Christoff Rudolff (1499–1543) was a German speaking *Rechenmeister*, a teacher of arithmetic and computing. Although born in what is now known as Poland, he lived most of his life in Vienna. In 1525 he composed a book on algebra entitled *Die Coss,* a term which can be traced to its Italian origin, *cosa,* meaning “thing” or “unknown.” Algebra was first conceived as a mathematical discipline of finding unknowns in a problem situation. The title page and the contents shown here are from the widely-distributed and -used 1553 edition, to which commentaries by Michael Stifel (1487–1567) were added. Other editions followed in 1571 and 1615.

Folio 6 (recto) provides examples of arithmetic progressions.

This page (folio 6, verso) provides the definition and examples of geometric progressions: e.g. 6, 18, 54, 162, 486, … or 3, 6, 12, 24, 48, 96, … .

On folio 63 (recto), Rudolff provided a listing of symbols for what we would write as 1 (Dragma, a unit, the “builder of numbers”), X (Radix), X^{2} (Zensus), … , X^{9} (Cubus de cubo).

On folio 63 (verso), Rudolff illustrated the squaring and cubing of numbers with examples: 4 x 4=16, 9 x 9=81. He also illustrated finding the *radix* or square root (e.g. 64 = 8 x 8) and finding the *cubicam* or cube root (e.g. 64 = 4 x 4 x 4).

Christoff Rudolff was the first person to introduce the modern radical sign, “\(\sqrt{\phantom{xx}}\) ". Here on folios 113 recto (above) and verso (below), he provided instruction on adding and subtracting expressions containing radical parts.

Here on folio 159 (recto), the author considered the equations X^{2 }+ 12X = 160, X^{2 }+ 8X = 240, and X^{2 }+ 96 = 20X. By a technique of “completing the square,” he demonstrated that the first has solution X = 8, the second X = 12, and the third both X = 8 and X = 12.

Here on folio 159 (verso), the new equations are X^{2 }= 8X + 240 and X^{2 }= 12X + 160. By a technique of “completing the square,” Rudolff demonstrated that both have the solution X = 20.

In 1526, Rudolff published *Kunstliche rechnung mit der ziffer und mit den zal* *pfenningen*, a reckoning book for merchants. He taught the new cipher arithmetic as well as counting table, or pfenning, calculation. This is the title page for the 1553 edition.

*These images from its George Arthur Plimpton Collection are presented through the courtesy of the Columbia University Libraries. *