(The VNR Concise Encyclopedia of Mathematics, Eds: Gellert, Kastner, Hellwich, Kastner, p. 80) "Together with numbers, equations belong to the first mathematical achievements of mankind. They occur in the oldest written mathematical documents, for example, in cuneiform texts of the old Babylonians, which go back as far as the third millenium B.C., and in ancient Egyptian papyri dating from the Middle Kingdom, about 1800 B.C.

"In accordance with the structure of the Babylonian society questions of sharing an inheritance were of great interest. The first-born son always received the largest share, the second more than the third, and so on. Here is one such sharing problem:

"[Translation, ancient text] '10 brothers, 1 2/3 mines of silver [1 mine = 60 sheckel, 1/2 mines = 100 shekel, a Biblical unit].... The share of the eighth [brother] is 6 shekel [sic]. Brother after brother, how much as he taken?' ....

"The problem leads to an arithmetic progression...." Of the form, a = 1st son's inheritance; d = difference each descending sibling:
a, (a —d), (a — 2d), (a — 3d), (a — 4d), (a — 5d), (a — 6d), (a — 7d), (a — 8d), (a — 9d).

The SUM is 10a — 45d = 100 shekel; 8th son's inheritance, a — 7d = 6 shekel.

Multiplying 2nd equation by 10 & subtracting, we have:
(10a — 45d) — (10a — 70d ) = (100 — 60) shekel. Or, 10a — 10a — 45d + 70d = 25d = (100 — 60)s — 40s; or 25d = 40s Þ d = 40/25s = 1 3/5s.

Substituting in a — 7d = 6s, we have a — 7(8/5)d = a — 56/5 d = 6, or a = 6 + 56/5 = 30/5 + 56/5 = 86/5s, or (1st son) a = 17 1/5 shekel. Then the apportions are:

INHERITANCE OF 10 BABYLONIAN SONS

1st Son	2nd	3rd	4th	5th	6th	7th	8th	9th	10th
17 1/5S	15 3/5S	14S	12 2/5S	10 4/5S	9 1/5S	7 3/5S	6S	4 2/5S	2 4/5S

(17 1/5 + 15 3/5 + 14 + 12 2/5 + 10 4/5 + 9 1/5 + 7 3/5 + 6 + 4 2/5 + 2 4/5)s = ((17 + 15 + 14 + 12+ 10 + 9 + 7 + 6 + 4 + 2) + (1/5 + 3/5 + 2/5 + 4/5 + 1/5 + 3/5 + 2/5 + 4/5))s = (96 + 20/5)s = (96 + 4)s = 100 shekel.

RETURN TO BABYLONIAN SCIMATH