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.

Word Problem of the Week: Wainscoting a Gentleman's House

From A Compendium of Algebra by John Ward, 1724.

p. 51:

Three Joiners undertook to wainscot a Gentleman's House in 150 days. One of then (being esteem'd the best) was to have 5 s. a day for every day he work'd; another was to have 4 s. 6 d. a day, and the third was to have but 4 s. for every day we wrought.
When the Work was finish'd, every one of them had just the same Sum of Money to receive; Quere, how many Days each of them work's? &c.

So, they all made the same amount, which makes me wonder: Is the highest-paid worker the best because he's the most efficient, or is he doing a bit of slacking off?

Word Problem of the Week: Arraying the Troops

From The Elements of that Mathematical Art Commonly Called Algebra, Expounded in Two Books by John Kersey, 1741.

p. 70:

A General of an Army having set his Soldiers in a Square Battel, there happened to be 500 (or b) Soldiers to spare; but to increase the Square so as that its side might consist of 1 (or c) Soldier more than the Side of the former Square, there would be 29 (or d) Soldiers wanting. The Question is, to find how many Soldiers the General had in his Army.