We’ve just received our shipment of the first printing of “From Truths to Tools” by Jim Tolpin and George Walker. We’ll start shipping out the people who placed pre-publication orders in the next seven days.
So this is the last call for people who would like to order the book and receive a free download of the book. Order by Oct. 30 so you can get the free pdf download with your printed copy of the book. After that date the pdf will cost extra.
— Christopher Schwarz
Yes it’s fine to think inside the box for a change, especially when there’s no need to think at all! At least not if your goal is to divide the box into any number of divisions. Thanks to the geometry of diagonal lines that occur inherently within a square, you need only a straightedge to reveal these fractions. This truth/tool allows you to lay out the baffles that will keep the bottles of spirits from rattling or worse. Of course there are easier ways to come up with these fractions (the sector springs to mind), but this is still a great way to construct by hand and observe by eye these geometric patterns as they spring to life.
See if you can follow the steps below which I’ve sketched on a sheet of graph paper. Why don’t you grab some paper and follow along too? Bet your kid can (and would like to) help you out! Get out a pair of dividers so you can confirm that the intersections do indeed produce perfectly spaced segments along a line.
Now isolate the first square…
…and continue by drawing a pair of diagonal lines as shown in green in the next sketch:
Next draw a horizontal line through the points that have been given to you where the (green) diagonals cross the first set. If you set your dividers to the intersections along this new horizontal line, you’ll find there are exactly/precisely/perfectly three equal-length segments. Now let’s draw another set of diagonals and connect their intersections with the original diagonals with a horizontal line:
As your dividers will reveal to you, that line is now automagically broken into four equal segments. Let’s continue the process with two more sets of lines (I really do highly recommend that you stop right now and grab a sheet of graph paper and watch this happen in real time through your own hands and eyes).
You’ll discover the red lines produces fifth segments while the yellow produce sevenths along their horizontals. Add another set (in blue here) of diagonals and horizontal and you come up with ninths:
Keep going if you like:
You’ll get ninths, elevenths and thirteenths – and on to infinity I suppose. What if you want an even number of divisions along a horizontal line, say tenths along the line of fifths? Well they are there waiting for you to discover with your dividers!
— Jim Tolpin, one of the co-authors of “From Truths to Tools“
Once upon a time, 10′ poles were a tool common to a number of trades including linemen and cemetery workers. What they touched with their 10′ poles – high-voltage power lines and corpses respectively – is not something most people would want to touch, not even with an 11′ pole.
Carpenters, however, were quite happy with the 10′, or as the drawing suggests, any length of stick divided into 10 equal segments. For in their hands lay a tool critical to the efficiency and accuracy of their layout work. As we discuss in our book “From Truths to Tools” a right angle can be formed by a triangle composed of three whole-number leg lengths. In the simplest triplet, the leg lengths are three, four and five “whatevers.” The 10′ pole simply employs a doubling of those numbers: six, eight and 10, which are measured in this case with the imperial feet of some long-dead king. (If you think feet stink, you could measure out the pole in the cubits {forearm lengths} of some even longer-dead pharaoh.)
As demonstrated below – lifted from the book – we can construct a “proof” of this particular triplet using a straightedge and dividers. Be aware that there are many more whole-number triplet combinations – perhaps an infinite amount.
The sketch below shows the pole in use aligning a post square (and therefore plumb) to a level floor:
It’s a simple enough procedure: After fixing the base of the post to the desired location on the floor system, you use the pole to lay out a mark 6′ up from the bottom of the post. Next, you lay out a mark 8′ away from the post on the floor. When the full 10′ length of the pole fits exactly between the mark on the floor and on the post face, your post will be exactly square to the floor. Turns out that this layout problem (among many others as you’ll discover in the book) can be beat with a stick!
— Jim Tolpin, byhandandeye.com
Or, translated from the Latin: “As Level as Water.” As we explored ancient layout tools at length in “From Truths to Tools,” it became quite clear that the artisans of antiquity were no dummies. For example, we see from their tools and works that they understood that there was a difference between the curved “level” of a horizontal line and the straight “level” of a sight line. In fact, when they used the term “horizontal” to name the latter they were alluding to “horos,” the horizon, the boundary between water and sky.
How did they know that the earth they stood on was a sphere? Two things for starters, according to source documents: They observed the arc-line shadow of the earth falling on the moon during a lunar eclipse, and they watched ships disappearing on the horizon from the hull to the top of the mast (as opposed to the ship simply getting smaller and smaller). Why is this so important? Try building an aqueduct so it works properly or digging a tunnel accurately through a mountain without accounting for this difference.
If the trough of the aqueduct were constructed to a sight (or laser!) line level, the water would flow toward the center because the center, relative to the earth’s surface, is downhill from either end. Another problem that could arise if the support columns were constructed to meet the trough at right angles, is that the columns would only be plumb in one location. They would all be parallel, but that doesn’t make them right! (Literally: the forces on the un-plumb columns would have some amount of shear in them, leading eventually to distortion and ultimately failure.)
In tunnel work, the opposite problem arises: if they relied solely on horizontal level as the digging progressed from start points established by sight lines shot around the mountain from surrounding benchmarks, the tunnel would not exit at the predicted opposing point. Digging from either end, one tunnel would travel above the other and they would never meet. Note that the gradient drop of the earth’s curve is about 8′ per mile – and it’s not an additive (linear) increase, but exponential to infinity. To grasp this intuitively, picture the earth constantly curving away from the sight line. Eventually, at a point just past a quarter of the way around the sphere, a line dropped down square from the sight line would never reach the earth’s surface.
The more George and I immersed ourselves in research for this book, the more we gleaned about the tools and works of the artisans of antiquity and the smarter they started looking to us. The corollary was the dumber the guys in the mirror looking back at us each morning started to look! Obviously, not only is there is still so much more to learn, there is so much more to relearn!
— Jim Tolpin, byhandandeye.com
Well that’s not entirely true, of course. I do use them when I need to make up a cut list from a full-scale drawing or story stick to tell a machine in numerical code (be it metric, Imperial or shaku) where to make the cuts. The cut list is, however, rarely necessary in the hand-tool approach to construction. So in typical layout work, I go with pin-point perfect real placements of cut or location lines.
For example, if I need to lay out the location of slats in a bed’s headboard, I simply stack the slats together against the post (or its location on a story stick) and find the intervening gaps by stepping out the number of gaps needed between the slats (number of slats + one). Layout follows as shown in the next drawing. The accuracy of the layout will be a function of however sharp I make the points of my dividers.
Of course, you can use algebra to generate dimensions with numbers:
As for me, I don’t want to spend the time doing it and then having to deal with reading tiny numbers on some ruler and coping with rounding errors!
As another example of rulers not always ruling: Say you want to locate placement buttons (the ebony plugs in the set shown here) on a pair of winding sticks so you can quickly locate the sticks on the edge of a board. In this case, the location is not a number at all (at least not until after the fact). You could, of course, measure the length of the sticks and divide by two to get a numerical center point. Or, to avoid rounding error, you could step off an even number of intervals to locate the middle division line and then enjoy the accuracy of a pin prick. But both would miss the point so to speak. What we are really looking for here is not the center of the stick, but its center of gravity. How do we find that? We just balance the stick on a sharp knife blade!
— Jim Tolpin, byhandandeye.com and one of the authors of “From Truths to Tools“