The Earliest Arithmetics in English

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(b) Division: direct. 100,000 ÷ 20,023. Here each counter in turn is a separate divisor.

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.)

Division. 7800 ÷ 166.

Thousands
H.	T.	U.	H.	T.	U.
				3	4	Differences (making 200 trial divisor).
			1	6	6	Divisors.
		[4]7	8			Dividends.
		1				Remainder of greatest dividend.
			1	2		Product of 1st difference (4) by 1stQuotient (3).
			9			Product of 2nd difference (3) by 1stQuotient (3).
		[4]2	8	2		New dividends.
			3	4		Product of 1st and 2nd difference by 2ndQuotient (1).
		[4]1	1	6		New dividends.
				2		Product of 1st difference by 3rdQuotient (5).
			1	5		Product of 2nd difference by 3rdQuotient (5).
			[4]3	3		New dividends.
			1			Remainder of greatest dividend.
				3	4	Product of 1st and 2nd difference by 4thQuotient (1).
			1	6	4	Remainder (less than divisor).
					1	4th Quotient.
					5	3rd Quotient.
				1		2nd Quotient.
				3		1st Quotient.
				4	6	Quotient.

Division. 8000 ÷ 606.

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 1stQuotient (1).
		[4]1	9	4		New dividends.
			3			Remainder of greatest dividend.
				9	4	Product of difference 1 and 2 with 2ndQuotient (1).
		[4]1	3	3	4	New dividends.
			3			Remainder of greatest dividend.
				9	4	Product of difference 1 and 2 with 3rdQuotient (1).
			7	2	8	New dividends.
			6		6	Product of divisors by 4thQuotient (1).
			1	2	2	Remainder.
					1	4th Quotient.
					1	3rd Quotient.
					1	2nd Quotient.
				1		1st Quotient.
				1	3	Quotient.

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.”