+-------------+-----------+
| Thousands | |
+---+---+-----+---+---+---+
| H.| T.| U. | H.| T.| U.|
+---+---+-----+---+---+---+
| | | | | 9 | | Difference (making 700 trial divisor).
| | | | | | 4 | Difference.
| | | | 6 | | 6 | Divisors.
+---+---+-----+---+---+---+
| | |[4*]8| | | | +Dividend.+
| | | 1 | | | | Remainder of dividend.
| | | | 9 | 4 | | Product of difference 1 and 2 with
| | | | | | | 1st Quotient (1).
+---+---+-----+---+---+---+
| | |[4*]1| 9 | 4 | | New dividends.
| | | | 3 | | | Remainder of greatest dividend.
| | | | | 9 | 4 | Product of difference 1 and 2 with 2nd
| | | | | | | Quotient (1).
+---+---+-----+---+---+---+
| | |[4*]1| 3 | 3 | 4 | New dividends.
| | | | 3 | | | Remainder of greatest dividend.
| | | | | 9 | 4 | Product of difference 1 and 2 with 3rd
| | | | | | | Quotient (1).
+---+---+-----+---+---+---+
| | | | 7 | 2 | 8 | New dividends.
| | | | 6 | | 6 | Product of divisors by 4th Quotient (1).
+---+---+-----+---+---+---+
| | | | 1 | 2 | 2 | +Remainder.+
| | | | | | 1 | 4th Quotient.
| | | | | | 1 | 3rd Quotient.
| | | | | | 1 | 2nd Quotient.
| | | | | 1 | | 1st Quotient.
+---+---+-----+---+---+---+
| | | | | 1 | 3 | +Quotient.+
+---+---+-----+---+---+---+
[Footnote 4*: These figures are removed at the next step.]
The chief Abacists are Gerbert (tenth century), Abbo, and Hermannus
Contractus (1054), who are credited with the revival of the art,
Bernelinus, Gerland, and Radulphus of Laon (twelfth century). We know as
English Abacists, Robert, bishop of Hereford, 1095, “abacum et lunarem
compotum et celestium cursum astrorum rimatus,” Turchillus Compotista
(Thurkil), and through him of Guilielmus R. . . . “the best of living
computers,” Gislebert, and Simonus de Rotellis (Simon of the Rolls).
They flourished most probably in the first quarter of the twelfth
century, as Thurkil’s treatise deals also with fractions. Walcher of
Durham, Thomas of York, and Samson of Worcester are also known as
Abacists.
Finally, the term Abacists came to be applied to computers by manual
arithmetic. A MS. Algorithm of the thirteenth century (Sl. 3281,
f. 6, b), contains the following passage: “Est et alius modus secundum
operatores sive practicos, quorum unus appellatur Abacus; et modus ejus
est in computando per digitos et junctura manuum, et iste utitur ultra
Alpes.”
In a composite treatise containing tracts written A.D. 1157 and 1208, on
the calendar, the abacus, the manual calendar and the manual abacus, we
have a number of the methods preserved. As an example we give the rule
for multiplication (Claud. A. IV., f. 54 vo). “Si numerus multiplicat
alium numerum auferatur differentia majoris a minore, et per residuum
multiplicetur articulus, et una differentia per aliam, et summa
proveniet.” Example, 8 × 7. The difference of 8 is 2, of 7 is 3, the
next article being 10; 7 - 2 is 5. 5 × 10 = 50; 2 × 3 = 6. 50 + 6 = 56
answer. The rule will hold in such cases as 17 × 15 where the article
next higher is the same for both, _i.e._, 20; but in such a case as
17 × 9 the difference for each number must be taken from the higher
article, _i.e._, the difference of 9 will be 11.