home email

Fra Luca del Borgo,

b. Borgo San Selpolcro, Tuscany c. 1445-1450
d. later than 1509

The Summa de Arithmetica, Geometria, Proportioni et Proportionalita (Summary of Arithmetic, Geometry, Proportions and Proportionality) of Pacioli, published in 1494 was the first great general work on mathematics to be printed. It is further distinguished by Pacioli's treatment of arithemetic, geometry and algebra from a scientific rather than a practical viewpoint. In this work we have a famous example of the problem of points.

The relevant text from the Summa. is available here in both Italian and English in facing columns.  

Pacioli's solution is easily generalized. Suppose a game is to continue until one of two players has won n rounds. The game is terminated when Player A has won a rounds and Player B has won b. By Pacioli's rule, the Players A and B receive shares of the stakes in ratio a to b independent of the number of rounds needed to win. Cardano thought this solution ridiculous.