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

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