Abraham Lincoln’s Cyphering Book and the Abbaco Tradition

Journal of the Abraham Lincoln Association, Vol. 36, No. 1, 2015 © 2015 by the Board of Trustees of the University of Illinois

Abraham Lincoln’s Cyphering Book and the Abbaco Tradition

MCKENzIE A. CLEMENTSNERIDA F. ELLERTON

There have been five recent developments with respect to the oldest extant handwritten manuscript of Abraham Lincoln—his cyphering book. The document comprises twenty-two pages (eleven leaves), which were prepared in Indiana, probably between 1820 and 1826.1 The first, and most important, development is the summary and analysis of an extra leaf (two pages) that we located and identified in Harvard University’s Houghton Library in 2009.2 An associated writ-ten document, in William H. Herndon’s hand and dated December 7, 1875, which confirmed that the leaf was indeed prepared by Lincoln, was also found in the Houghton Library, in the same folder as the leaf.3

The second development relates to the terminology used to describe the original manuscript that contained the eleven leaves. It is argued, from literature relating to the history of school mathematics in the United States, that the most appropriate term is “cyphering book” (or “ciphering book”), and that the expression “sum book”—used by most historians who have commented on the oldest of extant Lincoln manuscripts—is an anachronism.4 The third development is a chronological reordering of the eleven leaves based on the order in which the leaves were prepared. This reordering is consistent with the abbaco arithmetic curriculum, which controlled what arithmetic was studied and the order in which topics

1. Lloyd A. Dunlap, “Lincoln’s Sum Book,” Lincoln Herald 61, no. 1 (1959), 6–10. 2. See Nerida F. Ellerton, Valeria Aguirre-Holguín, and McKenzie A. Clements, “‘He Would Be Good’: Abraham Lincoln’s Early Mathematics, 1819–1826,” in Nerida F. Ellerton and McKenzie A. Clements, Abraham Lincoln’s Cyphering Book, and Ten Other Extraordinary Cyphering Books (New York: Springer, 2014), 158–61. 3. William H. Herndon to unidentified person, December 7, 1875, MS Am 1326, gift of Christian A. zabriskie, 1954, Houghton Library, Harvard University. 4. Nerida F. Ellerton and McKenzie A. Clements, Rewriting the History of School Math-ematics in North America 1607–1861 (New York: Springer, 2012), 3–4.

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2 Lincoln’s Cyphering Book

were studied in schools in the United States during the period 1776–1840.5 The fourth development is a reconsideration of Lincoln’s 1859 au-tobiographical statement that “when he came of age” he could cypher “to the rule of three.”6 The analysis presented here indicates that al-though that self-assessment was true, it represented an understate-ment of the future president’s arithmetic achievements at school. The fifth development relates to claims that Lincoln was wrong on all the “casting-out-nines” calculations he carried out when checking division tasks.7 It will be pointed out that, in fact, all his casting-out-nines checks were correct. In this article each of the above five points are discussed, in the order indicated. More detailed analyses of all extant twenty-two pages of Lincoln’s cyphering book are given elsewhere.8

The eleventh extant leaf of Abraham Lincoln’s cyphering book

Previous commentators on Lincoln’s cyphering book (often referred to as a sum book) have assumed there to be only twenty extant pages. In 2009, while examining cyphering books held by Harvard University, we discovered an additional leaf (two pages) prepared by Lincoln.9 The two sides of the leaf are reproduced in Figures 1 and 2. Neither page showed a date, but the Houghton Library holds a letter, dated December 7, 1875, by William H. Herndon, reproduced as Figure 3. The recipient is likely Willard Cutting Flagg (1828–78).10

On the letter Herndon noted, “Date of the leaf 1825, when Mr Lin-coln was 17 ys of age.” Since Lincoln was born on February 12, 1809,

5. Ibid, 1–156. The abbaco curriculum comprised a well-defined sequence of top-ics-numeration (based on Hindu-Arabic numerals), the four arithmetical operations, compound operations, reduction, practice, rules of three, loss and gain, simple and compound interest, discount, equations of payment, alligation, fellowship, position, vulgar fractions, decimal fractions, involution, evolution, arithmetical and geometrical progressions, mensuration, et cetera. This curriculum was translated from Europe to the North American colonies by the early European colonists. 6. Lincoln to Jessie W. Fell, enclosing autobiography, December 20, 1859, Roy P. Basler et al., eds., The Collected Works of Abraham Lincoln, 9 vols. (New Brunswick, N.J.: Rutgers University Press, 1953–55), 3:511. 7. Maurice Dorfman, “Lincoln’s Arithmetic Education: Influence on His Life,” Lincoln Herald, 68, no. 2 (1966), 61–80. 8. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good.’” 9. Ibid. 10. Herndon to unidentified person, December 7, 1875. Douglas Wilson, noted Hern-don scholar, agrees with the authors that Flagg was the likely recipient.

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Figure 3. Excerpt from a letter from William H. Herndon to William Cutting Flagg (?), December 7, 1875. MS Am 1326. Gift of Christian A. zabriskie, 1954. Houghton Library, Harvard University.

Figure 1. The first of the “double rule of three” pages held in the Houghton Li-brary at Harvard University. Page from Abraham Lincoln’s mathematical exercise book, c. 1825. MS Am 1326. Gift of Chris-tian A. zabriskie, 1954. Houghton Library, Harvard University.

Figure 2. The second of the “double rule of three” pages held in the Houghton Li-brary at Harvard University. Page from Abraham Lincoln’s mathematical exercise book, c. 1825. MS Am 1326. Gift of Chris-tian A. zabriskie, 1954. Houghton Library, Harvard University.

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the year 1825 is incompatible with the stated age of seventeen. How-ever, the last-prepared leaf of the eleven extant leaves, which dealt with “discount,” was clearly dated March 1826,11 and traditionally the double rule of three was studied well before discount. Thus, it is likely that Herndon was correct with the 1825 date and wrong with his “17 ys of age” assertion. Probably, Lincoln was almost seventeen when he prepared the leaf held in Houghton Library. According to that institution’s records, the leaf, together with Herndon’s letter, was acquired for the Library in 1954, through a gift by a Christian A. zabriskie.Lincoln’s preparation of the double-rule-of-three pages heralded a new “beyond-the-rule-of-three” period in his cyphering. On pages immediately before these two pages, Lincoln solved prob-lems from an arithmetic textbook authored by Thomas Dilworth,12 but on these double-rule-of-three pages, and on all remaining extant pages of the cyphering book, he grappled with exercises taken from a popular arithmetic by Stephen Pike.13 Louis C. Karpinski listed ten different editions and four associated keys of Stephen Pike’s Arithmetic published between 1811 and 1826.14 In what follows, reference will be made to the 1822 Philadelphia edition of Pike’s textbook (published by M’Carty and Davis), but nobody knows which edition provided the inspiration for the “Pike” examples in Lincoln’s cyphering book. Four of the five questions on the page shown in Figure 1 are identi-cal, or very similar, to questions on pages 84 and 85 of the 1822 edition of Stephen Pike’s Arithmetic. The question at the top left of the page

11. Abraham Lincoln, circa 1825, MS Am 1326, gift of Christian A. zabriskie, 1954, Houghton Library, Harvard University. 12. Thomas Dilworth was an Englishman whose text The Schoolmasters Assistant; Being a Compendium of Arithmetic Both Practical and Theoretical was widely used both in Great Britain and in North America during the period 1750–1825. Louis C. Karpinski, in his Bibliography of Mathematical Works Printed in America through 1850 (New York, Arno Press, 1980), noted that fifty-seven different American editions of Dilworth’s Arithmetic were published, by various publishers, between 1773 and 1825 (73–79). However, according to Ellerton, Valeria Aguirre-Holguín, and Clements, between 1819 and 1826, when Lincoln prepared his cyphering book, the influence of Dilworth’s text on U.S. school mathematics had waned, with the works of U.S.-born authors such as Daniel Adams, Warren Colburn, Nathan Daboll, Stephen Pike, and Michael Walsh being preferred. It is not known which actual edition of Dilworth’s Arithmetic was the source of the numerous “Dilworth examples” in Lincoln’s cyphering book (“‘He would be Good,’” 129). 13. Stephen Pike, The Teacher’s Assistant, or A System of Practical Arithmetic (Philadel-phia: M’Carty and Davis, 1822). 14. Karpinski, Bibliography of Mathematical Works, 185–87. All of the editions of Ste-phen Pike’s Arithmetic were similar.

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McKenzie A. Clements and Nerida F. Ellerton 5

shown in Figure 1 states, “If 100£ in 12 months gain 7£ interest what principal will gain 3£-18–9d in 9 months.” Question 5 in Pike’s 1822 edition states, “If 100£ in 12 months gain 6£ interest, what sum will gain 3£.7s.6d. in 9 months?”15 Some of the numbers differed slightly, but the structures were the same. These “inverse proportion” problems were more difficult than direct-rule-of-three problems. Pike stated the rule for inverse proportion tasks as “Transpose the inverse extremes; that is, set that which is in the first place under the third; and that which is in the third place under the first; then work as in Direct Pro-portion.”16 Despite the complex wording, Lincoln was able to proceed, and he applied the rule in his own way. All of the future president’s answers were correct, and although his solutions were set out according to a succinct problem-calculation-answer genre,17 his logic was easy to follow. Lincoln was usually a good speller, but he—or perhaps his teacher—misspelled “Application” in the calligraphic heading shown at the top of Figure 2. The example at the bottom right of Figure 2 (“If the tuition of 3 boys for two quarters of a year be $40–20 cts how much will the tuition of 60 boys amount to for 4½ years”) could be found in an arithmetic by Jacob Willetts.18 Although this is the only question from that source that we have found in the twenty-two extant pages, it is true, nonetheless, that throughout the final ten extant pages, problems from a number of different textbooks can be found. The most likely explanation is that at least one of Lincoln’s teachers during that pe-riod owned a cyphering book that included the problems, and either Lincoln copied problems from that cyphering book or the problems were dictated to him by his teacher. The two examples at the top of Figure 2 (“If 4 men in 5 days eat 7 lb of bread, how much will be sufficient for 16 men in 15 days” and “If 100 dollars in one year gain 3½ dollars interest, what sum will gain $38–50 cents in one year and a quarter”) were from Stephen Pike’s book.19 These problems were in a section that Pike headed “Promiscu-ous Examples.” They belonged to major classes of problems that had been dealt with earlier in the book. A student’s task was to decide, for each problem, which major class, which subclass, and so forth, was

15. Stephen Pike, The Teacher’s Assistant, 84. 16. Ibid, 83. 17. Ellerton and Clements, History of School Mathematics, 49–53. 18. Jacob Willetts, The Scholar’s Arithmetic Designed for the Use of Schools in the United States (Poughkeepsie, N.Y.: Paraclete Potter, 1822), 92. 19. Stephen Pike, The Teacher’s Assistant, 85.

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relevant, and then to solve the problem by the best method. In other words, the student was invited to reflect on which method was most appropriate to solve a problem. Sets of “promiscuous” problems,”20 sometimes called “miscellaneous problems,”21 were to be found in many arithmetics used in the eighteenth and nineteenth centuries. They were also in many students’ cyphering books.22 It was assumed that selecting appropriate methods, and developing and writing down correct solutions to problems, compelled students to “reflect” on prob-lem structures and made them likely to recognize and solve structur-ally diverse problems in the future.23

Many students solved double-rule-of-three problems without ap-preciating why the rule was appropriate. Lincoln systematically ap-plied the rules, and each of his solutions was carefully worked out before he committed it to his cyphering book—rarely did he cross anything out.24

Harvard University holds another cyphering book prepared in the mid-1750s by another Abraham Lincoln—a great half-uncle of the future president.25 The lineage of the young Abraham Lincoln who prepared a cyphering book in Spencer County, Indiana, in the early 1820s did include at least one person who had prepared a cyphering book.

Why use the term “cyphering book” and not “sum book”?

A cyphering book has been defined as a handwritten manuscript that focused on mathematical content and had the following four proper-ties:

1. Either the contents were written by a student or by a teacher who wished to use it as a model that could be followed by students preparing their own cyphering books.

20. See, for example, Titus Bennett, New System of Practical Arithmetic Peculiarly Cal-culated for the Use of Schools in the United States (Philadelphia: Bennett and Walton, 1815); William Butler, An Introduction to Arithmetic Designed for the Use of Young Ladies (London: Simpkin, Marshall, 1819). 21. See, for example, Consider Sterry and John Sterry, A Complete Exercise Book, in Arithmetic: Designed for the Use of Schools in the United States (Norwich, CT: John Sterry, 1795), 115–17. 22. Ellerton and Clements, History of School Mathematics, 143. 23. Warren Van Egmond, Practical Mathematics in the Italian Renaissance: A Catalog of Italian Abbacus Manuscripts and Printed Books to 1600 (Florence, Italy: Istituto E Museo di Storia Della Scienza, 1980). 24. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good,’” 163. 25. Ellerton and Clements, Lincoln’s Cyphering Book, 124.

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2. All entries, including handwriting and illustrations, were in ink, with headings being presented in calligraphic style.

3. Rules, cases, problems, and solutions to exercises associated with a well-defined progression of mathematical (usually arith-metic) topics were recorded in the book.

4. The content reflected the expectation that, normally, children would not begin to prepare cyphering books before they were ten years of age.26

Abraham Lincoln’s cyphering book can be regarded as an artifact that belongs to two major, but usually quite separate, domains of knowledge: Lincoln studies and the history of school mathematics. This present article belongs at the intersection of the literatures for those domains. It hardly needs to be said that that intersection has rarely been addressed by scholars with experience in both domains. Although we have examined hundreds of cyphering books in nu-merous archives across the United States, we have not found one labeled “Sum Book.” On the other hand, we have found many on which the words “Cyphering Book” or “Ciphering Book” were writ-ten. Almost certainly, in the 1820s the oldest extant handwritten manu-script of Abraham Lincoln would have been known as a cyphering book or a ciphering book and not as a sum book. Yet the entire page introducing the first set of documents in The Collected Works of Abraham Lincoln is devoted to “pages from Lincoln’s sum book, 1824–1826.” A “cyphering tradition” controlled European school mathematics for more than five hundred years, and that tradition was transplanted into North American schools during the period 1607–1861.27 In the 1820s almost all U.S. school students aged ten years or more who studied school arithmetic were expected to prepare cyphering books. There was a well-defined curriculum that historians have called the abbaco curriculum,28 and the topics in Abraham Lincoln’s book, as well as the order of dealing with those topics, reflected that curriculum. That said, in a paper written in 1894, Thomas Jefferson Chapman maintained that “sum-book” was the term he used for the handwrit-ten arithmetic text he prepared at a country school in Pennsylvania in the 1840s.29 In a footnote to Chapman’s article, the editor of the

26. Ellerton and Clements, History of School Mathematics, 3–4. 27. Ellerton and Clements, History of School Mathematics, 7; Van Egmond, Practical Mathematics. 28. Jens Høyrup, “Leonardo Fibonacci and Abbaco Culture: A Proposal to Invert the Roles,” Revue d’Histoire des Mathématiques 11 (2005), 23–56. 29. Thomas Jefferson Chapman, “Sum-Books,” Pennsylvania School Journal 42 (Janu-ary 1894), 297–98.

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Pennsylvania School Journal recalled that in the 1840s students at the school he attended—the zook school, a country school in Lancast-er County, Pennsylvania—such books were called “settling-down books.” It appears to have been the case that different terms were used in different places. But our evidence, based on analysis of more than a thousand manuscripts, points strongly to the conclusion that the most commonly used term, by far, was “cyphering” (or “cipher-ing”) book. Sometime in the nineteenth century, Lincoln scholars started refer-ring to Lincoln’s cyphering book as a sum book, and the term has taken hold among those who write about Lincoln’s early education.30 Although “sum” has been used colloquially for centuries to denote an arithmetic exercise, the term “sum book” has been much less used, and there is no evidence that Lincoln himself used that term. There can be no doubt, however, that in 1859 Lincoln recalled that when he was at school in the 1820s he had been engaged in “readin’, writin’, and cipherin’.”31 Furthermore, in 1891 Dennis Hanks, who lived with the Lincolns when Abraham Lincoln went to school, remembered him as “always reading, writing, cyphering, writing poetry.”32

From a historical perspective, the use of “sum book” takes atten-tion away from the cyphering tradition that, arguably, is worthy of much more scholarly attention than it has been given. North American families in the 1700s and early 1800s had very few books. Most had a Bible, and many managed to purchase an almanac, which provided them with meteorological and astronomical summaries, as well as anecdotal and topical information. The only other books commonly found in homes were cyphering books—which had been prepared by family members and were highly prized as guides for dealing with “figgers.”33

The term “copy book” (or “copybook”) has also often been used to describe a cyphering book, and that term was certainly used in the early nineteenth century. However, at that time it was used in a general way, describing any handwritten manuscript into which mate-

30. See, for example, William E. Bartelt’s “There I Grew Up”: Remembering Abraham Lin-coln’s Indiana Youth (Indianapolis: Indiana Historical Society Press, 2008), 30–31; Pages from Lincoln’s sum book, 1824–26, Basler, Collected Works, vol. 1; Dunlap, “Lincoln’s Sum Book,” 6–10; Lincoln Sesquicentennial Association, Abraham Lincoln Sesquicenten-nial, 1959 (Urbana: University of Illinois). 31. Lincoln to Fell, December 20, 1859, Basler, Collected Works, 3:511. 32. Isaac N. Arnold, The Life of Abraham Lincoln (Chicago: A. C. McClurg, 1891), 24. 33. James P. Wickersham, A History of Education in Pennsylvania (Lancaster, Pa.: In-quirer, 1886).

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rial had been carefully copied, especially manuscripts in which there had been an attempt to incorporate fine penmanship or calligraphy.34 Before 1800 in North America, a copybook focusing on arithmetic was usually called a “cyphering book” (with a y), but in the United States during the first half of the nineteenth century “ciphering book” (with an i) was increasingly preferred.35

A chronological reordering of the eleven extant leaves from Lincoln’s cyphering book

The order in which Lincoln prepared the ten extant leaves (excluding the double-rule-of-three leaf) has been a subject for discussion, with the main attempts at ordering being made by the editors of The Collected Works of Abraham Lincoln, Maurice Dorfman, and Louis Warren.36 In this section the eleven extant leaves are reordered, with placement being based on the standard ordering of topics in the abbaco curriculum that was central to the cyphering tradition. Table 1, which summarizes the arithmetic topics dealt with on the leaves, shows chapter numbers for those topics in four popular arithmetics—by Thomas Dilworth, Stephen Pike, zachariah Jess, and Francis Walkingame.37

Numerous editions of each arithmetic were published, and the ac-tual editions from which the problems Lincoln solved may or may not have corresponded to the exact years attributed to the texts in Table 1. There are two rows in Table 1 with “Simple Interest” in the left column; that is because there are two extant leaves (four pages) devoted to simple interest. The chapter orders for the Pike, Jess, and Walkingame texts are in agreement (statisticians would say that the rank-order correlations for the three ordered sets is 1.00). That is not surprising, because the main educational factor influencing the order in which the three authors

34. George Bickham, Youth’s Instructor in the Art of Numbers, a New Cyphering Book in Which Is Shewn Variety of Penmanship & Command of Hand: Engrav’d for the Use of Schools (London: William and Cluer Dicey, 1740). 35. Ellerton and Clements, History of School Mathematics, 5–6. 36. Pages from Lincoln’s sum book, 1824–26, Basler, Collected Works, vol. 1; Dorfman, “Lincoln’s Arithmetic Education”; Louis A. Warren, Lincoln’s Youth: Indiana Years Seven to Twenty-One, 1816–1830 (New York: Appleton, 1959). 37. Thomas Dilworth, The Schoolmaster’s Assistant; Being a Compendium of Arithmetic Both Practical and Theoretical (New York: G. Jansen, 1806); Stephen Pike, The Teacher’s Assistant; zachariah Jess, The American Tutor’s Assistant, Improved; or A Compendious System of Decimal, Practical Arithmetic (Philadelphia: M’Carty and Davis, 1821); Francis Walkingame, The Tutor’s Assistant; Being a Compendium of Arithmetic, and a Complete Question-Book (London: C. Whittingham, 1802).

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placed the chapters on the given topics was the assumption of appro-priate order in the abbaco curriculum. Thus, for example, one would not have taught compound interest before teaching simple interest, and one would not have taught discount before multiplication. It is reasonable to assume, therefore, that the order of topics in the left column of Table 1 represents the order in which Lincoln completed the eleven leaves. The order of topics for Thomas Dilworth’s text differs from that for the other three texts. For example, Dilworth dealt with compound addition of quantities immediately after, and in the same chapter as, simple addition. The other three authors dealt with compound op-erations for addition, subtraction, multiplication, and division only after they had completed all four simple operations. There is evidence (see Table 2) that Lincoln was influenced by Dilworth’s text early in his school years, but the influence of Dilworth was not sustained. Although we have chosen to order the eleven extant leaves on the basis of the abbaco order assumed by Pike, Jess, and Walkingame, we recognize that the order would have been different if Dilworth’s order was determinative. Table 2 shows our ordering of the topics on the eleven leaves and the orders given by Roy P. Basler and the other editors of The Collected Works, by Warren, and by Dorfman. In 1953 Basler and his editors did not claim to have established an appropriate ordering. Nevertheless, they appeared to put the leaves in an order that they thought suitable. In the right-hand column of Table 2 we note texts in which identi-cal, or almost identical, problems to those solved by Lincoln could be found. No text was associated with the first three leaves. The arithme-

Table 1. Order of Topics in Lincoln’s Cyphering Book Compared with Orders in Four Textbooks That Had Some Influence on Abraham Lincoln

Topic covered on leaf Dilworth Chapter number Jess Walkingame (1806) in Pike (1822) (1821) (1802)

Simple subtraction 3 3 3 3Simple multiplication 4 4 4 4Simple division 5 5 5 5Compound addition 2 8 6 7Compound multiplication 4 10 15 9Single rule of 3 7 13 20 13Double rule of 3 23 14 22 15Simple interest Leaf(1) 11 17 25 18Simple interest Leaf(2) 11 17 25 18Compound interest 13 19 26 22

Discount 14 20 27 23

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tic for those leaves was sufficiently standard that Lincoln’s teachers (or Lincoln himself) could easily have created the problems. Then for three leaves, problems from Thomas Dilworth’s text were identified. Problems from Stephen Pike’s Arithmetic could be associated with all the remaining leaves, and with the final leaf (on discount), problems from Jess’s 1821 text and Walkingame’s 1802 text were also identified. The changing pattern of textbook influences is striking.38

Spearman-rho rank-order correlations between our order for the then leaves known to Basler and his editors, to Warren, and to Dorf-man, and the orders indicated by these other scholars are 0.62, 0.84 and 0.96, respectively—statistics that suggest a convergence over time. We are not certain that the orders we have given for the simple subtraction and compound addition leaves are correct. Our ordering

38. See Ellerton, Aguirre-Holguín, and Clements “‘He Would Be Good,’” for a more detailed discussion.

Table 2. Our Ordering of Topics in Lincoln’s Cyphering Book Compared with Orders in Four Textbooks that Could Have Influenced the Order for Lincoln

Topic covered on leaf Our order Order implied by Basler et al.

(1953)

Warren’s (1959) order

Dorfman’s (1966) order

Linked with a textbook?

Simple subtraction 1 3 1 3Simple multiplication 2 5 3 1Simple division 3 1 4 2Compound addition 4 9 2 4 Dilworth (1806),

p. 13 and p. 35Compound

multiplication5 2 5 5 Dilworth (1806),

p. 43 and p. 44Single rule of 3 6 4 10 6 Dilworth (1806),

p. 65 and p. 66Double rule of 3 7 Did not list Did not list Did not list Pike (1822), p. 84

and p. 85Simple interest Leaf (1) 8 6 6 7 Pike (1822), p. 102

and p. 103Simple interest Leaf (2) 9 7 7 8 Pike (1822), p. 108

and p. 109Compound interest 10 10 8 9 Pike (1822), p. 111

and p. 111Discount 11 8 9 10 Pike (1822), p. 112

and p. 113; Jess (1821), p. 106 and

p. 107; Walkingame (1802), p. 63

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for the pages was based on the usual abbaco curriculum, on patterns arising from texts that can be linked to pages, on imprints from other pages, and on comparing patterns in which edges of pages are torn. If the orders were to be based on Dilworth’s curriculum sequence, then the compound addition leaf would be first. Although future analysis or technology could possibly lead to a further reordering of the pages, we are confident that our ordering is correct, or close to correct. Parenthetically, we should add that the textbooks to which refer-ence is made in the right-hand column of Table 2 do not include books by Nicolas Pike or by Nathan Daboll, authors of two popular North American arithmetics.39 Some writers have suggested that Lincoln may have had access to either or both the arithmetics written by those authors.40

We have not found any evidence that Lincoln had access to a copy of any arithmetic written by Daboll, but there is a possibility that he had access to Nicolas Pike’s text. Herndon and Weik recorded that all the students at Lincoln’s school learned to cypher “and afterwards used Pike’s arithmetic.”41 Furthermore, Leonard Swett stated that Lincoln made mention of an “old dog-eared arithmetic” in the Lincoln house, and it is possible that that arithmetic had been authored by Nicolas Pike.42 However, as Table 2 makes clear, Stephen Pike’s textbook was the source of quite a few of the exercises that Lincoln completed in his cyphering book. That Lincoln solved problems found in arithmetics authored by Thomas Dilworth, zachariah Jess, Stephen Pike, Francis Walkingame, and Jacob Willetts does not imply that he, or one or more of his teach-ers, owned arithmetics written by those authors. As Lloyd Dunlap pointed out, the teachers may have had their own cyphering books that included exercises from the textbooks.43 Careful analysis of the

39. For Pike, see Nicolas Pike, A New and Complete System of Arithmetic, Composed for the Use of Citizens of the United States (Newbury-Port, Mass.: John Mycall, 1788); for Daboll, Nathan Daboll, Daboll’s Schoolmaster’s Assistant; Improved and Enlarged; Being a Plain Practical System of Arithmetic Adapted to the United States (Auburn, N.Y.: Thomas M. Skinner, 1820). 40. Martin Luther Houser, Young Abraham Lincoln, Mathematician (Peoria, Ill.: L. O. Schriver, 1943). 41. William H. Herndon and Jessie W. Weik, Herndon’s Life of Lincoln: The History and Personal Recollections of Abraham Lincoln (New York: Albert and Charles Boni, 1936), 34. 42. Leonard Swett, “Mr. Lincoln’s Story of His Own Life,” in Allen Thorndike Rice, ed., Reminiscences of Abraham Lincoln by Distinguished Men of His Time (New York: Haskell, 1888), 458. 43. Dunlap, “Lincoln’s Sum Book,” 10.

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extant pages of Lincoln’s cyphering book has revealed that some of the problems recorded in book were dictated to him.44

Going beyond the rule of three

In Abraham Lincoln’s cyphering book the rule of three was reached about halfway through the book. After that, Lincoln studied the dou-ble rule of three, simple interest, compound interest, and discount—all of which were topics usually regarded as “beyond the rule of three” (and therefore treated in textbooks after the chapter on the single rule of three). That fact must be considered in relation to the following famous autobiographical statement that Lincoln prepared in 1859, for Jesse Fell, for possible used in a presidential campaign:

My father . . . grew up, litterally without education. He removed from Kentucky to what is now Spencer county, Indiana, in my eighth year. We reached our new home about the time the State came into the Union. It was a wild region, with many bears and other wild animals still in the woods. There I grew up. There were some schools, so called; but no qualification was ever required of a teacher, beyond readin, writin and cipherin to the Rule of Three. If a straggler supposed to understand latin, happened to sojourn in the neighborhood, he was looked upon as a wizzard. Of course, when I came of age, I did not know much. Still, some-how, I could read, write, and cipher to the Rule of Three; but that was all. I have not been to school since.45

There can be little doubt that Lincoln knew he had cyphered be-yond the rule of three, but he crafted his autobiographical statement so that the image it would project would be of a capable, determined, self-made man who, despite difficult surrounding circumstances, pro-ceeded as far in formal education as he needed to go but not further. Warren’s placement of the leaf dealing with the rule of three as the last-prepared of the extant leaves from Lincoln’s cyphering book was probably influenced by Lincoln’s autobiographical statement to Fell.46

Having cyphered to the rule of three was a definite achievement, one that would immediately be added to a young person’s portfolio and to his or her curriculum vitae. But, of course, having cyphered beyond the rule of three provided, for some, an even more impressive credential.

44. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good,’”129. 45. Lincoln to Fell, December 20, 1859, Basler, Collected Works 3:511–12. 46. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good,’” 156–59.

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Although the abbaco curriculum, which had developed over many centuries, defined the order in which arithmetic was studied in the schools, it should not be imagined that that curriculum was logically and mathematically well structured. For example, because he followed an abbaco curriculum during the period 1824–26, Lincoln found himself trying to solve problems concerned with simple interest, compound interest, and discount despite never having formally studied decimal fractions, common (i.e., vulgar) fractions, or percentages. Those topics came later in the abbaco sequence. Analyses of Lincoln’s efforts on the leaves dealing with simple and compound interest and discount have revealed that with some problems his lack of knowledge of decimal and vulgar fractions forced him to use algorithmic procedures that he did not fully understand.47

There are signs in Lincoln’s answers to the “discount” problems that he was becoming discouraged with his cyphering. There were no longer any pages left in his cyphering book, he did not fully under-stand what he was doing, and he was no longer always getting right answers. The date on his last leaf, on discount, was March 1, 1826. Like many other youths who lived on farms in the Midwest, school was for winter months only; the coming of the month of March marked the time when quills were replaced by hoes, and the land was prepared for crop.48 Young Lincoln would not have relished that thought. He had cyphered beyond the rule of three, and was probably beginning to wonder about what he might be able to achieve in the wider world beyond the horizons that had constrained him for so long.

The accuracy, extent, and presentation of Abraham Lincoln’s arithmetic

Having examined more than a thousand cyphering books prepared in North America between 1667 and 1861, it is with some author-ity that we can assert that from a purely arithmetic point of view, young Abraham Lincoln’s cyphering book was not very special. In the year 1826 there would have been hundreds, perhaps thousands, of young people, mostly male, cyphering, and many of them would have reached the rule of three—usually at ages younger than when Lincoln could say he had cyphered to the rule of three. There were also many other youths, often younger than Lincoln, who had proceeded

47. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good,’” 123–86. 48. John Mack Farragher, Sugar Creek: Life on the Illinois Prairie (New Haven, Conn: Yale University Press, 1988), 100.

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well beyond the topic discount to topics like alligation, fellowship, position, equation of payments, involution, evolution, mensuration, permutations, combinations, arithmetical progressions, and geometri-cal progressions.49 From the penmanship and calligraphy perspec-tive, too, Lincoln’s cyphering was nothing special. To his credit, he had consistently prepared attractive headings, and his writing was reasonably tidy. But, because he had only a small amount of paper, he had written as small as he could, and his entries always gave the impression of being crammed together. Any assessment of Lincoln’s cyphering prowess based solely on the amount of arithmetic he had studied, or on how attractive his entries looked, would be unfair. Although this article has not presented a care-ful question-by-question analysis of his cyphering book, that analysis has been done, and there can be little doubt that he was exception-ally strong in arithmetic and was willing to persevere to achieve an understanding of what he was doing.50

The frontier conditions surrounding the future president when he prepared his cyphering book should be recognized. He was living in a twenty- by eighteen-foot cabin, usually with at least six other people. At best, his teachers knew only a little more about arithmetic than he knew, and it is possible that some or all of them did not know as much. Flat surfaces for cyphering were hard to find, so most of the entries in his cyphering book had to be done during school hours. Even at school, conditions for cyphering were not ideal. At home, his family desperately needed his labor, as well as any income he could earn by assisting neighbors. Often, he was forced to find time for study when others were asleep or were enjoying themselves outdoors.51

When the conditions described in the previous paragraph are taken into account, it becomes obvious that Lincoln’s cyphering book rep-resents a huge personal achievement. We currently own more than 360 North American cyphering books and have examined many oth-ers. In our own collection there are only three cyphering books that originated in Indiana and none that originated in Illinois. By contrast, well over half of the cyphering books we have seen originated in New York, Massachusetts, or Pennsylvania. There can be no doubt that it was much easier to prepare a worthwhile cyphering book in New

49. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good,’” 181. 50. Ibid., 155–83. 51. See, for example, Arnold, Abraham Lincoln, 13–27; Bartelt, “There I Grew Up,” 28–30; Michael Burlingame, Abraham Lincoln: A Life (Baltimore: Johns Hopkins University Press, 2008), 1, 22–48.

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York City, Boston, or Philadelphia than it was in rural Indiana. From that perspective, Lincoln’s cyphering book begins to look impressive, even extraordinary. Both David Herbert Donald and Maurice Dorfman claimed that Lincoln made errors when casting out nines in order to check his calculations, but we and Valeria Aguirre-Holguín found only one casting-out-nines error, and that entry was probably made by a person other than Lincoln—perhaps his teacher. It was Donald and Dorfman who were wrong, not Lincoln.52

Concluding comments

Five important points have been made in this article with respect to the twenty-two extant pages of Lincoln’s cyphering book. The first was that as a result of a recent finding, there are now twenty-two extant pages, not twenty. The second was concerned with whether it has been wise for Lincoln scholars to refer to Lincoln’s cyphering book as his “sum book.” It was argued that the use of the colloquial term “sum book” has been erroneous because it is unlikely that it was used by Lincoln in the 1820s and because its use has made it easier for scholars to avoid mak-ing themselves aware of the cyphering tradition that controlled school mathematics in North America for over two centuries. The third was concerned with the order in which Lincoln prepared the eleven extant cyphering book leaves. Arguing mainly from our knowledge of the ab-baco curriculum, and from the pattern of textbook data associated with the extant pages, we put forward an order that, we believe, is accurate (see Table 2). The fourth contribution was to show that Lincoln clearly undersold his mathematical education when he informed Jessie Fell in 1859 that he only cyphered to the rule of three. Lincoln’s cyphering book included leaves dedicated to the double rule of three, simple in-terest, compound interest, and discount, and each of those topics was regarded as beyond the rule of three. The fifth main point refuted claims that Lincoln made errors whenever he cast out nines when checking his calculations. In fact, he got all of his casting-out-nines checks correct. Lincoln was strong arithmetically. Lincoln did not go far in mathematics while attending school. To leave school at seventeen having cyphered just to discount could be regarded, from national and international perspectives, as quite a modest achievement.53 Many students proceeded well beyond dis-

52. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good,’” 180. 53. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good,’” 179.

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count. However, the arithmetic that Lincoln completed at school was usually done in a convincing way. Almost always he appeared to understand what he was doing, and at all times he worked with an obvious determination not to proceed unless he did know what he was doing. Likewise, from a calligraphic and penmanship perspective, Lincoln’s manuscript was not particularly attractive—many cypher-ing books were much more striking in their appearance. But Lincoln’s book carried a message that the person who prepared it was efficient and down-to-earth, someone who tried hard to present his work in a compact, neat, and correct way. Lincoln had a penchant for writing poetry, and some who have pre-viously written about Lincoln’s cyphering book paid more attention to small pieces of doggerel that he inserted on some of the pages than to the mathematics.54 It is likely that Lincoln wrote some of the little ditties in his book several years after he actually did the mathematics on the pages.55 The ditty destined to be most quoted was “Abraham Lincoln, his hand and pen. He will be good, but God knows when.”56

The eleven leaves from Lincoln’s cyphering book deserve a special place in history. They represent the earliest examples remaining of his handwriting, and reflect the effort he put into filling the pages with appropriate rules, problems, and solutions. This was his book, created “by his hand and pen.” These leaves came from the formative years of his life, years that would prepare him in remarkable ways for what lay ahead. Understanding something of the structure and content of his early cyphering work gives us a small but powerful glimpse of the character, commitment, and thirst for knowledge of a lad from Indiana named Abraham. He would be good.

54. Bartelt, “There I Grew Up,” 43–51. 55. Ellerton, Aguirre-Holguín, and Clements, “‘He Would Be Good,’” 140, 147, 151. 56. Pages from Lincoln’s sum book, 1824–26, Basler, Collected Works, vol. 1.

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Abraham Lincoln’s Cyphering Book and the Abbaco Tradition

Documents

Transcript of Abraham Lincoln’s Cyphering Book and the Abbaco Tradition