Compass and Straightedge, Chisel and Saw

By Bobby Neal Winters

Working with hand tools has deepened my understanding of mathematics in some surprising ways.  Let me take some time to explain.

I know how to cut grooves and dados with hand tools now.  Yes, I know they can be done better and faster with power tools, namely a router, but I can cut them with hand tools.

You might be able to appreciate what I’m talking about if you knew what grooves and dados are.

Let’s start with groove because you probably have a good handle on that already.  A groove is, well, a groove. It’s a recess that you cut into a board so that you can slide another board into it. If you have a chest of drawers, it is quite possible that the sides of the drawers have grooves cut into them so that they can hold the thin board that forms the bottom of the drawer.

A dado is the same thing.  The reason we use the two different words is that a groove is cut parallel to the grain of the wood and a dado is cut perpendicular to it.

If you are using power tools, at least as far as I’ve experienced it, this makes little practical difference.  If, by way of contrast, you are using hand tools, it means that a different set of techniques are needed.

A number of years ago, I was approached by a friend and retired colleague of mine from the university. He is a very dear man.  Some one of his long acquaintance had claimed to have solved the problem of trisecting an angle with compass and straightedge.

This would be very interesting because this problem had been proven to be unsolvable a long, long time ago.  And I do mean proved.  It’s not just that mathematicians got to a point and said this is hard, I don’t think it can be done. They proved in a way that no one who has an understanding of the tools they used could reasonably deny.

It’s over.

Mathematicians had been interested in the problem for a number of reasons, one of which was they wanted an exact value for the sine of one degree.  Sine is a basic trigonometric function that has all sorts of practical uses.  If you could calculate the sine of one degree exactly, you could calculate the other trigonometric functions of one degree exactly, and this would make a lot of calculations sharper.

Just because it was shown that you couldn’t trisect an angle exactly didn’t make the practical problem go away.  Other method’s--say power tools if you may--have been invented that allow us not to calculate the sine of one degree exactly, but to calculate it (or any other trigonometric function) as close as we like, as close as we need.

To a mathematician, that is a big difference.  It is--as in the Princess Bride--the difference between being dead and almost dead.

I tried to explain this to the man who’d thought he’d solved the classic unsolvable problem.  I am not sure that I was able to convey the issue adequately. My toolbox of words was not up to the task.

A tool is not just a thing in itself.  It’s not made complete by what it’s made out of or even the form that it is in.  A tool is made by the person who knows how to use it.

I’ve said it before in this space, but it’s worthy of repetition. When you buy a chisel, you get what you pay for.  You can get 4 chisels for 8 dollars at Harbor Freight.  Or you can get a single Narex Richter chisel for $46.  There is no comparison in what you can do.  But--and this is key--someone who knows what to do with a chisel can do more with the 2-dollar chisel than someone who doesn’t know what to do with the 46-dollar one.

Let’s push it a bit further.  

It’s not only necessary that someone knows how to use the chisel, you need to have a community of knowledge.  The community teaches use, learns use, and passes on use to the next generation.

Even though we can calculate sine of one degree to as much accuracy as anyone could like, it is important that we keep alive the knowledge that we can’t solve it exactly.

Even though we can cut dados and grooves with a router, it is important that we keep the practice of working with hand tools alive.

We have to keep these practices alive.  There is a big difference between dead and almost dead.

Bobby Winters, a native of Harden City, Oklahoma, blogs at redneckmath.blogspot.com and okieinexile.blogspot.com. He invites you to “like” the National Association of Lawn Mowers on Facebook. Search for him by name on YouTube.