Tuesday, July 4, 2017

Newton's Problem of the Oxen in the Pasture

The Arithmetica Universalis of 1707 reproduces a set of lecture notes by Isaac Newton from the period 1669–1702 when Newton was Lucasian Professor of Mathematics at Cambridge. In 1720 there appeared an English translation, called Universal Arithmetick, which underwent various further editions, including one in 1769. On page 191 of the 1769 edition, the following difficult problem appears:


If 12 Oxen eat up 3-1/3 Acres of Pasture in 4 Weeks, and 21 Oxen eat up ten Acres of like Pasture in 9 Weeks; to find how many Oxen will eat up 24 Acres in 18 Weeks?
A version of the problem later appeared (with a disastrous typographical error) as the final challenge in Frederick Emerson's North American Arithmetic, Part Third, For Advanced Scholars (1834). The typographical error was the use of the value 3-1/2 in place of the original 3-1/3; this alteration causes the solution for the number of oxen to take a bizarre fractional value. Perhaps in part because of this twist, the problem "became famous overnight in the United States," according to Steven L. Jordan's article in Historia Mathematica (vol. 8, 1981, 145–160).

In 1835, the National Teachers' Association turned the problem into a contest, with a prize of $50 offered for the most lucid solution. Only 48 of 112 entries provided the correct answer. The prize was awarded to Mr. James Robinson, principal of the Department of Arithmetic at Bowdoin School, Boston. (His solution can be read in Ken Clements and Nerida Ellerton's account, which situates the Pasturage Problem in a larger discussion about mathematical elegance.)

The trend in the 1800s of posing difficult problems to schoolchildren was heavily criticized in 1890 by the great American historian Florian Cajori (1859–1930). (Cajori was a translator of Newton's Principia, and Cajori also wrote what must be the most abstruse book in my personal library, A History of the Logarithmic Slide Rule). In The Teaching and History of Mathematics in the United States, Cajori wrote

Think of it! Out of 112 of, presumably, the best arithmeticians in the country, only 48 got correct results; and yet this problem was intended to be solved by boys and girls. 

Clements and Ellerton however believe that the problem was fair, since it was the last problem in the last volume, intended for Advanced Scholars.

In any case, it's a hard problem. And beware, the text in the 1720 edition reads differently: "to find how many Oxen will eat up 36 Acres in 18 Weeks." (Compare the previous, "to find how many Oxen will eat up 24 Acres in 18 Weeks.") I don't know if this is a typographical error or an intentionally different version of the problem.

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I found the Universal Arithmetick strange. The book explains basic problems like 2099.6 ÷ 72.4, yet it also treats advanced problems of analysis using clever methods that were probably discovered by Newton himself. I wondered what kind of reader such a book was meant for. Could it be that Newton was forced to teach students with a range of mathematical knowledge? Or was this actually Newton's idea of a coherent course?

Newton's pedagogical method in the book is to take up different problem types in turn. He explains the general idea of the type, then he derives a formula that applies to any problem of the given type. To answer any particular version of the problem, just plug values into the formula. Here's an example of the approach:


Having worked out the reasoning once, Newton doesn't bother doing it again; the "reason" this problem has the answer 24 is that "if 8 be substituted for d, 15 for c, 405 for a, and 9 for b, the Number ad/bc will become 405 × 8 / 9 × 15, that is, 3240/135, or 24." Newton did however break his routine for the Pasturage Problem. After plugging numbers into the complicated general formula for such problems, Newton recapitulated the general reasoning in the specific context of the problem at hand.

All that said, in interpreting this book I don't think we should go very far in attributing authorial intent to Newton. The Arithmetic Universalis was published by his successor, William Whiston, and apparently Newton wasn't happy about that. Whiston claimed to have permission from Newton to publish the book, but it is also said that Newton tried to buy up all of the printed copies. I don't even know for sure that Newton wrote passages like this one, where we learn something of the "progressions" thinking behind the book:



(First the pupil learns formal manipulation, then he learns how to model in context.)

In any case, Newton's solution to the Pasturage Problem can be most easily found in this 1876 article by Alexander Evans (The Analyst, vol. 3, no. 3, 75–78), where the history of the problem is discussed. My own solution in a contemporary physicist's style is here.

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