Several Unknowns

Arabic algebra, and also Fibonacci, sometimes though rarely make use of several algebraic unknowns. The first abbacus writer to do it may have been Antonio de’ Mazzinghi, who reinvents the idea in a work in progress treating difficult second-degree problems. An anonymous near-contemporary does too, but for first-degree questions and in a way that suggests he has heard about the trick but not fully understood, and therefore tries his own way. In 1463, Benedetto da Firenze (also in a work in progress) develops symbolic linear algebra with up to five unknowns. Nobody seems to have been interested, Benedetto leaves no traces. In the Summa, however, Pacioli explains how to use thing and quantity in such a way that the quantity can be recycled, thus allowing operation with more than two unknowns (but only if they can be eliminated sequentially). Chuquet shows that Pacioli did not invent. Apart from one problem borrowed from Antonio, the technique is used for first-degree problems only. Rudolff takes up this “rule of quantity” and applies it often in his Coss from 1525. Stifel, in his Arithmetica integra from 1544, goes further, and gives to unknowns beyond the thing the names A, B, …, expressing powers and products by means of juxtaposition. His examples are almost exclusively of the first degree, but in his re-edition of Rudolff from 1553 he refines the system and offers many higher-degree problems. Not many took note, but Valentin Mennher did in 1556; thereby Stifel’s revised system reached French readers.