Category: By Hound and Eye

To develop the curves in the various brackets – here the support for the back fence on the lid of a desk – I followed the ancient practice of melding arcs of a circle along a straight line.

 

 

 

 

 

 

 

 

 

I begin by making a few concept sketches to get an intuitive feel for the curve I would like to see transition the horizontal lid surface to the vertical back fence. I’m going to go with the shape in the first drawing.

 

 

 

 

 

 

 

 

From the sketch, it reveals that the overall form suits that of a 1:2 rectangle. (An octave, by the way – but that’s another story). Next, I divide the horizontal length into four equal segments. The first of these segments defines the flat at the top of the curve. I then draw a baseline for the sine curve from this segment point to the lower right hand corner, then divide that baseline into three equal segments.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

To find the focal point of the arcs – which will each be one-sixth of a circle’s circumference – I set the dividers to the length of the segment (which is the chord of the arc) and swing out intersections to locate the focal point of the arc. Next, without changing the span of the dividers (because the chord equals the radius for sixth sector arcs as you may remember from Mr. Hammersmacker’s seventh grade geometry class), I swing the arc from the focal point to each segment point. The transition between the two arcs is seamless – proven to be so because a line connecting the two focal points will pass through the arc’s transition point.

— Jim Tolpin, byhandandeye.com (drawing by Andrea Love)

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.

— Jim Tolpin, byhandandeye.com

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).

— Jim Tolpin, ByHandAndEye.com

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