*Odds bobs, ladies, what am I?*

I'm at a distance, yet am nigh;

I'm high and low, round, short, and long,

I'm very weak, and very strong.

Sometimes gentle, sometimes raging.

Now disgusting, now engaging.

I'm sometimes ugly, sometimes handsome. . . .

I'm very dirty, very clean,

I'm very fat, and very lean,

I'm very thick, and very thin,

Can lift a stone, tho' not a pin. . . .

I'm at a distance, yet am nigh;

I'm high and low, round, short, and long,

I'm very weak, and very strong.

Sometimes gentle, sometimes raging.

Now disgusting, now engaging.

I'm sometimes ugly, sometimes handsome. . . .

I'm very dirty, very clean,

I'm very fat, and very lean,

I'm very thick, and very thin,

Can lift a stone, tho' not a pin. . . .

This riddle, written entirely in verse, goes on for about 30 lines. Submitted by a reader, it starts off the riddle section of the 1808 edition of

*The Ladies' Diary: or, Woman's Almanack*.

*The Ladies' Diary*was published annually, starting in 1704. Printed in London, it featured the usual stuff of almanacs: calendar material, phases of the moon, sunrise and sunset times, important dates (eclipses, holidays, school terms, etc.), and a chronology of remarkable events.

The little book's subtitle provides a glimpse of its purpose: "Containing New Improvements in ARTS and SCIENCES, and many entertaining PARTICULARS: Designed for the USE AND DIVERSION OF THE FAIR SEX."

Among those diversions were sections devoted to riddles (called enigmas), rebuses, charades, scientific queries, and mathematical questions. A typical volume in the series included answers submitted by readers to problems posed the previous year and a set of new problems, nearly all proposed by readers. Both the puzzle and the answer (revealed the following year) were often in verse.

In a paper just published in

*Historia Mathematica*, Joe Albree and Scott H. Brown of Auburn University review the mathematical contribution of

*The Ladies' Diary*, particularly the leading role it played in the early development of British mathematical periodicals.

"From the first decade of the 18th century and for many decades following,

*The Diary*was a pioneer in more than one respect," Albree and Brown write. "With unfailing regularity, starting in 1708, it presented an array of mathematical problems and their solutions to a wide range of readers."

"From the very start women were encouraged to participate fully in

*The Diary*'s mathematical program," they add.

*The Ladies' Diary*was founded by John Tipper (before 1680–1713), in part to advertise the Bablake School in Coventry, England, where he was master, and to help support his family financially. The publication also represented his recognition of a need for an almanac that catered to women.

"Tipper's attitude toward women was often remarkably progressive," Albree and Brown remark. The cover of each issue featured a picture of a prominent English woman, beginning with Queen Anne in 1704.

The publication was also a success. In its second year, about 4,000 copies of The Diary were sold; circulation reached about 7,000 in 1718 and, in the middle of the 18th century, amounted to around 30,000 copies a year.

The readership was diverse, and the publication attracted a wide range of contributors of both problems and solutions. However, because many contributor used pseudonyms, it's difficult to determine how many were female.

"The earliest mathematical questions in

*The Diary*were puzzles, enigmas solved by numbers, and such contrived problems continued to appear through the whole life of

*The Diary*," Albree and Brown say. "However, very quickly, the level of difficulty and the seriousness of many of the questions in

*The Diary*increased, and

*The Diary*became one of the participants in the popularization of mathematics and the Newtonian sciences in the 18th century."

Note that

*The Ladies' Diary*had been founded while Isaac Newton (1643–1727) was still alive.

By the 19th century, some of the proposed problems were quite sophisticated and occasionally called for knowledge of contemporary issues and advances in mathematics or physics.

*The Diary*even attracted some foreign contributions. Here's a geometry question submitted in 1830 by French mathematician C.J. Brianchon (1783–1864).

*The six edges of any irregular tetrahedron are opposed two by two, and the nearest distance of the two opposite sides is called breadth; so that the tetrahedron has three breadths and four heights. It is required to demonstrate that, in every tetrahedron, the sum of the reciprocals of the squares of the breadths is equal to the sum of the reciprocals of the squares of the heights.*

Some questions were open-ended. Here's one from 1822:

*Required a better method than has yet been published of finding*x

*in the equation*x

^{x}= a.

I came across the 1808 and 1809 editions of

*The Ladies' Diary*among the volumes in a remarkable collection of books at the library of the University of Calgary in Alberta. The Eugène Strens Recreational Mathematics Collection contains more than 6,000 items, including books, periodicals, newspaper clippings, and manuscripts devoted to recreational mathematics (in a very broad sense) and its history.

"The bulk of mathematics has really always been recreational," says mathematician Richard K. Guy, who was instrumental in bringing to the University of Calgary material collected by the late Eugène Strens, an engineer, amateur mathematician, and friend of the artist M.C. Escher. "Only a tiny fraction of all mathematics is actually applied or used."

Here's a riddle from the 1809 edition of

*The Ladies' Diary*:

*I'm a singular creature, pray tell me my name,*

I partake of an Englishman's freedom and fame;

I daily am old, and I daily am new,

I am praised, I am blam'd, I am false, I am true;

I'm the talk of the nation, while still in my prime,

But forgotten when once I've outlasted my time.

In the morning no Miss is more coveted than I,

In the evening no toy thrown more carelessly by.

Take warning, ye fair! I, like you, have my day,

And, alas! You, like me, must grow old and decay.

I partake of an Englishman's freedom and fame;

I daily am old, and I daily am new,

I am praised, I am blam'd, I am false, I am true;

I'm the talk of the nation, while still in my prime,

But forgotten when once I've outlasted my time.

In the morning no Miss is more coveted than I,

In the evening no toy thrown more carelessly by.

Take warning, ye fair! I, like you, have my day,

And, alas! You, like me, must grow old and decay.

A typical rebus provided clues for a starting word, which was then modified step by step to produce the final word:

*My whole's a small but luscious fruit;*

Take off my head and then you'll see

What sinful men sometimes commit,

That brings them to the fateful tree.

Two letters now you may transpose,

But place them both with care,

Another luscious fruit will then,

Most plainly soon appear.

Take off my head and then you'll see

What sinful men sometimes commit,

That brings them to the fateful tree.

Two letters now you may transpose,

But place them both with care,

Another luscious fruit will then,

Most plainly soon appear.

Or:

*Take two-fifths of what, seen on Delia's fair face,*

Always adds to her charms an ineffable grace;

These selected with care, and with art combined,

Will produce a Diarian of genius refin'd.

Always adds to her charms an ineffable grace;

These selected with care, and with art combined,

Will produce a Diarian of genius refin'd.

Scientific queries covered a wide range of subjects:

Query: A drop of oil let fall on a wasp, kills it in less than a minute; query: how is this effected?

Query: Thirteen years have elapsed since the Northern Lights have made their appearance. How is their absence accounted for?

The mathematical questions tended to focus on geometry, and they were reminiscent of the types of word problems found in math textbooks of a few generations ago.

*A circular vessel, whose top and bottom diameter are 70 and 92, and perpendicular depth 60 inches, is so elevated on side that the other becomes perpendicular to the horizon; required what quantity of liquor, ale measure, will just cover the bottom when in that position.*

The book didn't provide a diagram.

The math questions featured in

*The Ladies' Diary*were of sufficient interest that collections of them were published in four volumes covering the years 1704 to 1817.

I wonder if Jane Austen (1775–1817) or Ada Lovelace (1815–1852) ever pondered the puzzles posed by

*The Ladies' Diary*?

**Answers**

Riddles: water; newspaper.

Rebuses: grape, rape, pear; smile, smart.

Math question: Very nearly 189 gallons.

**References:**

Albree, J., and S.H. Brown. 2009. "A valuable monument of mathematical genius":

*The Ladies' Diary*(1704-1840).

*Historia Mathematica*36(February):10-47.

Almkvist, G., and B. Berndt. 1988. Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, pi, and

*The Ladies' Diary*.

*American Mathematical Monthly*95(August-September):585-608.

Guy, R.K., and R.E. Woodrow, eds. 1994.

*The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History*. Mathematical Association of America.

## 3 comments:

A circular vessel, whose top and bottom diameter are 70 and 92, and perpendicular depth 60 inches, is so elevated on side that the other becomes perpendicular to the horizon; required what quantity of liquor, ale measure, will just cover the bottom when in that position.

Trying to solve the above problem from the Ladies' Diary (inspired by http://www.maa.org/mathland/mathland_7_14.html) without using integration, I came up with the following solution (looking at the figure in the MAA note may be useful):

Let the angle between the plane cutting the cone and its axis be u and let the generated angle of the (full) cone be v.

Let the cone have the bottom radius R, top radius r and height H. The plane cuts the cone such that the ellipses generated has the eccentricity e = cosu/cosv = tanv = (R-r)/H.

The major axes of the ellipses are of length a and b (a > b) where a = Rsinu. With c = ae, we have that b = Sqrt[a^2 - c^2] = a Sqrt[1-e^2].

The asked for volume V is given by 2V = Pi abh for h = Rcosu since u + v = Pi/2.

So, V = Pi R^3 sin^2u cosu Sqrt[1-e^2] = Pi R^3 sinu sin2u Sqrt[1-e^2]. .

Using the numerical values R = 46, r = 35, H = 60, we have sinu = 60/61, cosu = 11/61, e = 11/60, so V = Pi (46/61)^3 x 7 x 11 x 60 x Sqrt[71], in cubic inches.

Using ale measure as asked for in the problem, we use ale gallon, which is 282 cubic inches according to http://en.wikipedia.org/wiki/Ale_gallon, so V = Pi (46/61)^3 x 770 x Sqrt[71]/47 = 185.976 gallon. I accept that this is very nearly 186 gallon.

So far so good, but the MAA note claims that the answer is very nearly 189 gallon.

Hmmmm. Something wrong?

Kent Holing,

NORWAY

From Wikipedia:

In 1824, Britain adopted a close approximation to the ale gallon known as the Imperial gallon and abolished all other gallons in favour of it. Inspired by the kilogram-litre relationship, the Imperial gallon was based on the volume of 10 pounds of distilled water weighed in air with brass weights with the barometer standing at 30 inches of mercury and at a temperature of 62 °F. In 1963, this definition was refined as the space occupied by 10 pounds of distilled water of density 0.998859 grams per millilitre weighed in air of density 0.001217 g/mL against weights of density 8.136 g/mL. This works out at approximately 4.5460903 L (277.4416 cu in).

Kent Holing:

Using 1 ale gallon as 277.42 cubic inches, V = 189.046 gallon - very nearly 189 gallon indeed!!

Kent Holing:

To avoid any confusion: In the general case, V=Pi R^3 sin2u Sqrt[|cos2u|]/2. Note v < u < Pi/2 since an ellipse is cut.

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