Moritz Cantor (1829-1920) was a German mathematician and historian of mathematics. His history of mathematics, *Vorlesungen über Geschichte der* *Mathematik *(*Lectures on the History of Mathematics*), in four volumes, was the first modern and comprehensive history of the subject. It still today remains a classic of historical scholarship: Volume I (1880) considers the developments of mathematics from ancient times until the year 1200; Volume II (1892) covers the period 1200-1668; Volume III (1894-96), the period 1668-1758; and Volume IV, completed with collaboration in 1908, the years 1759-1799.

Above: Title page of Volume I for the second edition of 1894

The excerpt above from Volume I, page 678, discusses Al-Khwarizmi’s* Algebra* (*ca.* 825). The problem being considered is one of “completing the square” to solve the algebraic equation, *X*^{2} + *bX* = *C.* This is the first solution demonstration given in the *Algebra*.

On page 828, Cantor gave a demonstration of Gerbert’s (*ca.* 980) method of division as performed on a surface abacus. The abacus is ruled in columns representing: units, S; tens, D; and hundreds, C. The diagram on the left illustrates “division with a difference,” Gerbert’s method in which he employed the complement of the divisor to carry out the operation. In the given illustration it is required to divide 668 by 6: instead, the division is performed dividing by 10 − 6 = 4. This process arrives at the correct answer, 111 R 2, but requires many steps (subdivisions) and seems overly complex to a reader today. The diagram on the right uses a modern approach.

In Volume II, above, Cantor discussed the construction of regular polygons, illustrating some construction methods of Leonardo da Vinci.

Above is the title page of the 1898 edition of Volume III of Cantor’s *History*.

*The images above were obtained from the Rare Book Collection at Franklin and Marshall College, Lancaster, Pennsylvania. The three images below of the fourth volume of Cantor's work were obtained through the courtesy of the University of California Libraries. A complete digital copy of this item can be viewed on the Libraries’ Internet Archive.*

The fourth volume of Moritz Cantor’s *Geschichte der Mathematik* covering mathematics history for the period 1759-1799 was published in 1908.

The noted mathematical historian Floran Cajori (1859-1930) gifted this copy of the book to his son and signed his name on a front page.

Page 72 contains a brief summary of the accomplishments in algebra for the historical period examined.

Index to Mathematical Treasures