Christopher Clavius S. J. (1537 - 1612) was highly respected in his time as a mathematical educator and curricular reformer. His textbooks were valued and widely used. This is the title page of the 1584 edition of his Practical Arithmetic, first published in 1583 in Rome. Among its notable users were René Descartes and Gottfried Leibniz. The missionary, Matteo Ricci S.J. (1552-1610) would eventually adapt and translate this work for the Chinese. It was published in China in1613 after Ricci’s death, and introduced Western arithmetic to the Celestial Empire.
Here are pages 76 and 77 of Clavius’ Arithmetic in which the author demonstrates shortcuts in using galley division and accommodating fractional remainders. On page 76, in obtaining the quotient of 6709456 and 2808, the division by 2808 is undertaken with the first four digits of the dividend. A remainder is obtained, 913, indicating the division process must continue. The large “X” at the side of the computation marks a “casting out of nines” was used to check the work. On page 77, the quotient of 13946007693 and 38000000 is sought. Here the abbreviated division process works as 3800, the “shortened” divisor, is an integral factor of the “shortened” dividend, 1394600. The “cut off tail” of the dividend, 7693, becomes the numerator of the remainder.