History of Mathematics Project | Lincoln's Cyphering Book

History of
Mathematics

Project

1824

Lincoln's Cyphering Book

President Lincoln's multiplication practice

Cyphering books (where "cyphering" is an archaic term for arithmetic) are practice books written by students. Their use dates back centuries and may have originated in the Middle East and India. A cyphering book was typically prepared by an instructor, used by a male student and written in ink. The subjects in the cyphering book began with basic arithmetic and progressed to higher levels of mathematics as the student advanced. Cyphering books were often shared among family members or passed down from parent to child; the two most valued books a family owned were often the Bible and a cyphering book.

Like many children, Abraham Lincoln, the 16th president of the United States, learned to read, write and do his 'rithmetic. This page (the fourth in the numbering of Ellerton and Clements) from Lincoln's cyphering book, shows how he learned to multiply numbers and "cast out nines" to check his work. In all, 22 pages of Lincoln's cyphering book survive but are spread across multiple locations in the United States.

Artifact dimensions

1.5 cm × 12.9 cm × 20.4 cm

Artifact origin

Indiana, United States

Current artifact location

Rare Book & Manuscript Library, Butler Library, Columbia University in the City of New York

Catalog number

various

Timeline

Interactive Content

Computational Explanation

The first multiplication exercise on the page "Multiplication Continued" (the fourth page in Lincoln's cyphering book in the numbering of Ellerton and Clements), multiplies 3456782 (the multiplier, denoted M for reasons unknown) by 30406 (the multiplicand, denoted B for reasons equally unknown):

In[]:=

multiplier=3456782;multiplicand=30406;

Since the multiplicand contains two zeros, only the 6, 4 and 3 must be multiplied by the multiplier, and these multiplications are shown in the three rows under the first horizontal line:

To verify Lincoln's multiplication, decompose the multiplicand into decimal digits:

MultiplicandDigits=IntegerDigits[multiplicand]

Out[]=

{3,0,4,0,6}

Now multiply back up by appropriate powers of 10 to get the actual sequence of numbers in the decomposition:

In[]:=

MultiplicandDecomposition=DeleteCases[MultiplicandDigits10^Range[Floor[Log10[multiplicand]],0,-1],0]//Reverse

Out[]=

{6,400,30000}

Taking the product of the multiplier by each part of the decomposition of the multiplicand gives the following three numbers, as correctly written in Lincoln's book:

In[]:=

rows=multiplierMultiplicandDecomposition

Out[]=

{20740692,1382712800,103703460000}

Adding these pieces gives the final product:

In[]:=

product=Total[rows]

Out[]=

105106913492

Performing the multiplication directly (with the aid of a computer) shows agreement:

In[]:=

multipliermultiplicand

Out[]=

105106913492

Writing all this out and comparing to Lincoln's cyphering book page reveals that all numbers correspond, except that Lincoln appears to have inadvertently included an extra 0 to the right of the 5, writing "05" instead of "5" when adding the third column of numbers from the left:

Out[]=

3456782

30406

20740692

1382712800

103703460000

105106913492

To check his multiplication, Lincoln used a technique known as "casting out nines," the result of which appears below and to the right of the multiplication:

This check uses the number theoretical fact that if

(n)

is the remainder after dividing

by 9, then

(ab)

(a)

(b)(mod9).

Furthermore,

(n)

is particularly easy to compute via its relation to the "digital root," defined as the result of iteratively summing the digits of a number until a single digit is obtained (and setting that final value to 0 if that digit is 9):

In[]:=

DigitalRootList[x_]:=Most@ReplaceAll[FixedPointList[Total[IntegerDigits[#]]&,x],90]

For example, repeatedly adding the digits of the multiplier gives:

In[]:=

DigitalRootList[multiplier]

Out[]=

{3456782,35,8}

As promised, this indeed converges to the value obtained directly by taking the remainder after dividing by 9:

In[]:=

Mod[multiplier,9]

Out[]=

Let

be a multiplier and

be a multiplicand. The casting-out-nines check in a multiplication problem proceeds by drawing an X and placing

(b)

to the right,

(ab)

on top,

(a)

to the left and

(

(a)

(b))

on the bottom. The first three of these numbers for Lincoln's problem are therefore:

In[]:=

{rb,rab,ra}=Mod[{multiplicand,multipliermultiplicand,multiplier},9]

Out[]=

{4,5,8}

… and the final one is:

In[]:=

Mod[rarb,9]

Out[]=

The check shows the final product to be consistent with the numbers being multiplied, as confirmed by the equality of the top and bottom numbers in Lincoln's diagram:

CastingOutNinesDiagram[a_,b_,s_:.25,dx_:.1]:=

Module[

]

In[]:=

GraphicsRowCastingOutNinesDiagram[3456782,30406],



Out[]=

Unfortunately for Lincoln, since the digital root of a number is unaffected by the addition of zeros, his inclusion of a spurious 0 as the third digit from the left went undetected by this check.

Additional Reading

Ellerton, N. and Clements, M. "He Would Be Good: Abraham Lincoln's Early Mathematics, 1819–1826." Ch. 6 in Abraham Lincoln's Cyphering Book, and Ten Other Extraordinary Cyphering Books. New York: Springer, pp. 123–186, 2014.
Ellerton, N. and Clements, M. Rewriting the History of School Mathematics in North America 1607–1861. New York: Springer, 2012.

Lincoln's Cyphering Book

Timeline

Interactive Content

Computational Explanation

Additional Reading

Additional Links