This week I am finishing my splay-legged coffee table… in the dining room, because the humidity is such that I don’t trust oil to dry in the workshop. I will have more to say about (and better photos of) this piece, which posed several, ah, interesting challenges, but for now let’s talk about this one, which I’ve faced before and will face again: constructing small arcs of large circles.

There are three long arcs of circles on this table, at the ends of the top and on the undersides of the aprons. The longest has a lengths of 23 inches and height of 1 inch — a radius of some five feet, so actually constructing the circle was straight out. I could probably have drawn the shortest one freehand to within sawing and shaving tolerances, but the longest moves so slowly that I didn’t trust myself. I’ve been known to use Affinity Designer (Adobe Illustrator for people without corporate budgets) to draw ogees when I couldn’t get what I wanted with French curves, but I can’t print something 23" long. What to do?

What George Walker and Jim Tolpin call artisan geometry has solutions for practically every design problem woodworkers face, including this one. What you do, is draw a half-circle with the height of the arc you want, then proportionally expand that arc using station points, from which you can build a form for bending a batten or simply sketch a template:

But while this method works well for doubling or tripling the length of the arc, it isn’t really practical for stretching it by a factor of twelve.

Finally, like all good craftsmen, I fell back on the training I received as a young apprentice: I used trigonometry. Knowing the length and height of an arc one can, using trigonometry, determine the radius of the parent circle, then plot coordinates of station points in terms of the horizontal distance from the center of the baseline and the height of the arc at that point. Of course, one doesn’t want to have to do those calculations thirty-odd times for a single project, so I wrote a bit of code to do it for me. And since I assume other people have similar challenges, I made the calculator available on my website.

To use the calculator, input the width and height of your arc, the horizontal interval (along the baseline) between the points you want to plot, and the desired precision. The coordinates will be generated in ordinary fractions of an inch, to the desired precision (up to 1/64 inch). If desired, you can also see decimal values.

A calculator for constructing small arcs of large circles

If you want to understand the math, my explanation follows.

For a point P on an arc, define x as its distance along the chord from the center and y as its distance from the chord. Given x, how do we find y?

If we’re given the width w and height h of the arc (as is common in laying out furniture), we first need to calculate the radius of the circle r. To do this, note that

w = 2rsinθ

and

h = r – rcosθ

where θ is *half* the angle of the arc.

Solving for sinθ and cosθ we get

cosθ = 1 – h/r

By trigonometric identity, we know that sin^{2}θ + cos^{2}θ = 1. Therefore,

w^{2}/4 + r^{2} – 2h/r + h^{2} = r^{2}

and solving for r we get

r = (w^{2} + 4h^{2})/8

(For our purposes, we don’t need to know θ, but calculating it at this point would be trivial.)

Now, for any point P on the arc, to find y given x, define φ as the angle of the arc from its center to P (see diagram). Then

x = rsinφ

y + (r-h) = rcosφ

Given x, we can solve for φ and then for y:

φ = sin^{-1}(x/r)

y = h – (1-cosφ)r

As a rule, I’d much rather use pencil-and-paper constructions, but it’s a craft, not a religion, and sometimes the old ways are best. “Old” being relative to the craftsman’s education. If, however, you want to use this new-old-fangled artisan geometry, I recommend you buy Walker and Tolpin’s book By Hand and Eye, which is really excellent.

]]>Stopping at the grocery store today to pick up something for lunch^{*} I was reminded that whatever our intentions, not to mention our policies and our laws, humanity will continue to screw itself. To wit: When the young man (I am now old enough to use that term) bagging groceries was about to pile everything into one paper bag, the clerk pulled out another bag and started helping him, with polite but pointed verbal correction. Everything would fit, yes, but it would make the bag too heavy, and lifted by its handles it would tear. “I’ll lift it from the bottom,” I said, “don’t waste a bag.”

Of course I got the groceries home just as well as I would have done before someone thought to put handles on paper grocery bags. So I started wondering how much more paper we now use bagging groceries because we thought they needed handles, or, rather, that *we* needed handles, either by splitting the groceries into smaller parcels or double bagging the larger ones. This is why although I am not opposed, in principle, to banning plastic shopping bags, I’m not enthusiastically for it, either: we’ll just find other ways of wasting resources.

Convenience costs, in other words — if not us then someone else or, more commonly, “the environment.” Of course it isn’t like anybody ever decided, yes, I want handles on my damn bags and I’ll cut down twice as many trees to get them. That we use more bags is the kind of thing that might have been foreseen but wasn’t and seldom is. Cost-benefit analysis might be a cure, but it won’t prevent further stupidity, because we can’t foresee all the consequences of our choices even if we were inclined to. What’s needed is a different ethic: to say, when confronted with a new convenience, *I don’t need that*. Not that all conveniences are ipso facto bad, but that our *default* ought to be to reject them; you can always change your mind later. Instead, our default is to accept without question any and all convenience. As long as that’s the case, we’ll keep destroying everything around us.

^{*}It was an unplanned trip, else I might have brought along one of my approximately two dozen reusable grocery bags. Emphasis on *might*: just as likely I’d have forgotten. I don’t claim not to be part of the problem.

Back in the early summer of 2013 I built this box to store my drawing and art supplies:

It’s the 19th-century schoolbox from *The Joiner and Cabinet Maker*, made with circa-1830 methods, resized and with a till for pencils. It’s functional and attractive sitting beside my desk, and the yellow pine has aged nicely, but it’s essentially a student project, without unnecessary adornment. The exposed dovetails aren’t contemporary ornament but evidence that the original wasn’t worth the effort of fancy moulding or veneer. The simple moulding around the top is simply nailed on, with historically accurate cut nails (headless brads, actually). Here’s what the top looked like when it was first built:

There’s a dab of glue at the center of the front piece and at the front end of the side pieces, but otherwise it’s just the nails holding that moulding on. To glue the moulding to the sides would invite disaster: wood, of course, expands and contracts across the grain as the ambient humidity changes, and if you force long-grained moulding to stay attached to the end grain of the top, something’s apt to break.

Novices and non-woodworkers may ask: *Does wood really move that much?* Moderately experienced woodworkers may ask, contrarily: *Won’t those nails keep the moulding from moving laterally and invite as much damage as glue?* When I first started working wood I was mildly skeptical on the first point; when I built this piece I was mildly skeptical on the second. But the answers are, quite definitively, yes and no respectively. Here’s proof.

The summer of 2013 in North Carolina was ludicrously rainy. Summer here is always humid; that summer was like living in a walk-in shower. While I was building the box I stored the wood in my workshop, which was an un-air-conditioned 12’x16′ shed. When I built the schoolbox, as you can see in the photo above, the moulding on the lid was trimmed flush with the back and the miter was square at the front.

After six years living in a dry, air-conditioned house, here’s what’s happened (click to zoom in):

I don’t have plans or notes from 2013, but to judge from the moulding, the top was originally 10 3/16" wide; now it’s exactly 10", about 2% shrinkage. Two percent does not sound like much, but it’s enough to visibly deform the moulding — but (ah!) not break it. I don’t believe I elongated the nail holes to allow for movement, but even if I had, I wouldn’t have elongated them *that* much. So, yes: wood moves, and nailing stuff together cross-grain works.

I’ve seen more than enough antiques to believe both points, but it’s fun to see it in my own work. Of course I could trim the moulding (or even replace it, since I used liquid hide glue), and it would, objectively speaking, look better — but I prefer to leave the evidence of this tiny little victory for historical methods. And as a reminder to let wood acclimate to a climate-controlled house for a few weeks before building it into furniture.

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