One share remains for the two paternal sisters, to be divided equally between them. Because this single share will not divide, we multiply the number of their heads—two—by the corrected base of the case, ten, and obtain a new base of twenty. The grandfather’s four shares are therefore doubled to eight; the full sister’s five shares become ten; and the paternal sisters’ single share becomes two, giving one share to each of them.
Schematic layout:
5/2 10/2 20
Grandfather 4 → 8
Full sister 5 → 10
Two paternal sisters 1 → 2 (1 each)
Alternatively, one may say: the root number is five. The grandfather, by co-division, receives two shares; the full sister takes one-half, that is, two and one-half shares; what remains—one-half share—goes to the two paternal sisters, giving each of them one-quarter. The denominator of a quarter is four, so we multiply it by the root, five, to reach twenty. Consequently the grandfather’s two shares become 2 × 4 = 8; the full sister’s half becomes 10; and the two paternal sisters receive 2, one share apiece.
- Zayd’s Abridged Case. The heirs are a mother, a grandfather, a full sister, a paternal brother, and a paternal sister. It is so named because solving it by strict co-division requires a base of 108, whereas by abridgement it can be settled on a base of 54. The original root is six: the mother takes one-sixth, i.e., one share; the remaining five shares are to be divided among the grandfather and the siblings by co-division. Their number of heads is six, which does not divide evenly, so we multiply the six heads by the root, six, obtaining thirty-six. The mother then receives—