Here’s a page from a forthcoming pamphlet I’m working on about the geometric truths that underlie – and led to the development of – all our layout tools, including (and ending up at) the sector. If you have kids around the house, you might find it interesting (maybe even enlightening) to put the sector into their hands to show them how they can physically (by hand and eye) manipulate a tool to illustrate the basic intuitive concept of the fraction.
As you’ll see when I get the pamphlet done, you can use every one of your layout tools to illustrate – that is, to perform – the reality behind the geometric math they are being injected with at school.
If you don’t have a sector, you can download (for free) a template to make one from the “shop” page at By Hand & Eye. Click here to visit that page. There you’ll find a manual for using a three-scale sector on the page as well…for a small fee to help George and I keep the lights on and the electrons flowing.
If you haven’t heard of the sector, it probably means you aren’t an artillery officer or a ship’s navigator working in the 17th century. An invention attributed to the great astronomer Galileo, the sector was a calculation instrument comprised of a pair of hinged plates engraved with a variety of scales that – coupled with a pair of dividers – enabled the operator to calculate proportions, polygons, trigonometric and numerous other table functions.
By the late 1700s, documents show that the sector was also taken up by architects and artisans to lay out designs based on the once ubiquitous whole-number segmentation and ratio-proportioning system of their trade. However, as 19th century machine-based manufacturing eclipsed the traditional practices of the artisans, their design and layout tools – dividers, sectors and applied Euclidian geometry in general – faded almost entirely from use.
I have discovered, however, that a simple version of the sector can be an incredibly useful and efficient tool for creating scaled drawings (or even doing direct layouts on the stock) when working within traditional design and layout systems. As you may know, George Walker and I describe this system in excruciating detail in our hard-bound book “By Hand and Eye,” and somewhat less-so excruciating in “By Hound and Eye” – the workbook.
With this three-scale sector in one hand and a pair of dividers in the other you’ll find that you can, literally in seconds, create equal segments between two points; derive harmonic proportional relationships along a line or between dimensions; generate angled lines to certain rise-to-run pitches; set out the facets of polygons (up to 12 sides); find the radius to draw arcs of these polygons between any two points; determine the circumference of a circle knowing its radius; and find out what your brother-in-law really does for a living.
Once you start working with our variant of this ancient calculator, you’ll wonder how you ever made do without it.
If you go to the “Shop” page of our By Hand & Eye website you’ll have access to a free-to-download template to make your own sector to play with. Here you’ll also discover a downloadable 40-page pamphlet on using the sector (offered for a small fee to defray expenses and keep George and me off the streets). For those who don’t want to cut out and assemble (i.e. hinge) the template, we also offer an assembled sector (with the “bonus” of being hand signed by George and myself).
The ancient “geometers” believed that geometry was the key to comprehending the incomprehensible; that an understanding of its inherent truths was the key to unlocking the mystery of how the “Gods” created order from chaos. The development of geometric constructions (the truths rather than proofs) became the foundational tool of the artisans to create a built world of inherently sound, durable and pleasing forms.
And it all started from nothing:
Across the Ionian Sea, a gentle wind blows from the West with the fading of winter–the breath of the God Zephyr, a harbinger of spring and the bringer of light. A God whose name would, as we will see, appropriately evolve to produce the word “zero.” The geometers did not use or represent zero as a number, but rather as a notation to show the location of the focus of a circle. Like the true center of a wheel, it is the only place that does not rotate, for it is a place of no dimension. There is nothing to rise, nothing to fall. All revolves around it. And like Zephyr, the renewer of life, the zero of the ancient artisans served as the seed of all shape and form. website
— Jim Tolpin, reprinted from the byhandandeye.com website, which explores artisan geometry
The authors of “By Hand & Eye” and “By Hound & Eye” have launched a beautiful and informative new website that you should add to your browser or RSS feed.
The website is appropriately called “By Hand & Eye Online” and you can get there via http://www.byhandandeye.com. At this point the website consists of an active blog maintained by Jim Tolpin and George Walker who are writing about the interesting pre-industrial design techniques they have been exploring together for several years.
At the moment the two are creating a six-part online series called “Tricks & Truths” that will further extend your knowledge of artisan geometry. While the online course isn’t ready yet, you can sign up to be notified when it is released here.
We at Lost Art Press have been greatly heartened and a bit surprised at how well George and Jim have connected with readers on the topic of design. (Design books are typically commercial duds.) Whenever I travel I hear customers rave about the techniques their books explore, which are based on simple whole-number ratios instead of magic formulas or art school mumbo-jumbo.
If you are interested in going deeper into their world, I highly recommend you check out their new site, read the blog and consider taking the online course when it becomes available. Jim and George are fanatics about this stuff, and it shows in the quality of their writing and teaching.
“By Hound and Eye” isn’t for everyone. If you’re a geometry whiz or an 18-century cabinet maker, you probably won’t learn anything new from the workbook. I’ve created a three-question quiz to help each blog reader decide whether or not “By Hound and Eye” will be worth its $20 price tag to you ($10 for the pdf alone):
Do you already use a sector and a divider to layout your designs?
Can you readily produce a set of full-scale measured drawing from the sketch of the writing desk below?
Can you write out the musical harmony literally embedded in the desk below?
If you answered “yes” to all three questions, “By Hound and Eye” might underwhelm you.
I probably should have learned all of these skills two years ago when I first picked up “By Hand and Eye.” But I didn’t. Tolpin and Walker’s prose was so eloquent that I imagined I was already seeing “the music frozen in furniture.” Thanks to their persuasiveness, I became a haughty, complacent reader. I didn’t do the exercises on page 28, let alone the dozens suggested in the following 150 pages.
By the final section, I felt completely underwhelmed by the simplicity of the projects, which seemed so straightforward that any electron-slaying woodworker could produce each of them in just a long afternoon. Instead of seeing design proportions, I saw easy joinery. Instead of hearing music, just the hum of my table saw.
In “By Hound and Eye,” Walker and Tolpin take the inverse approach. Instead of telling you about the music of furniture, they teach you how to read and write your own. After my first time through the book, I still cannot answer “yes” to questions 1 and 3 above. And I never had a singular “aha” moment of the sort Chris described. But I imagine I will, once I work my way through them another three or four times. This is math, after all, and translating theory into skills takes multiple rounds of practice (at least for me).
A final note: If you’re one of those brave souls who has already broken free of the rectangularity of arts-and-crafts designs, you might benefit particularly from Section III of the workbook, “Curves.” It was in those pages that I felt most inspired and, paradoxically, still more than a little ignorant.
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Image credits: Author’s photograph of hand forged dividers by Seth Gould; detailed crop from Raphael’s masterpiece “The School of Athens,” Images from By Hound and Eye, illustrated by Andrea Love.