It would seem probable that writers on the calendar like Bede (A.D. 721)
and Helpericus (A.D. 903) were able to perform simple calculations;
though we are unable to guess their methods, and for the most part they
were dependent on tables taken from Greek sources. We have no early
medieval treatises on arithmetic, till towards the end of the tenth
century we find a revival of the study of science, centring for us round
the name of Gerbert, who became Pope as Sylvester II. in 999. His
treatise on the use of the Abacus was written (c. 980) to a friend
Constantine, and was first printed among the works of Bede in the Basle
(1563) edition of his works, I. 159, in a somewhat enlarged form.
Another tenth century treatise is that of Abbo of Fleury (c. 988),
preserved in several manuscripts. Very few treatises on the use of the
Abacus can be certainly ascribed to the eleventh century, but from the
beginning of the twelfth century their numbers increase rapidly, to
judge by those that have been preserved.
The Abacists used a permanent board usually divided into twelve columns;
the columns were grouped in threes, each column being called an “arcus,”
and the value of a figure in it represented a tenth of what it would
have in the column to the left, as in our arithmetic of position. With
this board counters or jetons were used, either plain or, more probably,
marked with numerical signs, which with the early Abacists were the
“apices,” though counters from classical times were sometimes marked on
one side with the digital signs, on the other with Roman numerals. Two
ivory discs of this kind from the Hamilton collection may be seen at the
British Museum. Gerbert is said by Richer to have made for the purpose
of computation a thousand counters of horn; the usual number of a set of
counters in the sixteenth and seventeenth centuries was a hundred.
Treatises on the Abacus usually consist of chapters on Numeration
explaining the notation, and on the rules for Multiplication and
Division. Addition, as far as it required any rules, came naturally
under Multiplication, while Subtraction was involved in the process of
Division. These rules were all that were needed in Western Europe in
centuries when commerce hardly existed, and astronomy was unpractised,
and even they were only required in the preparation of the calendar and
the assignments of the royal exchequer. In England, for example, when
the hide developed from the normal holding of a household into the unit
of taxation, the calculation of the geldage in each shire required a sum
in division; as we know from the fact that one of the Abacists proposes
the sum: “If 200 marks are levied on the county of Essex, which contains
according to Hugh of Bocland 2500 hides, how much does each hide
pay?”[3*] Exchequer methods up to the sixteenth century were founded on
the abacus, though when we have details later on, a different and
simpler form was used.
[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).
+---+---+---+-----+---+---+
| | | | H. | T.| U.|
+---+---+---+-----+---+---+
| | | | | | 2 | Difference.
| | | | | | 8 | Divisor.
+---+---+---+-----+---+---+
| | | |[4*]9| | | +Dividend.+
| | | |[4*]1| 8 | | Product of difference by 1st Quotient (9).
| | | | | 2 | | Product of difference by 2nd Quotient (1).
+---+---+---+-----+---+---+
| | | |[4*]1| | | Sum of 8 and 2.
| | | | | 2 | | Product of difference by 3rd Quotient (1).
| | | | | | 4 | Product of difference by 4th Quot. (2).
| | | | | | | +Remainder.+
+---+---+---+-----+---+---+
| | | | | | 2 | 4th Quotient.
| | | | | 1 | | 3rd Quotient.
| | | | | 1 | | 2nd Quotient.
| | | | | 9 | | 1st Quotient.
+---+---+---+-----+---+---+
| | | | 1 | 1 | 2 | +Quotient.+ (+Total of all four.+)
+---+---+---+-----+---+---+
[Footnote 4*: These figures are removed at the next step.]