Circular reckoning (hisab al-dawr)

"Circular reckoning" (hisab al-dawr) is a heading that al-Khwarizmi places in the Book of Bequests of his Book of al-Jabr and al-Muqabala before a group of problems in which each unknown depends on the other, so that the reckoning circles between them. ¹ Its chapters in the Cairo edition are: marriage in final illness, manumission in final illness, acknowledgement in circular cases, and forward sale (salam) in final illness. ^{1 2 3 4}

A man in his final illness marries a woman for a dower of a hundred dirhams, owning nothing else, while her fair dower is ten dirhams; then she dies, bequeathing a third of her property, and then the husband dies. ¹ What exceeds the fair dower is a bequest to her from the sick husband; of what reaches her, half of the remainder after her own bequest returns to the husband as inheritance, increasing his heirs' share, which in turn changes how much may lawfully go to her. ¹ Al-Khwarizmi calls her bequest a "thing", follows what ends up in each party's hands, and sets the husband's heirs' share equal to twice her bequest, because a bequest is valid only up to one third; by al-jabr and al-muqabala the thing comes out at thirty-five dirhams. ^{1 5} He then varies the problem, with the wife owing a debt of ten dirhams, or with the husband also bequeathing a third of his property to another man, and reworks it on the same principle. ⁵

"Manumission in final illness" has problems of a man who frees two slaves during his illness, after which one of them dies leaving property, ² and problems of a slave woman given away during illness with whom the recipient then has relations, so that he owes the 'uqr compensation, which enters the reckoning of the bequest; ^{6 7} under the heading "acknowledgement in circular cases" comes the problem of a man who gives away a slave woman in his final illness, owning nothing else, and then has relations with her himself, so that he owes the 'uqr. ^{3 8} In "salam in final illness" a sick man pays thirty dirhams in advance for a measure (kurr) of grain worth ten and then dies; the favour is twenty dirhams, and the part that may stand, found by al-jabr, is ten dirhams. ⁴ The whole book ends with a problem from this chapter. ⁹

These problems rest on the jurists' rules on bequests, among them that favouritism, gifts and manumission by someone in his final illness are treated as bequests and stand only within a third of his property; this is why al-Khwarizmi makes the heirs' remainder twice the bequest in each problem. ^{1 4} The book does not discuss these rules; it takes them as given and supplies the method of calculation.