Vol. VII, No. 11.] References to Indian Mathematics. 805 
[N.S.] 
ILI, 
Libri has eo in Latin the text of a work with the 
following title 
Inber “augmenti et ese eit vocatus numeratio divina- 
o quod sapientes Indi posuerunt, quem 
Abraham comptlait et secundum librum qui Indorum 
dictus est composuit 
Of this Abraham practically nothing is known, but it i 
been supposed that he is the same as Ibrahim b. Ezra,! 
learned Jew, who lived in the twelfth century (1093-1168 a.p. . 
His work consists of some thirty-three algebraic problems which 
he solves in various ways. After the brief introductory remarks 
the author mare no reference to India. Of the shanty three 
The rule of two Aas or regula elchatayn,’ or regula 
duorum jfalsorum, or method of eA balance, or method of 
increase and decrease as it is variously called, occurs in no 
known early Hindu wor 
The rule enables us to solve problems that can be expressed 
in the form 
f (xz) =axr+b=k. 
For if we set k—f (2) =e, the ‘first error’ and k—f (8)=e, 
the ‘second error’ we have the rule 
on Re—ae, 
€,—e, 
which is so largely employed by Abraham. 
The following is a fairly typical example taken from ed 
Liber augmenti et diminutionis expressed in modern notation * 
f (a) =2—4-} (@—4)-5-}{x-4-3 (ead) Bie l0 ~ 
First method: f (16)=3 and e,=7 
f (32) =12 and e,= —2 
32.7+2.16 
whence x= ie Ss. ees = 28 
Second method: —4—} (w—4)=3 47-3 
p2—3-5=} 2-8, b(}2-8)=42+} § e-2 
4 mee a+. st v= 16 
‘Rabbi ben Ezra, but it is very doubtful. 
2 Elkhata’ ayn. 
8 Libri, vol. i, pp. 310-311. 
