The Usefulness of Mathematical Learning - Part 2

Part 1

Continuing on the the second page, we find that

In all ages and countries where learning has prevailed, the mathematical sciences have been looked upon as the most considerable branch of it. The very name "Mathematician" implies no less, by which they were called either for their excellency or because of all the sciences they were first taught, or because they were judged to comprehend "all things mathematical". 

I'm no scholar of Greek, so I'm taking a bit of a liberty with what I think Arbuthnot was trying to say here. I do know that the original Greek word from which the term "mathematics" is derived has senses of meaning "what is learned." So it's not really surprising that he goes on to write

And amongst those that are commonly reckoned to be the seven Liberal Arts, four are mathematical, to wit, Arithmetic, Music, Astronomy, and Geometry.

He is referring here to the Quadrivium, which was the second course of study in the medieval university. The first was the Trivium, and consisted of Grammar, Logic and Rhetoric. Of course, contemporary people don't really think of music as a mathematical study, even though things like rhythm, pitch, tempo, and duration can all be quantified, and despite the explicitly mathematical origins of the art in the Western tradition starting with Pythagoras.

Not only that, logic actually is a branch of mathematics now, but wasn't in the 17th century. It started to be formalized as a mathematical study in a serious way by thinkers such as Boole, Pierce, and Russell.

In the next part, the author starts to describe why he thinks people don't learn math as much as they should.

But notwithstanding their excellency and reputation, they have not been taught nor studied so universally as some of the rest, which I take to have proceeded from the following causes:

The reasons he gives all sound very modern: people don't like to think so hard, they don't realize how useful math is, they think that only geniuses can learn it, it is not encouraged, and there aren't enough good teachers.

The Usefulness of Mathematical Learning - Part 1

I have, for quite some time, admired "An Essay on the Usefulness of Mathematical Learning in a Letter from a Gentleman in the City to his Friend in Oxford."

Though it was first printed in 1701, the arguments it makes are remarkably modern sounding - possibly because the time we live in actually has a lot in common with the late 17th and early 18th centuries.

So I think that this book is worth reading, but the format, typesetting, and 17th century usage make it tricky. Therefore, I will transcribe it into slightly more modern English while doing my best to preserve the tone and rhythm of the prose. Also, I mean to offer such commentary as may be useful to a student who endeavors to fully comprehend the import of the argument.

I think I'm in the spirit now, so here we go.

AN

ESSAY

ON

The Usefulness of

MATHEMATICAL LEARNING, &c.

Sir,

I am glad to hear from you that the study of the Mathematics is promoted and encouraged among the youth of your university. The great influence, which these sciences have on Philosophy and all useful learning, as well as the concerns of the public, may sufficiently recommend them to your choice and consideration: and the particular advantages, which you of that place enjoy, give us just reason to expect from you a suitable improvement in them. I have here sent you some short reflections upon the usefulness of mathematical learning which may serve as an argument to incite you to a closer and more vigorous pursuit of it.

Word Problem(s) of the Week(s) - Meat!

Meat related Algebra Problems!

When I showed this to my students, one comment was, "There is more than one kind of bacon?"

Squeeze Theorem

What did you do last Friday night? Went to the club? Dancing? I see.

It wasn't on the floor, was it? Table? I thought so.

And you were asked to leave? Naturally.

But you didn't, did you? Why not? Oh yes, a sustained period of bad judgment. Because....

Of course.

How big were the security people? Six-foot-seven or so, with weights approaching four hundred pounds?

And I suppose none of it was fat, either.

They got you by the arms? And headed for the door? I'm sure you didn't want to.

You had no choice about it though? I suppose not. They had you between them and they went to the door, so you did too.

What are you doing this Friday night?

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: 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.

Calculus Made Easy

The prologue of a Calculus book from 1914:

This is still true today.

A new blog.

I created this to blog about the history of math and science, which is a subject I really enjoy. I teach math in a high school, and it is not always possible to talk much about this sort of thing, because it is not actually in the curriculum. Science has historical developments which are covered in science class, but the development of math is not.

One of my favorite parts of the subject is the personalities of the people who made historical developments. I like to tell my classes that anyone smart enough to make a lasting contribution to mathematics is likely to also be a bit colorful. One of my favorites is John Napier, hence the name of the blog.

If you have noticed that the description mentions comics, let's just say that public domain images from old artwork are very handy.