Yes, the answer to last week’s Crocodile mystery is as obvious as it seemed. We were looking for a number which unites the table, the fractions, and the superfluous but artful penmanship. Answer: **60**, of course!

What we are actually looking at here is nothing more than a simple division sum from the 17th century where A = 1/2 = 30, B = 1/4 = 15, C = 1/5 = 12, D = 1/3 = 20, E = 1/6 = 10. If we look above the table we see a direction serving as an introduction to our colorful and neatly presented mystery.

Divide 60^{li} Amonghst 5 men and Tell me Each mans

Part thereof giveing to –

Hence the answer to this written problem is C: 1/5: **12**.

This crocodile mystery is from Sarah Cole’s Arithmetic book (V.b.292) or as we are told at the beginning of the manuscript (on an inserted page, in a later hand that has apparently misread the “A” for a “7” from the original title page):

S.7.RAH COLE.

Her Book

Scholler to Elizabeth Beane

Mrs in the Art of Writing

Anno 1685.

Who ever inserted that page apparently liked the poem well enough to copy it, for it appears in the manuscript, in Sarah’s own hand:

Arithmeticks The Art of Computation

By Numbers which brings many Consolation

Those who True Reckonings from false Discern

Arithmetick Let Them Compleatly Learn

By This the Merchant and *th*e man of Treade

By Ignorance or Skill are marr’d or made

Yet in this Art Therse none thats so accute

As all Its Excellencies To Compute

This rather large book, measuring over a foot in length and about eight inches in width, is a fascinating combination of art and arithmetic. It contains several sums and tables on the subjects of addition, multiplication, division and subtraction, or as Sarah Cole writes ‘substraction’.

A section on bartering is included as well as interesting pages on the measurement of sugar, tobacco, ‘oyles’, and more. There are a number of pages under the heading “The Golden Rule,” of which the page featured here is one. In the Early Modern period girls did not generally have access to a formal education, but some were educated in topics which could help them in the running of a household, such as arithmetic. The problems and solutions in this book do tend to pertain to domestic concerns. Throughout the book colorful pictures, borders, doodles, creatures, faces and various swirls (such as the one included in the mystery) provide decoration. Sarah Cole was probably copying a template from her teacher, whom she names as Elizabeth Beane. Another pupil, Mary Serjant, identifies Elizabeth Beane as a tutor in “The Art of Writing and Arithmetick” (currently held by the Beinecke Library: Osborn Shelves MS fb98).

On the rest of the page there are other interesting workings out using pounds, shillings and pence.

For example just underneath our mystery table, one’s attention is drawn to the calculations 60 x 30 = 1800 or 1800 / 60 = 30, 1800 / 30 = 60 : 60 x 20 = 1200, or 1200 / 60 = 20, 1200 / 20 = 60 and so forth… We are not entirely sure what is going on in Sarah Cole’s head at times from the way the multiplications are written, so if you fancy trying out some of the harder sums on the page, let us know how you get on!

**Edit, 9/8/15, 3:30pm: **Clarified that the arithmetic poem appears in the manuscript and was then later copied onto a page inserted at the start of the manuscript.

That is some beautiful quill pen calligraphy. That initial S in “Subtraction” is especially exquisite.

Yes, indeed, we came across many beautiful majuscules throughout Sarah Cole’s manuscript book!

The page title gives a clue to what Sarah is doing in the other exercises on the page. The fact that I’ve been doing some arithmetic with pre-decimal currency recently also helped me to identify what was going on.

The Golden Rule is also known as the Rule of Three, and is a formula for solving equations of the form a/b = c/x, or word problems of the type ‘if a yards of cloth cost b shillings how much would c yards cost?’. The solution is given by c x b / a. (See e.g. https://en.wikipedia.org/wiki/Cross-multiplication#Rule_of_Three)

What Sarah is doing below the table from the crocodile is dividing £60 among the five men in proportion to the amounts in the second column of the table. That is, A gets 30/87, B gets 15/87 etc, where 87 comes from the sum 30+15+12+20+10. Hence the questions of the form ‘if £87 comes from £60 what will £30 come of?’, i.e. a Golden Rule form.

The answer to the first such question following the Golden Rule is 30×60/87, (to which we would simply answer 20.69) however this is complicated by the fact that currency at the time was non-decimal (20 shillings to 1 pound, 12 pennies to 1 shilling, 4 farthings to 1 penny), so Sarah has to work out whole numbers of each of the different units of currency. Her process is as follows:

* 60×30 =1800 (from the first part of the Golden Rule)

* 1800/87 = 20 pounds remainder 60 (from the second part of the Golden rule) [This division is surrounded by red dots]

* 60×20 = 1200 (convert remainder £60 into shillings)

* 1200/87 = 13 shillings remainder 69 (continue dividing by 87 to get the next part of the answer) [this division surrounded by red and yellow dots]

* 69×12 = 828 (convert remainder 13 shillings into pence)

* 828/87 = 9 pence remainder 45 [this division surrounded by red dots]

* 45×4 = 180 (convert remainder 45 pence into farthings)

* 180/87 = 2 farthings remainder 6 [the 2 is circled]

at which point Sarah has run out of smaller units of currency, so her answer comprises the integer results of dividing by 87, i.e.

£20 13 shillings 9 pence 2 farthings

Farthings were normally written as a fraction of a penny, so Sarah writes her answer as £20-13-9 1/2

She calculates the amounts for the rest of the men in the same way, and totals up her results at the bottom of the page to check if they equal £60. They are half a penny off (£59-19-11 1/2) because some of the quantities don’t yield a whole number of farthings (as we saw above).

Thank you Philip! We read your detailed comment on the mathematical exercises which occur below our Crocodile mystery table with much interest. There are a number of neatly presented pages on ‘The Golden Rule’ in the manuscript, with similar eye-catching calculations.

More about Sarah Cole’s manuscript can be found in an essay by Emily Bowles Smith, who participated in Steven May’s NEH summer institute on manuscript culture. That institute was in 2005. Here’s a link to her essay: http://wayback.archive-it.org/2873/20140920060043/http://www.folger.edu/html/folger_institute/mm/EssayES.html#

My wife (Nerida Ellerton) and I authored a book on the early arithmetc manuscripts, The title is “Rewriting the history of school mathematics in North America, 1607-1861, and it was publishedby Springer in 2012. Suggest you google “ellerton and Clements.” Currently, we have 550 such manucripts in our collection, dating from about 1667 through 1861. All that said, the Sarah Cole manuscript is especially beautiful and historically significant. We wish we’d known about it when we wrote the book. Thanks for making it available on the web.