[Footnote 3*: See on this Dr. Poole, _The Exchequer in the Twelfth Century_, Chap. III., and Haskins, _Eng. Hist. Review_, 27, 101. The hidage of Essex in 1130 was 2364 hides.] The great difficulty of the early Abacists, owing to the absence of a figure representing zero, was to place their results and operations in the proper columns of the abacus, especially when doing a division sum. The chief differences noticeable in their works are in the methods for this rule. Division was either done directly or by means of differences between the divisor and the next higher multiple of ten to the divisor. Later Abacists made a distinction between “iron” and “golden” methods of division. The following are examples taken from a twelfth century treatise. In following the operations it must be remembered that a figure asterisked represents a counter taken from the board. A zero is obviously not needed, and the result may be written down in words. (_a_) MULTIPLICATION. 4600 × 23. +-----------+-----------+ | Thousands | | +---+---+---+---+---+---+ | H | T | U | H | T | U | | u | e | n | u | e | n | | n | n | i | n | n | i | | d | s | t | d | s | t | | r | | s | r | | s | | e | | | e | | | | d | | | d | | | | s | | | s | | | +---+---+---+---+---+---+ | | | 4 | 6 | | | +Multiplicand.+ +---+---+---+---+---+---+ | | | 1 | 8 | | | 600 × 3. | | 1 | 2 | | | | 4000 × 3. | | 1 | 2 | | | | 600 × 20. | | 8 | | | | | 4000 × 20. +---+---+---+---+---+---+ | 1 | | 5 | 8 | | | Total product. +---+---+---+---+---+---+ | | | | | 2 | 3 | +Multiplier.+ +---+---+---+---+---+---+ (_b_) DIVISION: DIRECT. 100,000 ÷ 20,023. Here each counter in turn is a separate divisor. +-----------+-----------+ | Thousands | | +---+---+---+---+---+---+ | H.| T.| U.| H.| T.| U.| +---+---+---+---+---+---+ | | 2 | | | 2 | 3 | +Divisors.+ +---+---+---+---+---+---+ | | 2 | | | | | Place greatest divisor to right of dividend. | 1 | | | | | | +Dividend.+ | | 2 | | | | | Remainder. | | | | 1 | | | | | 1 | 9 | 9 | | | Another form of same. | | | | | 8 | | Product of 1st Quotient and 20. +---+---+---+---+---+---+ | | 1 | 9 | 9 | 2 | | Remainder. | | | | | 1 | 2 | Product of 1st Quotient and 3. +---+---+---+---+---+---+ | | 1 | 9 | 9 | | 8 | +Final remainder.+ | | | | | | 4 | Quotient. +---+---+---+---+---+---+ (_c_) DIVISION BY DIFFERENCES. 900 ÷ 8. Here we divide by (10-2).