Word Problem of the Week: Whiskey Rations Aboard Ship

There are two things I find interesting about this problem. First, the detail offered gives it a level of realism that is not typically found in these sorts of texts. On the other hand, the quantities are all represented as variables, which adds a level of abstraction. Thus, the answer gives us a formula for any possible scenario of this type. The problem is number 23 on page 479:

Word Problem of the Week(s): Classical History and Mythology

These problems come from Mathematicall Recreations, or

A Collection of many Problemes extracted out of the Ancient and Modern Philosophers, as Secrets and Experiments in Arithmatick, Geometry, Cosmographie, Horologiographie, Astronomie, Navigation, Musick, Opticks, Architecture, Staticks, Mechanicks, Chemistry, water-works, Fire-works, &c.

Basically, a seventeenth century Dangerous Book For Boys.

Several of the arithmetic questions have to do thematically with the stories of ancient Greece. These are:
Of the number of Souldiers that fought before old Troy.
Of Pythagoras his Schollers.
Of the number of Apples given amongst the Graces and the Muses.
Of the Cups of Croesus. (He gave the Temple 6 golden cups weighing a total of 600 drams, with each one a dram heavier than the next.)
Of Cupid's Apples. (The Muses steal them.)

Word Problem of the Week: Map the Ohio River

It's nice to see a little connection between geometry and geography.

From Graphic Algebra, by Andrew Wheeler Phillips and William Beebe, 1904

p. 8:

Draw a map of the Ohio River from the following latitudes and longitudes, which are reckoned from the Equator and the meridian of Washington respectively:

This is the earliest book I have seen that emphasizes graphing equations as an algebraic tool. Then, 80 years later, the graphing calculator is invented.

Word Problem of the Week: Problem of the Lights

From Elements of Algebra, by Charles Davies, 1854.
p. 162:

"Find upon the line which joins two lights, A and B, of different intensities, the point which is equally illuminated by the lights."

This is a great little simple sounding problem that lends itself, in this case, to three and a half pages of discussion of various cases. At the end of this, the author states that "the preceding discussion presents a striking example of the precision with which the algebraic analysis responds to all the relations which exist between the quantities that enter a problem." I should say so.

Word Problem of the Week: Say, Lovely Woman, the Number of Bees.

From Lilavati, translated by Henry T. Colebrooke, 1817.
p. 211:

The square-root of half the number of a swarm of bees is gone to a shrub of jasmin ; and so are eight-ninths of the whole swarm : a female is buzzing to one remaining male, that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.

The whole book isn't like this though; the very next problem is about a guy shooting arrows at his enemy.

I highly recommend this review of the twelfth century Indian algebra text Lilavati, by Bhaskara. It references the Colebrooke Translation from 1817, which also mentions this gem from a commentary:

Whilst making love a necklace broke.
A row of pearls mislaid.
One third fell to the floor.
One fifth upon the bed.
The young woman saved one sixth of them.
One tenth were caught by her lover.
If six pearls remained upon the string
How many pearls were there altogether?

Although Colebrooke refers to her as a "wench" which is a bit less romantic.

Word Problem of the Week: French Degrees

I was telling the students the other day how dividing a circle into 360 pieces is a bit arbitrary. In fact, it comes from the Babylonians, and it could have been different. There is nothing special about 360.

The French, it turns out, actually did propose a different system of angular measure: one in which the fundamental unit was one percent of a right angle. Thus there are four hundred of them in one full revolution.

I remember years ago I saw a calculator with the familiar "degrees" and "radians" settings but also something called "gradians." This is the name for the "French Degrees." They are also called "grades" in a lot of old textbooks. I personally like what I called them in class, which is "People's Revolutionary Degrees."

Anyway, there seems to have been some confusion as to whether gradians were actually ever used. Some are of the opinion that the unit was "frequently used in France and ocasionally elsewhere" whereas others are convinced that they were not.

I suspect they must have been, because it is otherwise difficult to explain the enthusiasm shown for the French Degrees by a certain Mr. Isaac Todhunter. His trigonometry text is full of strange abstract problems involving equivalencies between the English and French units. Fairly typical is this one:

Divide two-thirds of a right angle into two parts, such that the number of degrees in one part may be to the number of grades in the other part as 3 to 10.

This combines the slightly obscure and uncommonly used unit with the proportionality question so common in 18th and 19th century texts to produce a masterpiece of bizarre irrelevance. Really, this is an interesting puzzle, but trig texts are usually much more practical than this.

Word Problem of the Week: From the Notebook of Benjamin Banneker

This morning I had the pleasure of attending a talk by John Mahoney concerning his experience teaching at Benjamin Banneker Academic High School in the District of Columbia. Part of the talk concerned Banneker himself, who is a fascinating personality about whom I'm going to have to write more later.

Banneker kept a notebook in which he recorded interesting problems. This one is interesting because there is an elegant solution that is unintuitive to someone educated in the modern way. At least to me, it was unintuitive. Problems such as this seem to have been popular in the 18th century, though.

Question by Elliot Geographer General
Divide 60 into four Such parts, that the first being increased by 4, the Second decreased by 4, the third multiplyed by 4, the fourth part divided by 4, that the Sum, the difference, the product, and the Quotient shall be one and the Same number.

You start by making a guess as to the number mentioned at the end, then adjust things based on the error you get.