, derived from
the Greek, read: even, odd, evenlyeven, unevenlyeven, and
evenlyevenodd. Ibn Khaldun does not mention oddlyodd, i.e., odd
numbers multiplied by each other:
(2n + 1)(2n + 1).
Cf. alBiruni,
op. cit., p. 25.

611 
That is, in theoretical mathematics. 
612 
Cf. 1:238, above, and
pp. 123 and 137, below. 
613 
Ibn Khaldun is said to
have written a work on the subject himself; cf. 1:xliv, above. The first
two paragraphs of this section are quoted by J. Ruska, "Zur iiltesten
arabischen Algebra and Rechenkunst," in
Sitzungsberichte der Heidelberger Akademie der
Wissenschaften, Philos.hist. Kl.
(Heidelberg, 1917), pp. 19 f. 
614 
A "root"
is ,
not ,
which might be irrational. The following discussion of rational numbers
and surds appears in the margin of C and the text of D, and is not yet
found in the earlier texts. 
615 
The word mabsul
as such can also mean "extensive," but in view of the
character of the work, Renaud (see n. 616, below) suggests the above
translation. 
616 
Muhammad b. 'Abdallah b. 'Ayyash.
Cf. GAL, Suppl., II,
363; and esp., H. P. J. Renaud, "Sur un passage d'Ibn Khaldun relatif a
l'histoire des mathematiques," Hespiris, XXXI,(1944), 3547,
where Renaud corrects statements he had made earlier in Hespiris, XXV
(1988), 24 (n. 6). Renaud shows that a
large work by alHassar, whose existence we
should expect from Ibn Khaldun's reference to the "small" work, actually
did exist. 
617 
The work referred to is
Ibn alBanna"s wellknown Talkhis a'mal alhisab. 
618 
At the beginning of the
Raf , Ibn alBanna' states that
the work was intended to "explain the scientific contents and comment
on" the apparent difficulties of the Talkhis.
Cf. MS. or., Princeton, 1092A (80 B). 
619 
The pronoun found in the
Arabic text must refer to the Raf ,
but the statement would seem to apply rather to the
Talkhis. 
620 
The following eight lines are not found in Bulaq, and
in A they are still in the form of a marginal note. There is no
reference in Ibn alBanna"s works to the effect that he used the sources
mentioned. However, it is clear from Ibn Khaldun's attitude toward Ibn
alBanna' that he would not think of accusing him of plagiarism. 
621 
Mubammad b. 'Isa b. 'AbdalMun'im,
who lived at .the court of Roger II of Sicily. Cf. H. P. J. Renaud in
Hespiris, XXV
(1938), 8885. Nothing is known about alAbdab and his
work. 
622 
The translation follows that suggested by Renaud in
Hespiris, XXXI
(1944), 42 f. 
623 
Cf. Qur'an 24.35 (35). 
624 
Adad:
i.e., n, the part of the equation that is not a
multiple (or fraction) of the unknown. 
625 
Shay': Latin res,
that is, x (the unknown). 
626 
Mal: Latin
substantia, census, that is,
x^{2}. 
627 
That is, the higher powers are expressed by multiplying
two or more times the second and third (ka'b "cube," not mentioned by
Ibn Khaldun) powers. Thus, x^{4} is mal mal, x^{5} is mal ka'b,
etc. For uss, cf. P. Luckey, Die
Rechenkunst bei Gamssid b. Mas'ud alKasi (Abhandlungen
fur die Kunde des Morgenlandes, No. 31') (Wiesbaden, 1951), pp. 59, 70
f., 104 f. According to Luckey, uss has two meanings, that of exponent,
and another referring to the position of the numbers (one for the units,
two for the tens, three for the hundreds, etc.). Cf. also 1:241, above,
and pp. 203 ff., below. 
628 
Jabr, hence
Algebra. 
629 
I.e., n, x,
x^{3},
and the three basic equations:
ax = n , bx^{2} = n , and
ax^{2}
= bx. Cf. L. C. Karpinski,
"Robert of Chester's Translation of the Algebra of alKhowarizmi," in
Contributions to the History of Science
(University of Michigan Studies, Humanistic Series,
No. 11) (Ann Arbor, 1930), p. 69. 
630 
ax = n; bx^{2
}= nx 
631 
x^{2
}= bx; x = b, b being the multiple of
the "root." 
632 
ax^{2
}
+ bx = n ,
or rather:
x^{2} + n = ax. The geometrical solution for the
equation x^{2} + 21 = l0x is explained by alKhuwarizmi;
cf. Karpinski, op. cit., pp. 83 f. The expressions
tafsil addarb and addarb almufasfal are not quite clear to
me. They have been rendered tentatively by "multiplication in part,"
since they seem to refer to the addition of
^{ }which
is necessary for finding the value of x. 
633 
Apparently, ax^{3}
+ bx^{2} = cx + n , or the like. 
634 
The simple
equations are: ax = n; bx2 = n; and ax^{2}
= bx. The composite equations are:
ax^{2 }
+ bx = n;
ax^{2}
+ n = bx;
and ax^{2} = bx + n.
Cf. Karpinski, op. cit., pp.
69 and 71. 
635 
Muhammad b. Musa, who lived in the first half of the
ninth century. Cf. GAL, 1, 215 f.; Suppl., I, 381 f. 
636 
Ca. 900? Cf. GAL, Suppl., I, 390. 
637 
He is referred to as
AbulQasim alQurashi of Bougie, and was a source of
Ibn alBanni's Talkhis. Cf. H. P. J. Renaud in Hespiris,
XXV (1938), 3537. 
638 
Cf. 'Umar alKhayyam, Algebra, ed. F.
Woepcke (Paris, 1851). 
639 
Qur'an 35.1 (1). 
640 
I.e.,
calculation (elementary arithmetic) and algebra. 
641 
Ali b. Sulayman. Cf. Sa'id alAndalusi,
Tabaqat alumam, tr. R.
Blachere (Publications de l'Institut des Hautes Etudes
Marocaines, No. 28)
(Paris,
1935), pp. 131
f.

642 
Agbagh b. Mubammad, d. 426
[9791035]. Cf. GAL, 1, 472; Suppl.,
I, 861; Sa'id alAndalusi, Tabaqdt, pp. 13o
f., where his age is incorrectly
given as
fifty (instead of fiftysix) solar years.

643 
'Amr ('Umar?) b. Ahmad, d. 449 [1057/58], a
member of Ibn Khaldun's family. Cf. Sa'id alAndalusi, Tabaqdt, p.
133, and above, 1:xxxiv. Ibn Fadlallih al'Umarl, d. 749 [1349],
states in his Masdlik alabdr that he had seen very good
astrolabes signed by Ibn Khaldun, and had personally copied a work of
his, which, however, he lost later on. ( MS, Topkapusaray, Ahmet III,
2797, Vol. V, p. 417.) 
644 
The subject
was treated as a part of jurisprudence, pp. 20 ff.,
above. 
645 
In such cases, the process called 'awl
"reduction," mentioned below, is applied. The total of the inheritance
shares, as stipulated by the Qur'an, may be greater than the entire
estate. Thus, according to the famous example, if a man leaves two
daughters, his two parents, and one wife, the daughters would be
entitled to twothirds, the parents to onethird, and the wife to
oneeighth of the estate. Qur'an 4.11 f. (1214).
Therefore, the following procedure is used. The fractions are reduced
to their common denominator:
the
new numerators are added up (16 + 8 + 3 = 27); and the total is
made the new denominator. Thus, the new shares are
The
wife's share, which was oneeighth, is "reduced" ('awl) to
oneninth, but the proportion of the shares to each other is preserved.
In our
symbols, the procedure can be expressed as follows (for the sake of
simplicity only two fractions are assumed):
The
correctness of the procedure can be proven as follows:

646 
Sic
C and D. 
647 
Cf p. 22, above. 128 
648 
All these scholars were mentioned above, p. 21. This
passage was used by Hajji Khalifah, Kashf
az zunun, ed. Flugel
(Leipzig & London, 183558), III, 64. 
649 
This Berber name is
spelled with a ,s into which a small z is inserted. Cf. 1:67 (n. 183),
above. He is 'Abdallah b. Abi Bakr b. Yahya, who was born
ca. 643 [1245/46] and who was
still alive in 699 [1299/1300]. Cf. Ahmad Baba,
Nayl alibtihaj (Cairo,
1951/1932, in the margin of Ibn Farhun, Dibaj), pp. 14.0 f. 
650 
Cf. the
Autobiography, pp. 31 f. 
651 
This remark
would seem to imply that Hanafite and Hanbalite works are inferior. 
652 
Cf.
Qur'an 2.142 (136), 213 (209), etc. 

