I suspect that I learned more about math, civics and language from “Schoolhouse Rock!” than I did from school itself. The magic of “Schoolhouse Rock!” is the magic of all good education. That is: It is so engrossing that you don’t realize you are learning heavy and important lessons.
Practical geometry is one of those topics that can be intensely boring – just look at the beginning chapters of any 18th- or 19th-century text on woodworking, which almost always begis with chapters on geometry. And they are all as dry as a popcorn fart.
The fact is that practical geometry is one of the most exciting subjects for a woodworker. If you can find someone to explain it to you in the right manner, geometry will change your life and your work at the bench.
And that’s why we are dedicated to the work of Jim Tolpin and George Walker, who have made it their life’s work to explain workshop geometry in as many ways as possible – from the academic to the absurd (a talking dog?).
Of all their books, “From Truths to Tools” is probably my favorite. I can give this book to anyone, from a third-grader to an octogenarian, and they will enjoy the heck out of it. It is like a graphic novel for geometry. Each chapter is a discrete short story – heavily illustrated – that explains a very complex subject in simple terms. When I edited this book, I had to stop every few chapters and clear my head. It’s that good.
— Christopher Schwarz
The following is an excerpt of “From Truths to Tools,” by Jim Tolpin and George Walker, illustrated by Andrea Love.
English oak coffer; 16th century. (Image from Wiki Commons, public domain.)
The once ubiquitous coffer (from the Greek “kophinos” – a basket; later from the French “coffre” – a chest) was also referred to as a “strong box” – because it was. (Later the term coffer would refer to an institution’s financial reserves.) This stout, often highly ornamented, chest reached its pinnacle of design and construction in the mid 1600s and was likely the first, and perhaps the only, piece of furniture that a commoner family might own. Likely used every day as a bench, its primary purpose was to keep the family’s valuables safe and private. Its thick oak walls and lid could often even keep its contents safe from a fire.
Accurate drawing of a similar 17th century English chest (by John Hurrell, published in 1903).
For George Walker and me, what is truly fascinating about these coffers is that they clearly demonstrate the traditional, artisan design process we have described in excruciating detail in “By Hand and Eye” (and decidedly less excruciatingly in “By Hound and Eye”).
For those unfamiliar with this process, here it is in a nutshell: Unlike modern builders who think primarily in terms of measurements to an external standard such as inches or centimeters, pre-industrial artisans took their cues from the builders of antiquity and thought more in terms of proportions. They would start by selecting a simple rectangle of harmonic proportions (literally from the audible harmonic musical scale) to govern the overall form. For example, the front elevation of height-to-length were commonly ratioed at 1:2 (an octave); 2:3 (a perfect fifth); 3:4 (a perfect fourth) or 3:5 (a perfect sixth). Within this rectangle they would select the span of some prominent element of the structure to act as a module (an internal index measurement often based on a element of the human body) and then tie all the other details proportionally to it.
The coffer is a perfect example of this ancient design process: a straightforward layout based on the geometry of a cuboid defined by simple whole number ratios of height, width and depth. Like the proportions embedded in the design of Grecian columns (which deeply influenced the design methodology of the joiners and cabinetmakers of the 17th and 18th centuries) the designer of this coffer clearly used the width of the chest’s leg in the front elevation as the “module” for the design. (The Greeks used the diameter of the base of a support column’s shaft, which happens to be the span of the human body, as the module for all the other elements of the temple.) We encourage you to print out the above drawing by Hurrell, take a sharp pair of dividers, set it to span the width of the leg (we label it “M”) and go exploring with us:
The first thing we’ll discover is that the height of the leg (to the underside of the lid) is exactly seven times its width (again, the module for this design). By eye, it looks like the length of the chest may be twice its height. When we step the module between the outside of the legs, however, we don’t come up with that nice whole-number ratio. On our second shot at it, we discover the lid from edge to edge is a precise 14 modules long. So there’s our 7:14 ratio – or to simplify 1:2. Which is a perfect octave harmonic, and a common choice for the coffers of this era (and later for highboys in their vertical extension).
Further exploration reveals that the mid-stiles and bottom rail are also a module wide, as is the height of the carved inscription of the date 1689. If you continue poking around, you’ll unearth all manner of modular-indexed relationships buried in the intricate geometric carvings. Be aware that the spans and radii, if not exactly a module-length, will be a whole number fraction above or below that length. For example, the module (plus a third of the module) serves as the spacing for the positioning of the lower rail from the baseline as well as the width of the top rail.
As in the ancient temples of Greece, every single element of this coffer enjoys a whole-number relationship with each other – and with the overall geometric form. As such, you can scale this piece of furniture up or down by simple changing the span of the module – no measuring to numerical dimensions is necessary – just an adherence to the ratios.
Now let’s explore the design of this coffer’s frame-and-panel lid. We were excited to discover how some anonymous 17th-century artisan made clever use of the module to add subtle, eye-pleasing asymmetry to the layout. Before you see how they did it (below), try to discover it for yourself. We find this sort of thing fun, and we bet you will too.
So here’s what we found: The module is utilized in four different ways: For the middle frames, it defines its overall width; for the hinge-side frame it defines its width, but it does not include the lid edging; for the end frame it does include the edging; and for the latch-side frame the module defines its width from the inside of the edging to the edge of the bevel next to the panel. Subtle, but just enough to make the design lively to the eye.
For your further entertainment, below are a couple more of Hurrell’s drawings of 17th-century English coffers for you to print out and explore. To see what we unpacked with our dividers, check out our blog at www.byhandandeye.com. One hint/reminder: The module for each of these designs is the width of the leg.
To learn more about the construction and carved ornamentation of these traditional coffers (also called a “joined chest” in America), you can do no better than to watch Peter Follensbee’s video “Joined Chest” available from Lie-Nielsen here or to read “The Artisan of Ipswich” by Robert Tarule, available from John Hopkins University Press here.
This excerpt from our latest book, “From Truths to Tools,” speaks to a rather esoteric, but highly useful, rule for use with scaled drawings:
Here’s a typical, traditionally drawn small boat plan:
To find the dimension of any particular part of the boat, we simply set the divider to the part – here the cap on the centerboard trunk:
Then we transfer the dimension (what the Greeks called, more precisely, a “magnitude”) to the graphic rule…
…and read the numerical distance of 3′, 6″. This technique eliminates the need for an awkward and hard-to-read scaled ruler and, furthermore, works no matter what scale the drawing is made to.
We’ve had a few folks ask about the “hidden hexagon” mentioned in the text, and we think it’s time to share the answer with everyone. This also means revealing a little bit more about what is going on (and not going on) in this geometric construction.
What is going on is this: Drawing lines from and through certain points seems to magically create a representation of one of the most important, not to mention useful, theorems for artisans in geometry: the Pythagorean Triplet. In the geometry of this particular interaction of a circle with a square, a triangle is formed in the upper half of the circle whose legs go on to generate a pair of squares that, when their areas are added together, equal the area of the lower square — and they do that in what looks to be a simple triplet ratio of leg lengths of three to four to five.
To arrive at the correct root lengths of the upper two squares to make this simple ratio happen, the trick from antiquity is to generate a hexagon inside the circle (by stepping the radius of the circle around its circumference) and to then draw a line from the lower left hand corner of the lower square through the vertex of the closest hexagon facet. Next, you continue the line to intersect the upper portion of the circle. This provides the point to which you then draw the legs of the triangle.
The results are leg length relationships of three segments to four segments to the five segments of the diameter line. We have just revealed the simplest of the countless Pythagorean triplets. But have we really? The answer is: Almost, but not really.
We had our friend Dr. Francis Natali take a look at it, and after a couple pages worth of quadratic equations, the truth was outed: The whole-number relationship just isn’t there – though it is, inexplicably, amazingly close. Another friend, Kit Africa, generated the drawing above via CAD, also revealing an oh-so-close 3-4-5 triplet. The bottom line: This drawing from antiquity is apparently symbolic: It celebrates the interaction of easily generated shapes that allowed artisans to intuitively design and build beautifully proportioned and aligned forms on the principles of simple plane geometry.
