The `Unknown Heritage´: Trace of a Forgotten Locus of Mathematical Sophistication

The "unknown heritage" is the name usually given to a problem type in whose archetype a father leaves to his first son 1 monetary unit and 1/n (n usually being 7 or 10) of what remains, to the second 2 units and 1/n of what remains, etc. In the end all sons get the same, and nothing remains.

The earliest known occurrence is in Fibonacci's Liber abbaci, which also contains a number of much more sophisticated versions, together with a partial algebraic solution for one of these and  rules for all which do not follow from his algebraic calculation. The next time the problem turns  up is in Planudes's late 13th-c. Calculus according to the Indians, Called the Great. After that the simple problem type turns up regularly in Provençal, Italian and Byzantine sources. It seems never to appear in Arabic or Indian writings, although two Arabic texts (one from c. 1190) contain more regular problems where the number of shares is given; they are clearly derived from the type known from European and Byzantine works, not their source. The sophisticated versions turn up again in Barthélemy de Romans' Compendy de la praticque des nombres (c. 1467) and, apparently inspired from that, in the appendix to Nicolas Chuquet's Triparty (1484). Apart from a single trace in Cardano's Practica arithmetice et mensurandi singularis, the sophisticated versions never surface again, but the , the sophisticated versions never surface again, but the simple version spreads for a while to German practical arithmetic and, more persistently, to French polite recreational mathematics.

Close analysis of the sources shows that Barthélemy cannot have drawn his familiarity with the sophisticated rules from Fibonacci. It also suggests that the simple version is originally either a classical, strictly Greek or a medieval Byzantine invention, and that the sophisticated versions must have been developed before Fibonacci within an environment (located in Byzantium, Provence, or possibly in Sicily?) of which all direct traces has been lost, but whose mathematical level must have been quite advanced.