Easy as one, two, three

The rule of three

Abraham Lincoln wrote a letter in 1859 to Jesse W. Fell of Springfield, Illinois, giving a little autobiography of himself. In it, he explained his education as a boy:

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 so-journ in the neighborhood, he was looked upon as a wizzard. There was absolutely nothing to excite ambition for education. Of course when I came of age I did not know much. Still somehow, I could read, write, and cipher to the Rule of Three; but that was all.1

Hmmmmm… “the Rule of Three”…

That phrase came up again at the Association of Professional Genealogists’ Professional Management Conference in Salt Lake City last week, where The Legal Genealogist used an apprenticeship record where the master of a boy being bound to him was required to ensure that the boy learned to “read, write and cypher to the rule of three.”2

And afterwards one of the attendees came up and asked where he could find a definition of the rule of three.

Here’s a hint: you’re not going to find it in Black’s Law Dictionary.

That’s because it’s not a legal term. It’s a mathematical term. And yeah, actually, I had to go look it up too.

Start first with the dictionary definition. It’s “a method of finding a number in the same ratio to a given number as exists between two other given numbers”3 or, more precisely, “a mathematical rule asserting that the value of one unknown quantity in a proportion is found by multiplying the denominator of each ratio by the numerator of the other.”4

That seems easy enough. So how was it described in the school books our ancestors might have used?

Well, in the 17th century a man named Edward Cocker wrote a book called Cocker’s Arithmetick : being a plain and familiar Method suitable to the meanest Capacity for the full understanding of that Incomparable Art, as it is now taught by the ablest School-Masters in City and Country. There’s a 1702 edition available on Google Books. And there, the method explained for folks like me, of the “meanest Capacity” for math, is this:

The Rule of Three (not undeservedly call’d the Golden Rule) is, that by which we find out a fourth number, in proportion unto three given Numbers, so as this fourth Number sought may bear the same Rate, Reason, or Proportion to the third (given) number, as this second doth to the first, from whence it is also called the Rule of Proportion.5


Okay, let’s try Ask Dr. Math. There’s it’s explained this way:

The Rule of Three is an ancient mechanical method for solving proportions, which we can do fairly easily (and with more understanding) using algebra. Briefly, it says that if you know three numbers a, b, and c, and want to find d such that

a/b = c/d (that is, a:b::c:d)


d = cb/a6

And Wikipedia explains it best to this non-mathematical mind in this graphic:7

Whew. Now back to the easy stuff like scire facias and entries ad terminum qui praeteriit and…


  1. A. Lincoln, Letter to Jesse Fell, December 1859, in Roy P. Basler, ed., Collected Works of Abraham Lincoln, Vol. III (New Brunswick : Rutgers University Press, 1953), 511; digital edition, Abraham Lincoln Association, University of Michigan Digital Library (http://www.hti.umich.edu/l/lincoln/ : accessed 26 Mar 2013).
  2. Minute Book, Burke County, North Carolina, Court of Common Pleas and Quarter Sessions, January 1804 – April 1807, Part II, July 1806 session; call no. C.R.014.301.4; North Carolina State Archives, Raleigh.
  3. Oxford Dictionaries Online (http://oxforddictionaries.com/ : accessed 26 Mar 2013), “rule of three.”
  4. The Free Dictionary (http://www.thefreedictionary.com : accessed 26 Mar 2013), “rule of three.”
  5. Edward Cocker, Cocker’s Arithmetick : being a plain and familiar Method suitable to the meanest Capacity for the full understanding of that Incomparable Art, as it is now taught by the ablest School-Masters in City and Country (London : John Hawkins, 1702), 102; digital images, Google Books (http://books.google.com : accessed 26 Mar 2013).
  6. Rule of Three,” Ask Dr. Math: Questions and Answers from our Archives, Math Forum, Drezel University (http://mathforum.org/dr.math/ : accessed 26 Mar 2013).
  7. Wikipedia (http://www.wikipedia.com), “Cross-Multiplication: Rule of Three,” rev. 15 Mar 2013.
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10 Responses to Easy as one, two, three

    • Judy G. Russell says:

      Tell me about it, Bill! My college placement tests put me in advanced English, language, social studies and even science. And remedial math.

  1. John says:

    It’s interesting that this is a mathematical term.

    The first thing that came to my mind for “The Rule of Three” is the rhetorical device – where dividing an argument into three main points is often more persuasive than two or four points.


    Now all we need is a third usage…and there are many to choose from.

    It seems rules of three are as popular as sets of three:

  2. Dave says:

    May I offer a suggestion for citations of mathematical formulas in future? In the blog, where the citations are a different color (albeit still barely distinguishable), citation number 6 works. I read the blog in my email, though, where colors don’t come through. It took me a minute to figure out that the superscript 6 was a reference, and not indicating a to the 6th power! Might be a good idea to add a space before the reference in cases like this (or even two spaces, although anything rendered in HTML will likely remove the extra space).

    • Judy G. Russell says:

      I will certainly take that under advisement in the (exceedingly unlikely) event that I ever again cite a mathematical formula, Dave! (It didn’t occur to me that, ulp, math uses superscript numbers too!)

  3. Annick says:

    I knew right away what it was! Not difficult: I am from France and we learn “la regle de trois” as soon as we are able to do our first arithmetic problem in 3rd grade. I also tought it since I was a elementary school teacher before coming to the US. Of course we didn’t use the letters (looks too much like algebra!), but we talked the problem out.
    If the problem was (to use the data above): How many miles will you be away from your home at 7PM tonight knowing that you departed at noon and that you have clocked 90 miles and it is now 3PM in the afternoon?
    In 3rd grade the problem would be done in two steps. In later grades the students would talk out the problem in their heads and write just one formula. The key is to ALWAYS put first what you are looking for, like here the MILEAGE. It works EVERY TIME!
    How many miles did I drive in one hour
    90 miles/3hours= 30 miles
    How many miles will I drive in 7 hours
    30 miles x 7 = 210 miles

    A fourth grader would have writen:
    I will drive (90/3)*7= 210 Miles
    Even though this is a little long, I hope this proves how easy and useful the rule of three is.

  4. Dawn Bingaman says:

    I think the important takeaway here (in context of the PMC talk, which I attended virtually and enjoyed) is whether this was considered a typical amount of schooling, above average, or something else?

    • Judy G. Russell says:

      It appears to have been an emerging norm in the 1800s, Dawn — that children should be schooled enough to be able to achieve this even if nothing more.

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