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## Search Loci: Convergence:

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Attaching significance to invariants is an effort to recognize what, because of its form or colour or meaning or otherwise, is important or significant in what is only trivial or ephemeral. A simple instance of failing in this is provided by the poll-man at Cambridge, who learned perfectly how to factorize a^2 - b^2 but was floored because the examiner unkindly asked for the factors of p^2 - q^2.

In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.

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# Using Historical Problems in the Middle School

## Problems from Bonnycastle

Here are some problems used by Karen Dee Michalowicz of the Langley School.  She reports that her students have enjoyed solving these and have created similar problems of their own.   The problems that follow come from the 1807 London Edition of John Bonnycastle’s Introduction to Mensuration and Practical Geometry.  Two problems are in rhyme.  The third problem is one that would never be printed in a book today, because of its “political incorrectness.”

In the midst of a meadow well stored with grass,

I took just an acre to tether my ass:

How long must the cord be, that feeding all round,

He may’nt graze less or more than acre of ground.

One ev’ning I chanc’d with a tinker to sit,

Whose tongue ran a great deal too fast for his wit:

He talked of his art with abundance of mettle;

So I ask’d him to make me a flat-bottom’d kettle.

Let the top and the bottom diameters be,

In just such proportion as five is to three:

Twelve inches the depth I propos’d, and no more;

And to hold in ale gallons seven less than a score.

He promis’d to do it, and straight to work went;

But when had done it he found it too scant.

For though it held right, the diameters fail’d it;

Thus making if often too big and too little,

The tinker at last had quite spoiled his kettle;

But declares he will bring his said promise to pass,

Or else that he’ll spoil every ounce of his brass.

Now to keep him from ruin, I pray find him out

The diameter’s length, for he’ll never do’t, I doubt.

Ans. Bottom 14.44401, top 24.4002

Two porters agreed to drink off a quart of strong beer between them, at two pulls, or a draught each; now the first having given it a black eye, as it is called, or drank till the surface of the liquor touched the opposite edge of the bottom, gave the remaining part of it to the other:  what was the difference of their shares, supposing the pot was the frustum of a cone, the depth being 5.7 inches, the diameter at the top 3.7 inches, and that of the bottom 4.23 inches?

Ans. 7.05 cubic inches.