De Burgo, the most noted mediæval writer on arithmetic, thinks this last process—our long division—much less pleasant than the following method. Surely tempora mutantur, et nos mutamur in illis.
Example 9. Divide 97535399 by 9876.
In this work the divisor is placed next below the dividend, and removed one place to the right since it is not contained in the first four figures of the dividend. The process with the first figure of the quotient, placed as usual at present, is as follows: The first number of the divisor, 9, is contained in 97 nine times with a remainder 16. The first figure of the divisor having been used is canceled; as are also the first two figures of the dividend. (The "scratches" or canceling-marks are omitted in the illustration.) The remainder, being of the same denomination as the first two figures of the dividend, is put directly above them. The next number to be used is 165. Multiplying the second figure of the divisor, 8, by 9, and subtracting from 165, 93 remains; 165 and 8 are now canceled, having been used. The remainder 93 is placed above in the proper orders, the 6th and 7th places. So it continues, leaving, after completing the work with the first figure of the quotient, the remainder 8651399. The divisor is now set down again, taking one place to the right as it should to correspond to the highest order now in the dividend: the last figure being raised to the line above, probably for symmetry. The process is continued as before.
All writers upon arithmetic appear to have agreed in commendation of this method as late as the end of the seventeenth century. It was, in fact, the only method thought necessary to notice. The English arithmeticians, from evident cause, called it the "scratch way" of division. Our present method was known specifically as Italian division, and was not introduced until the beginning of the last century.
One writer on arithmetic, a pious monk, furnishes a good illustration of mediæval logic. He is embarrassed by the usage and meaning of the term "multiplication" in the case of fractions in which the product is less than the multiplicand, and he proposes the question,
