An Investigation of Subtraction Algorithms from the 18th and 19th Centuries - Definitions and Algorithms

Definition of Subtraction

The definitions of subtraction presented in the arithmetic books and cyphering books from 1700 to 1900 are not the same as those given in our modern classrooms and textbooks because they exclude the possibility of negative numbers. Edward Cocker (1702, p. 29) wrote,

Subtraction is the taking of a less Number out of a greater of like kind, whereby to find out a third Number, being or declaring the inequality, excess, or difference between the numbers given.

Similarly, John Ayres (1711, p. 33) wrote,

Subtraction teacheth to take the lesser number from a greater, or an equal from an equal; whereby we discover the remainder excess, or difference. 

The phrase a “lesser number from a greater number” is present in nearly all of the books during this time period (e.g., Brookes, 1776; Dilworth, 1810; Adams, 1848). The definitions of subtraction from 1700 to 1900 are stated in a manner that excludes negative numbers. It was unusual to find any mention of negative numbers within the subtraction sections of the books of this time period. However, there were some interesting exceptions. Cocker (1702) did mention the existence of negative numbers. Thomas Weston (1729) included “negative quantities and imaginary numbers” in his explanation of subtraction. These kinds of entries were rarely to be found, which makes Cocker’s and Weston’s presentations particularly interesting as they are from the late 1600s and early 1700s, the early part of our period of study.

The types of subtraction in both the arithmetic books and cyphering books from the 1700s to the early 1900s commonly came in two forms: simple subtraction and compound subtraction (e.g., Dilworth, 1810). Simple subtraction is the subtraction of whole numbers. In some cases, simple subtraction also included the subtraction of decimals (e.g., Colburn, 1855). A section on compound subtraction would either immediately follow the simple subtraction section in the books or would appear slightly later in the book. Compound subtraction, which was a direct application of simple subtraction, typically included problems about quantities relating to dry measures, land measures, money, and other important contexts of the time period. The subtraction was called “compound” because it was composed of a variety of units. For example, suppose that there are 12 pence in a shilling. Subtracting 1 shilling and 7 pence from 3 shillings and 5 pence is an example of a compound subtraction problem. Much rarer was Ayres’ (1711) definition of simple subtraction as subtracting single digit numbers and compound subtraction as subtracting compound numbers, which he defined as whole numbers composed of two, three, four or more digits.  The research focus of this paper is on simple subtraction of whole numbers with any number of digits, although similar algorithms can be found in discussions of compound subtraction as well.       

Minuend, Subtrahend, and Remainder

Figure 3. Minuend, subtrahend, and remainder in Adam’s [sic] New Arithmetic, by Daniel Adams, published in Keene, New Hampshire, in 1830. (This image has been reproduced, with permission, from the arithmetic book belonging to Nerida F. Ellerton and M. A. (Ken) Clements.)

Language issues in subtraction and indeed in all of arithmetic deserve historical investigation, but will not be developed in this paper. We introduce only the terms the reader needs in order to understand the statements present in the texts and cyphering books.  Authors of this time period utilized words such as “minuend,” “subtrahend,” and “remainder,” largely absent from our modern curriculum. The minuend is the larger number and the one that will be “lessened.” The word “minuend” comes from Latin “minuere” which means “to lessen” (Wingate, 1865). The subtrahend is the smaller number or the number being subtracted, and the remainder is the difference between the two numbers.

Although it was a very familiar sight to see the language of minuend, subtrahend, and remainder used in the 1700s to early 1900s, not everyone applied this terminology. Many books used “lesser number” and “greater number” or “lower figure” and “upper figure” rather than “minuend” and “subtrahend” (Brookes, 1776; Colburn, 1824; Daboll, 1829; Dilworth, 1802; Dilworth, 1810; Farrar, 1818; Lee, 1797; Pike, 1809). The words minuend and subtrahend did appear in early texts during this time period, and by the mid-1800s this terminology not only appeared, but was preferred over other terms. All of the texts examined after 1848 used the terms “minuend” and “subtrahend” (Adams, 1848; Colburn, 1855; Ray, 1856; Colburn, 1858; Wingate, 1865; Ray, 1877), but, again, these words are not often used in the modern classroom.

Algorithms Revealed

Four subtraction algorithms were used in different parts of Europe and America in the 18th and 19th centuries. They are the equal additions, decomposition, complement, and Austrian algorithms, and all four algorithms have been identified in the arithmetic books and cyphering books from this time period.  Twentieth and 21st century historical research on subtraction algorithms has largely overlooked the complement algorithm and has not emphasized the scarcity of the Austrian algorithm in North America (Osburn, 1927; Johnson, 1938; Ross & Pratt-Cotter, 2000; Smith, 1909).  

Mathematics educators of the past have recognized five or six subtraction algorithms. David Eugene Smith (1905), an influential historian of mathematics education, wrote in his Handbook to Smith’s Arithmetic: “There are at least three or four methods in common use throughout the country” (p. 56). By 1909 he had increased his total to five algorithms, adding the “left-to-right method” to his earlier list of methods (Smith, 1909). Smith claimed, without development of the algorithm, that the left-to-right method "is more adapted to the needs of a professional computer, however, than to those of the average citizen, and may therefore be dismissed with this mere mention" (Smith, 1909, p. 77).

W. J. Osburn (1927) classified subtraction into three main categories: additive, take-away, and complementary. Osburn’s “additive” approach is referred to in this paper as the Austrian algorithm. Surprisingly, Osburn grouped both the decomposition and equal additions algorithms into his “take away” category. Osburn’s definition of the complementary method of subtraction is the same definition that is used in this paper. Within these general categories Osburn identified sub-methods based on the syntax of the subtraction problems. He considered “2 from 8” and “8 take away 2” as “upward” and “downward” subtraction, respectively, and constructed six different methods of subtraction based on this language. Based on the actual data from the arithmetic textbooks and cyphering books, I will discuss the four established algorithms:

• the equal additions algorithm,
• the decomposition algorithm,
• the complement (or complementary) algorithm, and
• the Austrian algorithm.

Figure 4. The four subtraction algorithms to be discussed in this article.