Many of the computing patterns used today in elementary arithmetic, such as those for performing long multiplications and divisions, were developed as late as the fifteenth century. Two reasons are usually given to account for this late development, namely, the mental difficulties and the physical difficulties of such work.

The first of these, the mental difficulties, must be somewhat discounted. The impression that the ancient numeral systems are not useful for calculations is largely based on lack of familiarity with these systems. It is clear that addition and subtraction in a simple grouping system require only the ability to count the number symbols of each kind and then to convert to higher units. No memorization of number combinations is needed. In a ciphered numeral system, if sufficient addition and multiplication tables have been memorized, the work can proceed much as we do it today.

The physical difficulties, however, were quite real. Without a large supply of something to write on, any very extended development of the arithmetic process was bound to be limited. It must be remembered that our common machine-made pulp paper is little more than a hundred years old. The older rag paper was made by hand and was consequently very expensive.*NOTE*