In our book* By Hand and Eye*, we showed a simple straight taper–common enough in Roman columns and quite easy to generate. But some columns from Greek antiquity display a taper that follow a curve. As shown in the drawing below, the curves get more radical as you move from parabolic to elliptical to hyperbolic. All were developed, not from a numerically dimensioned layout, but from the generation of a relatively simple geometric construction familiar to ancient artisans.

The parabolic curve is the simplest and fastest to execute. As shown in the drawing, it is simply a matter of dividing up (with dividers of course) the inset amount of the top of the column into equal segments, than running straight lines (with a straightedge or string) from these points to the corner of the column shaft at the base. You then create station points at evenly spaced, horizontal intervals drawn across the length of the column. (I show only four intervals here for clarity–plus I’ve compressed the height-to-width ratio to exaggerate the curve).

To create the elliptical curve, the artisan drew a half circle to the diameter of the bottom of the shaft, then segmented the half sector into six even slices. Lines drawn vertically from the intersection of the horizontal segments with the rim of the arc create your station points above.

The hyperbolic curve station points arise from evenly spaced segments stepped along the circumference of the half circle.

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Step 1: Establish the baseline (via a stretched string) and set a pin (a sharpened stick works) at the focal point to which the angle will converge.

Step 2: Make a loop at the end of a non-stretchable rope (i.e. avoid nylon) and run it out along the baseline from the base pin. Measure out 57-feet, 2 ½-inches from the pin along the rope and make a mark with a sharpie or tie on a piece of string. Also set a pin at the baseline at that distance.

Step 3: Now arc the rope away from the baseline in the direction you want to lay out the angle.

Step 4: Set the base of the ten-foot pole at the base line pin and orient it to the rope. When the eight-foot mark on the pole passes over the mark on the rope the angle to the baseline is (drum roll) eight degrees.

So how does this work you might ask? As my friend Joe Youcha of buildingtoteach.com explained to me: “The answer is buried in the math we were all injected with in grammar school.” We were all told about the “transcendental number” called “pi” which when inputted into your calculator would provide you with either the circumference of a circle based on its diameter or vice versa.

Artisans of antiquity, however, had no knowledge of the decimal number pi. In fact decimal numbers in general had not been described in detail in the western world until the late 1500’s by the mathematician Simon Stevin. But artisans did have an excellent working relationship with the straightforward (non-cendental?) proportional ratio system. In the case of the relationship of the diameter of a circle with its circumference, they would just step out the diameter into seven segments and know that twenty-two of those segments would, to a high level of accuracy, give them the length of the circumference. Good enough for government work (such as the Parthenon) as they say.

Since we apparently need to work with degrees (probably because the architect speced out the angle in degrees instead of the length of a chord as they would have in antiquity) we would need to know what number of segments the diameter would be if the circumference were stepped out to 360 segments. (That number is, of course, an arbitrary but widely accepted convention since Babylonian times as a convenient way to divvy up a circle. We like it as it can be evenly divided by so many whole number divisions–though for a time Europeans were quite fond of 400 degrees.

But I digress, back to how it works: If you go to the trouble of physically stepping out along the circumference of a circle with dividers, you’ll discover that when three hundred and sixty segments do the trick, one hundred and fourteen and five twelfths of another segment will define the diameter. Of course, using al-Jabr ( given to us by the Islamic mathematicians), we can quickly solve for this result using an algebraic equation to solve for an unknown.

For this purpose we’ll use half of the diameter segments–fifty seven and three and one half twelfths–to lay out the radius length on the rope. The bottom line: We find that a radius of 57 feet, 3 ½ inches produces a circumference length of 360 feet. So for every foot we swing the arc, we produce an angle of one degree.

]]>I make two versions of the straightedge–one three to four hand-spans long for testing typical furniture components and another two handspans for use with small components and at the drawing board.

The try square’s handle is a hand-span long while the blade is a third again that length. I use this square for testing and laying out right-angle cuts on furniture components such as rail and stiles and drawer parts.

To lock the blade permanently to the handle I use a bridled, through mortise and tenon and triangulate ebony pins. There is a bit of hide glue in there as well, but that really serves mostly as a lubricant to slide the parts together and to create a humidity barrier. As with any hide glue joint (which is not a joint by the way), I can’t guarantee the chemistry lasting for more than a couple of centuries. The physics of the pinned bridle and tenon joint should last as long as the wood itself (which, in the case of the Douglas fir, is already well over 500 years old).

I sign and date each of these one-off layout tools. Several of each are currently in stock (as of December 15). I’m asking $85 for the small straightedge, $185 for the larger one, and $250 for the try square. These prices include shipping within the continental US. Please contact me directly at jimtolpin@gmail.com for payment details.

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I just finished up work on this traditional tool chest that was started by Chris Schwarz during a workshop he taught with us here at the Port Townsend School of Woodworking several years ago. Chris built the box, I (Jim Tolpin) built the lid and interior sliding till and applied the milk paint. All the work done was a product of our hands–no machines involved. A boatbuilder in Vermont, Keith Mitchell, made the lifting beckets of rope and leather. The chest is signed under the lid by Chris and I.

We all donated our time to this project to raise money for the school’s youths-in-woodworking scholarship fund and is posted for auction over at eBay. You can do a search under “joiner’s tool chest” on the eBay site. Here’s a video tour:

Toolchest from Jim Tolpin on Vimeo.

]]>Want the “proof”? All you need is a couple sticks and a bit of string as in the photo below. Have your four year old lend you a hand…she’ll immediately intuit what an equation is really all about! (No, this is not your rigorous algebraic proof, or even a Euclidean logic proof…Instead it’s what me bandmates used to call: “Good enough for rock and roll”.)

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Amongst this collection of Masonic “Past-Master’s Jewels” medals, two show a representation of the pythagorean theorem. It is reported that its presence on the owner’s medal indicates that person was what we would likely now call a crew foreman. Once of his many responsibilities was to ensure that all the layout tools were true–a clue to why the homage to Pythagoras. This theorem, codified later by Euclid into his “Proposition 47”, offers a logic proof that the area of the squares erected on the legs of a right triangle would equal the area of the square erected on its hypotenuse. That’s all well and good, but why would that particular equation be of vital interest to the foreman of a joiner’s or mason’s crew? To try to find out, I decided to construct an exact-as-possible, large-scale drawing of the graphic upon which I could explore with a pair of dividers.

The first thing I discovered was that the vertical line CL, which is fixed by the inherent base line’s intersection points C and D, forms a right angle with the hypotenuse. Even though this result is likely nothing more than symbolic (there are a lot easier ways to generate a right angle with a compass and a straightedge), I believe this right angle–hidden in plain sight–is probably as important to the metal (and its wearer) as the theorem itself. The right angle (“recto” in Greek) is simply the right way to set a vertical post. (Wood’s superb resistance to compression happens when, and only when, the post is set at a right angle to level–an orientation that aligns the grain parallel to the force of gravity). It’s also the right angle to create symmetry to a baseline in common rectilinear structures (think cathedrals).

No reason to stop there, though. Exploring further revealed other attributes to this graphic that offer additional symbolic (and real) representations of the truths inherent in Geometry (note the traditional capital G). Print out the template (you’ll find it on the shopping page of this site) and take a look around on it for yourself. You’ll discover triangles with perfect 2:3 base-to-height proportions (one of the fundamental harmonics in music and architecture of the medieval era); you’ll find sequences of the infamous triplet (the 3-4-5 triangle) revealed in the hypotenuse and even in the circumference of the circle that started it all; and you may find the module upon which the entire construction revolves. Have fun with this–I sure did!

By the way: If you want to buy the medal shown, you can find it here

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Finish is several coats of tung oil followed by several coats of shellac, hand rubbed. Notice the little chamfer at apex of ellipse–this allows for finger tips to slide under to lift it from flat board. If you’d like to purchase one directly from Jim, the cost will be $85 including shipping. Shoot an email to jim.tolpin@gmail.com for purchase details.

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The metric man cannot behold

all the wonders nature holds.

For hath the ruler in his grip

no index for relationships.

Square and plummet fail to mark,

Matters felt and of the heart.

No level, rod, nor referenced gauge

Convey what only comes with age.

Now we see, that we’ve resigned

our quest to map the grand design.

Points that lie in perfect place,

the space between them flowing grace.

© 2015 Garrett Smith

]]>Shop Tour from Jim Tolpin on Vimeo.

]]>The thickness of the blade next to the bevel comes to 11/64ths–which is less than half the length of the bevel. So the bevel angle is less than 30 degrees, probably close to the typical 25 degrees. Out of curiosity, let’s see how a quick bit of artisan/intuitive geometry reveals why a 30 degree bevel produces a length exactly twice that of the thickness of the blade:

On a piece of graph paper erect a right angle and draw a quadrant of a circle. Divide the radius in half–this represents the thickness of the blade and is indicated by the “X”.

Now connect a line as shown (the line is the radius length–which represents twice the thickness of the “blade”.)

Finally, set your dividers to the marked angle and step around the quadrant. You’ll discover that it steps exactly three times to reach the perpendicular. Which means the angle is describing one-third of the quadrant, and therefore one-twelfth of a full circle. If you divide 360 degrees by 12 you come up with 30 degrees. Of course, you could always just look up the answer in a trig table, but what fun is that?

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