Introduction to Arithmetic

definite rules of procedure in such problems, leading to the difference 5½, if unity be taken as the first term, under the conditions proposed. Our common procedure, in analytical solution of this problem, leads to the result, d = 5½ or d = 5½ if a is 1. Even if the method of arriving at this value for d be that of ‘false position,’ the procedure which, being adaptable to similar problems, arrives definitely and surely at the complete solution of the proposed problem must be regarded as scientific.

From this point the solution follows the lines of previous problems. With 1 as the first term and 5½ as the difference, the terms are 1, 6½, 12, 17½, and 23, having 60 as a sum. To complete this to the required 100 loaves there must be added 40, or ⅔ of 60. After it has been noted that this is the case, there is added to each of the numbers in the discovered series ⅔ of itself, a process that gives 1⅔, 10 5/6, 20, 29 1/6, and 38 1/3 as the series fulfilling the required conditions.

A second problem involving an arithmetical series is entitled “Instructions for the difference in distribution.” The solution opens with the phrase, “If you are told,” which was later adopted by Arabic mathematicians, and is not uncommon even today. “If you are told, [distribute] 10 measures of grain to 10 people so that the difference in [the amount received by] each person as compared with the next one is ⅛ of a measure of grain. I take the mean, one measure. I subtract 1 from 10, leaving 9. I take ⅛ of the difference, ⅛, and take it nine times. This gives 1 ⅛, which I add to the mean. From this take away ⅛ measure for each person in order to arrive at the goal. Following instructions: 1 7/16, 1 5/16, 1 3/16, 1 1/16, 15/16, ½ ¼ ⅛ 1/16, ½ ¼ 1/16, ½ ⅛ 1/16, ½ 1/16, ¼ ⅛ 1/16, together 10.” The solution of this problem as given by the Egyptian manual should be compared step by step with the solution by the ordinary procedure with the formulas of our elementary algebra; the close correspondence is too striking to be regarded as wholly accidental.

No one could ask that the ancient Egyptians should have modern formulas with a literal symbolism, for this advance was not made in Europe until the end of the sixteenth century of the Christian era. The similarity in method is, however, highly significant, revealing a development in analytical thinking that is not equalled for many centuries. In effect, we have in these problems the first term of an arithmetical series regarded as a function of the common difference, under given conditions, and the last term as a function of the mean.