Joy of Sectors II: Revisiting "The Arte & Misterie of Coopering"

You may remember this picture of a persnickety Perkins working the angles on a coopering demonstration.

With a bit of math and the able assistance of my consulting engineer, we determined that the eight perfectly-equal staves of my Mary Rose tankard needed to have 67.5 degree bevels so that they would align to form a perfect circle of watertight, oaky goodness.

This assumed if you didn't round off your tankard/bucket/barrel or whathaveyou that you'd have an equilateral polygon. All sides (eight sides in this case) were the same and would be worked with the typically-modern obsession with precision and symmetry.

As we discussed yesterday, that wasn't quite right. Coopers make their staves whatever width their wood allows and it's not always symmetrical and there aren't always an even number of staves. This is important since they were cutting wood straight from the log and doing a lot of their shaping with an axe. (Axes are awesome and speed up woodworking considerably, but they're not really precision instruments.)

All of which throws off my equation a bit. You could still figure it out, of course, but it just got a bit harder.

Part of the problem here is that I tend to think of barrels and tankards as segments of a ring rather than pieces of pie. I've been peeling all these apples, so let's make a pie.

Not a meat pie, you goofball, an apple pie.

Much better. (The ice cream's a nice touch, but we won't be needing it today so I'll just eat it and get it out of the way...)

Like a period tankard or bucket, the average pie slicing cook isn't going to make every slice exactly the same size. The piece for the baker is big and the piece for the kids are small. Also Aunt Agnes, who always says she's on a diet, but takes twice as much ice cream as everyone else.

The nice thing about this way of imagining a coopered vessel is that you could (if everyone can hold their forks for a minute) rearrange those pieces any way that you want and you'd still have a pie the same size as you started with. The pieces would all fit together no matter how you rearranged them in the pie plate.

But with coopering, you're starting with a pile of uncut sticks and an entirely imaginary pie that you're trying to craft out of thin air and oak. So how do you do this when you don't start with a pie?

Enter the Sector.

If you remember, at its simplest form, the sector is two straight pieces of wood hinged at one end. Pretty much exactly like a pair of dividers, except made of wood.

The geometry of the sector is devilishly simple: The two legs are equal, meaning that the legs will form an isosceles triangle. If you place it at the center of your imaginary bucket (or whatever you're making), it forms an imaginary piece of pie. If you want to make a ten-inch bucket (measured across the bottom) you just mark your sector five inches from the hinge.

That's the inside diameter of your bottom.

The second mark you see there indicates the desired thickness of the walls of my imaginary bucket.

Now, no matter where the sector crosses the the outer wall of your bucket, or how far apart the the legs of the divider have to be (to accommodate your rough-cut variations in stave width) the legs form the angle that you're shooting for. 

You found your stave bevel without writing a single number on a single piece of paper.

As you can see, this is also helpful if you decide to repair or knock off an existing bucket.

But wait, there's more!

It's not just a measuring device for finding the angle of your bevel either. It's also a jig you can use to monitor your progress as you create the bevel on your staves. Hold the bevel up to the stave as you work and align it to your marks as you go. 

The angle of the sector will tell you how far you need to lean the stave over as you plane the bevel. When the stave sits snugly in the sector between the marks, you're golden.

The importance of this cannot be exaggerated for the working cooper. And this is the part that I forgot when I was doing my demonstration.

I almost wish that building that bakery at the Washington Midsummer Renaissance Faire had been the first step in this project rather than happening in the middle. I suspect that I've fallen down on most of these projects simply because I'm thinking in terms of one-off bespoke items rather than the economies of production.

The practicing cooper isn't making one bucket at a time. This isn't bespoke bucketing on Bond Street, it's mass production because a cooper (and his apprentices) gotta eat. So if they're making as many buckets at a time as they can reasonably get away with, there's no room to be fiddling about with equations and making these exacting symmetries that we've become so accustomed to post industrial revolution.

A cooper's shop is making piles and piles of rough-cut staves and assembling them into a pile of ten-inch buckets and they all have to fit together no matter which bucket they're getting shoved into.

Yesterday I asked where I should stop. I don't know that I'm going to go back and repeat every experiment and demo that got us this far, but going forward I will certainly be taking a hard look at each of these trades and asking the question that should be obvious: How did they do this fast enough to make a living at it?

Because therein lies the real secret of most of these trades.

Now go eat some pie. You've earned it.

-Scott 

Postscript

The "Aha!" moment for this didn't come from thin air. (They never do, no matter what anyone tells you.) I've been reading about sectors recently, mostly using Jim Tolpin's books and articles so I was in the right mindset when I had the good fortune to catch an episode of Roy Underhill's "The Woodwright's Shop" featuring white cooper Norm Pederson who actually demonstrated many of the things that past coopers on that show had previously waved away as "something you learned with experience."

The episode is not available online anywhere I can find it, but you can download it for $3.99 from Popular Woodworking here: https://videos.popularwoodworking.com/courses/the-woodwrights-shop-s24-ep04-norm-pederson-white-cooperage  

Or you can buy the whole season on DVD like I did. I highly recommend it if you have any interest in trying this out yourself with better results than I had on my first outing.

Further watching...

This is not that video, but it's an excellent demonstration of white cooperage filmed at an outdood event, so the sound isn't always so great.

Thoughts from the peeler: The artisan obsession and where does it end?

There is in just about every artisan, a touch of obsessive compulsion. Whether or not it's a disorder depends on how you feel about being both obsessive and compulsive at the same time with sharp implements in hand.

I don't want to make light of a genuine medical disorder. As someone who suffers the black periods of lost joy and time that is depression, far be it from me to make light of someone else's affliction.

Obsession and compulsion exist on a sliding scale, which is set by the same people who have categorized an affection for coffee as a mental disorder.

So let's ignore those folks for a bit.

For all practical purposes, it boils down to whether your obsession/compulsion is positive or destructive influence on your life.

Be ye moderate in all things except moderation.

So it is with caffeine and beer and so too it is with handicrafts.

But where is that moderate line? When do I stop? How far do I take each of these explorations of a craft? When do I tie it off and call it good? Do I keep going until I've got it perfect? Is perfect the enemy of the good?

I discovered recently when I began exploring the uses of the sector, that I was wrong in a very important way when I discussed the many ways for finding the angle at which the staves of a bucket or tankard meet.

My methodology was modern. For one thing, we started with an equation. For another, it depended heavily upon looking at the tankard as an equilateral polygon and we did some really sweet math based on that assumption.

That was an inaccurate assumption.

Even though it worked.

As I examine more coopered buckets and tankards, it because clear to me that the old coopers didn't think that way. The staves of a bucket are rarely all the same size, and no two identically-sized buckets seemed to have the same number of staves.

My math was accurate, but my method was wrong.

The period method is really cool. It's easier. And it involves a sector and some different neat math having to do with isosceles triangles and dividers.

My assumptions were wrong and even my successful result was... I don't know. Was it a failure to achieve the goal by apparently modern means?

A period item was created, but it was based on best guesses made with a modern mind. My methods of arriving at that item were modern even though I used my best period tools to achieve the result.

I know all of this because I didn't finish exploring coopering when I finished writing about it. I kept going. I kept talking to other coopers. I examined barrels and buckets in antique shops. I made a bucket. Then I made a butter churn. Then I repaired some damaged buckets and barrels and tankards back to working order.

I know I was wrong because I didn't stop.

I made an ale pail that would hold ale, but did I succeed or did I fail.

Sorry, that was Seussical. Sometimes I can't resist.

At some point do I stop going back and adding to these projects?

Or is this exploration of artisans a reflection of artisanship itself in that the learning never actually ends? And if that's so, is my quest really impossible after all? Will I ever have more than the most surface knowledge of any of these crafts if I cannot devote more than the duration of a few blog posts to each of them?

How do I know when to stop?

And if I'm honest with myself, can I stop even if I want to?

- Scott

The Joy of Sectors: Getting our Galileo on...

"For the eye is always in search of beauty, and if we do not gratify its desire for pleasure by a proportionate enlargement in these measures, and thus make compensation for ocular deception, a clumsy and awkward appearance will be presented to the beholder."   

- Vitruvius, De Architectura 

Of course, a major part of the "rebirth" heralded by the renaissance was a revival of the mathematics and geometries of the Arabs and the ancients. By harkening back to the glories of their Hellenic ideal with their domes and pillars, the Renaissance brought with it a new and almost slavish devotion to finding the sacred in geometry and symmetry. Not just buildings, but furniture and textiles began to push painted, woven, and carved decorations to ostentatious heights.

I'm not particularly well known for being good at math and certainly didn't receive high enough marks in school to give one the feeling I would go on to write fluently about engineering and architecture. Thankfully, our typical renaissance artisan wasn't particularly well known as a mathematician either.

Please note that here I am drawing a line between the theory and the application of maths. Although the loftier theories may have passed him by, the practical maths of proportion and symmetry were alive and well in 16th century workshops. The average Elizabethan joiner may or may not have known who Euclid or Pythagoras was, but he could apply their theories well enough to please the eye and the customer.

We've discussed some basics of dividers before, when we were coopering. Add a sector and by their powers combined, you can accomplish an amazing number of tasks with very little actual number-crunching.

I first learned the magic of the sector in the same math class where I learned about the Fibonacci and the various permutations of the Golden Mean. Then I didn't think about it much for several decades.

Like most woodworkers, I've always kept a set of dividers. Dividers are handy for drawing circles and arcs for those fantastically symmetrical carvings I mentioned, also transferring dimensions from a ruler or a drawing to the wood. I've used them for laying out dovetails and for finding center and a host of other simple tricks.

But when they're accompanied by a sector, they can do much, much more.

My geometry teacher knew that the wickedly-sharp compasses we were equipped with as part of our standard kit were capable of more than stabbing us through our canvas bookbags. When paired with a sector, they could be used to accomplish great feats of proportion and scale

And she had no less a personage than Galileo Galilei backing her up on that.

I didn't care, I was nine; I wanted to draw circles and stab ants with the damn thing. Education is wasted on the young. Sometimes, I think adults should be required to repeat primary school periodically to pick up all the sharing and math and social studies that we missed, never mind the history. We seem so determined to keep repeating our history anyway, it might as well be in a classroom.

"I'm sorry, boss, I can't come in today, I have geometry class and then detention because I said I was thinking about voting for Donald Trump..."

Anyway... flash forward to a 2011 issue of Popular Woodworking magazine I picked up at the newsstand because of a cool cover article about Thomas Jefferson's stacking bookcases. Inside was an article by Jim Tolpin on the use of the dividers combined with a sector (see the video below) to derive a host of useful proportions and measurements for cabinetry design.

Like my teacher before him, Jim attributed the invention of the the sector to Galileo. I'm a big Galileo fan, going way back, and ere the end of things, we might even get into some of his experiments with optics because I enjoy that sort of thing.

Galileo's Sector displayed in the Putnam Gallery -- Image via Wikimedia Commons, Creative Commons CC-BY-SA 3.0

They were both likely wrong about the inventor. The basic principles were first proposed by Euclid and put to various uses since. It seems more likely that he was the Bill Gates or Steve Jobs of the late Renaissance. He was a technological entrepreneur who envisioned new and popular uses by combining existing technologies and concepts in unique ways. That said, who initially turned a compass into a more complex instrument matters little, because ere the end of the 16th century, the concept broke out in a Big Way in the manner that technological leaps always seem to.

The sector as Galileo created it is partly well known because of who he was, and partly because it was enormously successful as a commercial product. The sales of the instruments made his fortune long before he started tweaking the beards of the Inquisition with his planetary models.

Galileo primarily sold his sector as a military tool, an instrument which in addition to its more basic Euclidean functions carried additional scales useful for the gunner in the trenches.

I have no use at the moment for determining powder loads and trajectories. There just aren't that many armies out there right now that need that sort of thing done the old fashioned way. I will be making a simpler, significantly less schmancy, workingman's sector along the same lines as Jim Tolpin's.

If nothing else, I have a lot of period carving and surface decoration on my project list, so we can look forward to seeing great granddad's dividers and sectors come out for that.

And for now -- since sectors weren't all that widely used until the 17th century anyway -- that will be the soft limits for our use for the things. I'll make a couple in different sizes and we shall see what use can be made of them without gunpowder getting involved.

That said, the Honorable Artillery Company was knocking about, but they weren't really what you'd call a trade guild. Nevertheless, I picked up a copy of Galileo's instruction book that was sold alongside his sector because you never know when you might need to hit something a long way away with a ball of something fired out of a tube full of grey powder.

- Scott

The Arte & Misterie of Coopering: Making a Mary Rose tankard, Part Two

This being the part where we turn math into a drinking vessel...

You may remember this bit of algebra from last week. Contrary to what I've been seeing on an alarming number of YouTube videos and woodworking forums, it's the way to find the correct angle to cut your bevels on the staves of your bucket/tankard/butter churn, or whathaveyou.

n = the number of staves you want in your vessel.

And it doesn't change no matter how big or small your vessel gets.

There seems to be a lot of confusion about that out in the Interwebs. The angle is the same whether you have a diameter of ten feet or ten inches or two inches. For an octogon (or an eight-piece circle) the angle for the cut is 67.5 degrees.

Geometry is awesome like that.

The mystery of the bevel solved, it only falls to us to decide how to cut that angle accurately. As I said in the lead-in to this project, every source I have just says "the cooper eyeballed it." And the tool they used for that eyeballed angle was a big jointer plane mounted upside down on the floor of their shop.

You can see one being used in this engraving from my muse and tormentor Jost Amman. That's one seriously large piece of equipment. I checked around and those things are expensive even if you make your own (mostly the cost of getting the blade made).

Image source: Wikimedia Commons
Cleaned-up digitally by yours truly.

You can see it in use in this video I posted the other day of Ramona Vogel, journeyman cooper at the Colonial Williamsburg living history village: http://vimeo.com/11313428 I think that beyond some basic geometry, the real "art and mystery" here is how to finagle the right tools to make it all work.

Thankfully, the woodworking blog I got that video from (The Village Carpenter) came to my rescue once again with this post about a friend of hers who makes buckets.

He doesn't use a huge floor-mounted jointer; he uses a hand plane mounted upside down and it seems to work just fine. He also uses a jig, but I wanted to try my hand at this "eyeballing" thing that everyone seems so impressed by, so I grabbed my largest bench plane and a handful of clamps and my eyeballs this is what I ended up with...

I've already leveled the sole of the plane, now I'm making some fussy adjustments on the blade.

Taking the first few swipes across the blade.

Note that I'm wearing a glove on my pushing hand and keeping my fingers well back. I want to keep my fingertips on my hand and not on the ground where they'd get dirty and I'd have to hire someone to sew them back on...

Thankfully, the upside down plane worked a treat and I didn't so much as trim a fingernail on that razor-sharp blade that made such short work of that dense oak stave.

Take another look at the Mary Rose tankard we're imitating here and you'll note that the top is narrower than the bottom. Sort of an inverted pint glass shape. So I have to not only trim a bevel on either side of each stave, I needed to make each stave slightly smaller at the top than at the bottom. 

Forgot to mention that I also made it thinner when I was shaping them on the shaving horse. The intent being to give a more substantial foot to this thing since it's going to be quite tall and on the rolling deck of a ship no one wants to spill their beer.

The tool I'm using to check the angle in the photos below is called a sliding bevel.  I pre-set it to the correct angle and locked it off. I've only to hold the piece up to the light and slide the blade of the tool along the wood, watching for gaps.

  

I tried to find out how they did this in the 16th century. A lot of tools like dividers and the like predate the period, but I can't find anything concrete on the topic of the sliding bevel. If anyone has anything on this, I'd love to hear it.

The results are eight staves, evenly shaped to form a circle which will, under compression from its hoops, swell to become water-tight.

It's almost but not really discouraging to note that if you have a sliding compound miter saw, a couple of router bits, and a table saw, this project really would take you an afternoon. But it wouldn't be nearly as cool as this one.

Or so I keep telling myself...

Still a lot of work to do with the scrapers on the inside and quite a lot of fiddly work aligning the staves and making everything work out just right.

Then I can make a handle and a lid.

Note: Don't worry, it only looks like I'm behind. I'm also learning to knit, making a leather bottel, and getting started on hornwork.  More updates on all that stuff later this week... I hope. 

~ Scott

The Arte & Misterie of Coopering: A method to the mathness...

This is going to be a longish post and it's going to involve some basic math. If you've a fear of either, I feel for you because though I'm not skittish about reading novel-length posts as long as I'm learning something, math isn't my bestest friend in the whole wide world.  I may be a nerd, but when I was a kid there were math nerds and there were art nerds and there were comic book nerds and band nerds.

The venn diagram of my life intersects all of those, but didn't cross the border of math nerddom until I married one.

And if I want to make it as a cooper in even so small a way as this, I'm going to have to shake hands with a math book. This is what we're going for with the coopering portion of the project.

Image Source: Wikimedia Commons

Image Source: Wikimedia Commons

It's a tankard that was brought up from the wreck of the Mary Rose.  It is one of the many tankards of a similar sort that are knocking around museums in Europe. Several were brought up from the Vasa. They seem especially common in seafaring contexts (shipwrecks and port cities) and are hooped with wood or cane, which is easier to fabricate without a smithy (mine isn't built yet) than iron or copper.

One of the key "mysteries" of the cooper is judging the correct angle for the staves to meet in order to make a perfect circle. The resources online and in the books I have available to me aren't a lot of help on this one. Like the cooper in that video I posted, most sources simply say that a cooper learned to 'eyeball' it over the years.

Obviously, I don't have years to devote to this. I've achieved a measure of acceptance for the idea that I'm never going to get past the apprentice level for most of these trades. The eternal apprentice am I.

The inner and outer curves are easy enough to make up as I go along; the bevel is another story all together. If the bevel's wrong, this thing isn't going to fit together or it's going to leak like a sieve.

Which is to say that I need to figure it out using math.

Somewhere a math teacher that I told "I'm never going to need this after I get out of school" is laughing uproariously.

It's nice that I can continue to amuse them so many years later.

Step One: Ask an Engineer

I asked my wife how to figure out the correct angle for the sides of each stave of the tankard.  She looked me right in the eye and said "AutoCAD". After a moment's reflection, she changed her mind. "Actually, Solidworks is better for this kind of thing."

Hmmm... Thanks, honey.

Okay, for the record, I studied technical drawing "the old fashioned way" in art school and can use CAD if I have to. I could also have my dearest darling draw it up for me. And she offered to do so.  It would certainly be easier for her since numbers aren't really my favorite thing and tech drawing wasn't my favorite class.

The problem with teaching anything, is convincing the student that the subject will be important enough for them to pay attention to. In my past experience, math teachers were especially bad at this. Mine certainly never really did a very good job of telling me why I needed to know this stuff.

To be honest, I didn't do a very good job of listening either, so a plague on both our houses, I guess. Like almost everyone in the world, I use algebra and physics on a daily basis. Only as an adult did I come to really appreciate that fact and had to go back and teach myself (or humble beseech my wife to teach me) those subjects.

Pay attention in math class, kids.

There are several ways to do this without using a computer, or even using a calculator.

Step Two: Do It the Easy Way

Years ago when I started making wheelbarrows, I had to figure out how to cut felloes of the correct arc and length. ("Felloe" is the technical term for the wheel sections of a wooden wheel.) I made my template by cutting out a circle and folding it in half several times to make something that resembles a pie.

If your scale is 1:1, then each pie piece is the correct size and the edges are at the correct angle for a section of a wheel or the stave of a tankard. Cut the center out and the widest point of each pie piece is a template for every angle you're going to need.

If you run on a smaller scale, this method is almost infinitely scalable. Just use a sliding bevel to translate the angles from the template to the wood and you should get it done with a minimum of futzing around.

Step Three: Find a Method Closer to Period:

One of my favorite measuring devices is an ancient tool called dividers. They appear in museums (these supposedly belonged to Michelangelo) and paintings and drafting sets and the bookbags of school children.

Most of those children grow up to be adults who call them "compasses" and think that all they're good for is drawing circles. And for most people, that's enough.

About the only places I still run into them being used in their original capacity is in navigation where they're used to mark out distances on charts and in woodworking. Woodworkers tend to use them to scribe circles and copy relief surfaces.

This is historical woodworker Peter Follansbee's article on using dividers: http://pfollansbee.wordpress.com/tag/compasses/.

They can be very expensive and finely made or they can be cheap as chips, bought in the school supply section of your local Target.

I own several pairs of varied vintage and before the project's finished, I suspect you'll see all of them. The ones in the photo below were purchased for a couple bucks at a Harbor Freight in Tacoma. 

If you stand your staves in a circle and set your dividers to the width of the farthest point, you can draw the line of the bevel as illustrated below.

This gives you your bevel as well as letting you "eyeball" how much material you're going to need to remove in order to get the walls to their desired thickness for your bucket, tankard, barrel, or butter churn. All of those things were made by coopers, and all demand slightly different wall dimensions as dictated by their end use.

I admit it. I needed geometry and math and all the things my high school math teachers had to put up with me complaining I'd never use. I not only needed them, I also needed to know how to apply them when the time came. My apologies to math teachers everywhere.

~ Scott