There has been a resurgence of interest in pre-industrial design techniques over the past ten or so years. George Walker and Jim Tolpin have now published three books (*By Hand and Eye*, *By Hound and Eye*, and *From Truth to Tools*, all published by Lost Art Press) that catalog many of the nearly forgotten design techniques that rely on simple tools and an eye for finding or establishing the rational relationships between dimensions in the design. Moreover, through teaching workshops, publishing, and blogging, they’ve done much to justify why these techniques have value and why people who build things should be interested in them.

In short, Tolpin and Walker have gone a long way toward addressing the questions of what is pre-industrial design and the how do you practice its techniques. I believe there is also a lot to learn from the question of “why?” — as in “why does it all work?”

In this article, I’ll briefly list an inventory of some of the more commonly used design techniques. Many of these techniques are things we’ve learned and practiced as recipes. We can certainly get a lot of mileage out of these recipes simply by learning how to perform them. However, I believe that we can gain even more from understanding why the recipes work. In short, this kind of understanding gives us the potential to creatively develop new techniques of design that we had not conceived of before. It can allow us to expand the field of design itself. One of the best places to go in order to cultivate that understanding of why it all works is Euclid’s Elements. Therefore, much of this article (and those that follow it in this series) will take a look at what the Elements can tell us about why our pre-industrial design techniques work.

Listed below are several operations that will become second nature if you establish a regular practice of using pre-industrial design techniques. On their own, they are well worth practicing and mastering. However, they will also inform our study of Euclid’s elements.

- Draw a straight line segment.
- Draw line segments whose lengths satisfy a given proportion or divide a line segment into pieces that satisfy a given proportional relationship to each other (or to the length of the original line).
- Draw segments that intersect at right angles.
- Draw segments that intersect at a given angle.
- Draw line segments that are parallel to each other.
- Subdivide a line segment into a given number of equal sub-segments or into a given number of sub-segments whose lengths satisfy a “harmonic” progression.
- Draw a rectangular box whose side lengths satisfy a given proportion
- Draw various regular polygons.
- Draw a line segment that is tangent to a circle
- Draw compound curves that are constructed from circular arcs.
- Draw elliptical curves and other compound curves that are constructed from elliptical arcs.

More than any other book published, Euclid’s Elements provide us with the reasons why our pre-industrial design techniques work. The following outline describes the parts of the Elements we will include in our study to support the validity of our pre-industrial design techniques. In coming articles, I will look at the ideas laid out in this outline in detail. Mastering will not only strengthen your understanding of why the existing set of design techniques work, but it should also provide you with the potential for devising new ones.

- Book I establishes the majority of the ideas behind design that makes use of rectilinear (not curved) figures. Mastering this book by really understanding
*all*of its definitions, postulates, common notions, and postulates is a great way to build a critical mass of geometric understanding that will later allow you to pick and choose ideas from the remaining books in a way that will hopefully allow you to begin developing your own, creative design techniques. In any case, here is broad overview of some of the more critical parts of Book I. We will dig into them in more detail later:- Definitions 1-23 establish the foundational building blocks of plane geometry.
- Postulates 1-5 assert the starting points of plane geometry as a logical system. They provide us with initial facts about lines, circles, right angles, and parallel lines. From these facts, essentially all other facts about plane geometry in Book I can be deduced.
- Common Notions 1-5 These are also facts about plane geometry. They are less earth shattering than the postulates, but they are facts all the same. They might as well have been called postulates.
- Proposition 1 outlines the process for constructing an equilateral triangle.
- Proposition 2 proves that line segment can be copied and transferred.
- Proposition 3 proves that a line segment can be cut down to a given, lesser length.
- Propositions 4, 8, and 27 prove the four triangle congruence conditions (SAS, SSS, AAS, ASA) but only by using a concept known as superposition. This is controversial because superposition was is not something that was already put into place in the form of a postulate or a common notion. In more modern views of geometry, these conditions are taken as a group to be an axiom. They also have been shown to be logically equivalent (all of them must be either simultaneously true or false). Therefore, it is sufficient to take any one as an axiom of geometry and deduce the rest. This is the approach I’ll take.
- Propositions 5 and 6 are statements about isosceles triangles.
- Proposition 9 proves a recipe for bisecting an angle.
- Proposition 10 proves a recipe for bisecting a line segment.
- Propositions 11 and 12 offer two techniques for constructing right angles.
- Propositions 13, 14, and 15 are theorems about supplementary angles and vertical angles.
- Propositions 16 and 17 are statements about the measure of angles in triangles
- Propositions 18 and 19 describe the relationship between lengths of sides and measures of angles opposite to them in triangles.
- Propositions 20 and 21 establishes the triangle inequality.
- Proposition 22 establishes when you can construct a triangle from three given segments.
- Proposition 23 proves a recipe for copying a given angle.
- Propositions 27-30 establish a variety of parallel line properties.
- Proposition 31 proves a recipe for constructing parallel lines
- Proposition 32 proves that the sum of angles in a triangle equals two right angles.
- Propositions 33-41 establish several relationships between parallels, parallelograms, and triangles)
- Proposition 46 finally establishes and proves a recipe for the Construction of a square.
- Proposition 47 is the Pythagorean Theorem.

- Book II, in many senses, lays the groundwork for what is meant by area for rectangular, parallelogramic, and triangular figures. From a design point of view, this won’t be so critical for us. However, there are a few aspects of this book we will draw upon in future work.
- Definitions 1-2 essentially answer the questions, “what is a rectangular area” and “what is a gnomon?” We’ll use the first definition in some situations. We’ll use the second one to a much lesser degree.
- Proposition 11 provides a recipe for constructing line segments that are proportional in the golden mean. From a design point of view, I believe much of the importance and applicability of the golden mean has been overblown. however, it is critical to be able to construct a regular pentagon, and this is something we will make some use of.

- Book III begins to explore the properties of circles and circular arcs. We will draw from this book because of its applications to laying out curved features in our designs.
- Definitions 1-11 state the meaning of what we now refer to as circular arcs, chords, sectors, etc.
- Proposition 1 describes how we can reconstruct the location of the center of a given circle.
- Propositions 11 and 12 describes the relationships between circles that are tangent to each other and the line that connects their two centers. These propositions are useful in constructing cyma curves, which are piecewise smooth networks of circular arcs.
- Propositions 16-19 establish methods of drawing lines that are tangent to a given circle (or a circle that is tangent to a given line). This can be useful for rounding the corners of a polygonal figure.
- Propositions 18, 22, 25, and 27 are utility propositions that can be used in proving the mathematical validity of a certain template for drawing large scale circular arcs that are contained in a rectangular region. This machine is a good option for drawing such arcs on full scale work pieces… especially at the architectural scale.
- Proposition 25 demonstrates how to locate the center of a circular arc.
- Proposition 29 demonstrates how to bisect a given circular arc.
- While not in the elements, there are techniques that you can use to both approximately and exactly lay out elliptical curves. These techniques fall outside off Euclidean geometry, but they are extremely useful in adding curved features to your designs.

- Book IV introduces the concepts of inscription and circumscription. It also introduces some constructable, regular polygons.
- Definitions 1-7 establish the meaning of inscription and circumscription relative to circles and polygons.
- Proposition 1 establishes a technique for fitting a chord of a given length to a circle.
- Propositions 2-5 are all related to inscription and circumscription of circles and triangles relative to each other.
- Propositions 6-9 are all related to inscription and circumscription of circles and squares relative to each other.
- Propositions 10-14 are all related to the construction of a regular pentagon.
- Proposition 15 describes how to construct a regular hexagon.
- There is not a proposition in the elements for laying out regular octagons. It is useful to create our own so that we can lay out structures with octagonal cross sections (such as chair or table legs). This proposition can also lead to a technique for building a template that helps us to accurately lay out the facets of an octagonal leg, but we’ll need concepts from books V and VI in order to demonstrate why this template will always work.

- Book V introduces the concepts of ratio and proportional relationships. These concepts are fundamental to the techniques we use in pre-industrial design. However, most of what appears in this book is somewhat abstract and sets the framework for concepts like
*similarity*. Therefore, other than the definitions that appear at the beginning of Book V, we’ll use its concepts only indirectly.- Definitions 1-6 all work to establish the meaning of the words
*ratio*and*proportion*in the sense that we use them.

- Definitions 1-6 all work to establish the meaning of the words
- Book VI is the last book of the Elements that we will work with while building the foundations for pre-industrial design techniques. Our main interest in this book is that it establishes the concept of similarity and provides us with the similarity tests for triangles. This can be very useful in scaling our designs up or down in size and it also lays the foundation for how one of our design tools, the sector, actually works.
- Definitions 1-4 establish the meaning of similarity and related concepts.
- Propositions 1-8 establish the interplay between proportionality and similarity facts for triangles.
- Propositions 9 and 10 justify the techniques that we use for cutting a segment into proportional pieces.

Nearly two years ago, after reading “By Hand and Eye” by George Walker and Jim Tolpin, I had the opportunity to take a course by the same name (taught by Jim at the Port Townsend School of Woodworking… if you are a woodworker and have never been there, I highly recommend signing up for one of the classes they offer. They’re first rate). It is almost cliche to say this, but the experience was transformative. My designs have evolved more quickly. They are better both aesthetically and structurally. As a result, I believe my woodworking has improved as well.

Still, throughout this experience, I’ve often found myself thinking that there are things I should contribute to what can only be described as a revival of interest in design. I make my living as a mathematician but keep some measure of daily composure by retreating into my shop in order to make things from wood. As such, I know an awful lot about geometry. I’d like to share some of what I know (and some of what I can teach myself on the fly) through an ongoing series of articles on this blog.

I hope to keep the articles somewhat grounded somewhere more concrete than the Euclidean plane by relating many of them to the drafting table pictured above. I designed and built it in the summer of 2015. Since I immersed myself in the design techniques described in “By Hand and Eye,” the table seems like a good device for illustrating some of the geometric ideas I hope to explore. At the moment (and this could very well evolve as I go), I envision organizing this sequence of articles into at least three distinct chapters:

**Chapter 1** will probably be the most familiar. I’ll walk through the design process I followed in order to create the plan and elevation drawings (and a story stick) I used while building the table. However, I will give a fair amount of attention to the fundamental concepts of geometry in the Euclidean plane that make the techniques of proportion based design work so reliably. There will be math and logic involved here, but it will be math and logic with a context that should be meaningful to woodworkers and designers. In addition to laying a reasonable (but not overbearing) geometric foundation for the techniques of design I used on my table, I plan to explore the difference between exact and approximate layout methods. As an example, consider the common design task of dividing a distance into some number of equally sized pieces. There is a well known Euclidean construction that will perform this division exacly (at least in principle). However, many designers will frequently use a set of dividers or a sector (or both) to accomplish this task. The latter approach is by nature an approximation. Which approach is better? Which is more efficient? Are the answers to these questions the same in all circumstances? Some? Are they the same for all designers? Also, I haven’t decide upon this yet completely, but I suspect I will put some time into the topic of creating isometric drawings in this chapter.

**Chapter 2** will be a first step into uncharted territory for me from the perspective of design (less uncharted from the perspective of geometry). As it turns out, I’m pretty proud of my drafting table. I’m rather happy with the way it turned out. I’ve thought about the idea of writing an article about its construction. In addition to my belief that it is a neat table, there are some nonstandard design features hidden in it that I think others would find useful. Unfortunately, I did not take enough “in process” photos of the table while I was building it. I was rather focused on the building and less focused on the documenting. In addition, I suffer from a measure of ineptitude when it comes to using a camera. Even the pictures of the finished table end up looking less like what I’d like to see if I were going to read an article about it in a magazine or online. However, something I *can* do is draw in proper perspective. There is a rather rich field of geometry that forms the basis for doing this (projective geometry), but depending on your purpose for constructing a perspective drawing, you might need to know anything ranging from none of its fundamentals to a great deal of them. Chapter 2 will begin at the simple end and explore drawing in proper perspective the way artists do: I will walk through the process of drawing an object (such as my table) in perspective when I have the object in front of me (ugh… I’m going to need another table for that). This artists’ approach is very intuitive and can be successfully undertaken without much knowledge of geometry at all (certainly without any real projective geometry). It is an approach that is particularly useful when you want to create a hand drawn representation of an object that already exists. To me, an obvious reason to do this is if you want to publish a representative illustration of your work in place of a photograph for a book or article. Perspective drawings look more natural than our plan and elevation view shop drawings. They provide the viewer with a better understanding of how our work looks when a high quality photograph is not available.

**Chapter 3** will dive much deeper into projective geometry and perspective drawing. Imagine you have produced a set of plan and elevation view shop drawings for a proposed piece of furniture. Before building it for your client (or even for yourself), you might be wise to get a more accurate representation of how it will look after construction. The artists’ approach will not work here. You do not have the object itself to base your perspective drawing upon. There is no physical object for you to view in perspective yet! All you have is a geometric model for the object (in the form of your plan and elevation view drawings). The problem before you is this: How can you transform a set of elevation and plan drawings into a single image that is drawn in proper perspective? We’ll need at least a rudimentary understanding of projective geometry (and the projective plane) in order to solve this problem. Projective geometry is a rather vast subject, so I’ll provide the focus upon the parts we need.

**Chapter 4**, if I ever get there, will address what I would call an inverse problem. Suppose you are interested in building an accurate reproduction of an antique piece of furniture, but the curator of the collection in which it resides forbids you from approaching it or touching it with your measurement devices. You can take pictures from various vantages, measure the distance from your camera to the piece, etc. but that’s about it. Your photographs can be thought of as perspective images of the piece of furniture. Do you have enough information to deduce a set of plan and elevation views of the piece? If so, how? If not, how much more information do you need? What are some strategies for obtaining it without running the risk of enraging the curator? We’ll need an even deeper understanding of projective geometry and the projective plane in order answer these questions, but the effort could be worth it in orrder to gain an accurately proportioned set of drawings of the piece you want to reproduce

As I’ve said, I have a day job. Because of this, I suspect what I’ve described here will be a long term project. At first, I’d like to try to stick to a schedule of an article a week (perhaps releasing them each weekend), but I can almost guarantee I will deviate from that schedule from time to time. I’ll make a serious effort to publish them frequently enough that anyone interested in reading them will be able to maintain focus/interest. I’ll file all of the articles under the Geometry and Design category so that they can be found easily. I have also placed a menu item on the top of this blog that will take you directly to the Geometry and Design articles. Comments and feedback are welcome. Hopefully we can all learn something from this effort.

]]>I originally intended to build a replica of the “swinging saw vise” featured on The Woodwright’s shop a season or so ago, but I decided that even though I’ve allowed my handtool operation to spill into the livingroom of my house, I couldn’t really justify the space (and with time flying by, I couldn’t really justify the added complexity either). Instead, I designed a smaller vise that is held in place by my leg vise and clamps (for now) with a pair of hex bolts and wingnuts. I will probably still throw a coat of finish on it, but it is basically ready to go to work now.

Things I would do differently if I were to build a second one:

- I would move the vertical posts closer to the center to allow the biggest handsaws to be clamped along the full length of their toothline. As it stands, the left post interferes with the tote on my D-8, so I will be filing the last for our five teeth unsupported. Oops.
- I might make the posts longer and leave some extra material at the bottom where the hinges attach. I believe this would help create beter clamping pressure on the sawplate. I’m not sure if this is a problem yet. The vise seems to hold the plate securely, and it is unlikely I will be bearing down on a file so hard that the plate would shift, but I won’t know how well things perform until I start sharpening. Luckily (I guess), I have six saws or so that need work, so I’ll find out soon.
- I would consider making the vertical posts a little wider. I was thinking of using the Moxon vise hardware from Tools for Working Wood instead of the (much cheaper) hex bolts and wingnuts. I know this would give me better clamping pressure, but I worried that half inch holes might be a bit much for the narrow posts (my hex bolts are 3/8″).
- I’d use something stiffer than alder for the posts. It’s pretty soft and flexible. However, I had enough scrap to use on this project (and not really enough for something more substantial), so here we are. At least it’s pretty. The hard maple jaws will certainly hold up to some abuse.
- Even though I would make a smaller vise for sharpening backsaws (especially my dovetail and carcass saws, which have too shallow of a plate to be accomodated by jaws as large as these), I would probably cut some recesses on the insides of the vertical posts. Without doing this, this vise will not accommodate the back of a backsaw. This would be an easy enough modification for me to make to this vise, and I still might do it. I could then use this vise to sharpen my larger tenon saw.

Lately, I’ve been burning through some leftover scrap by making some layout tools and bench caddies. These include a long straight edge and two sets of long and short winding sticks made from padauk and a bow for laying out curves made from maple and rawhide. The bench caddies are both made from hard maple and white oak. One of them will eventually find its way to my father’s shop in Colorado. Also pictured are a carving axe made by John Switzer at black bear forge in the foreground (It saw a fair bit of use in roughing out the curve on the straight edge) and my recently completed resaw beast in the background. I’m hoping make a pile of sawdust with the saw soon.

Click to view slideshow. ]]>This side table is well sized to stand in your entryway or to serve as an accent piece in a sitting room. I’ve constructed it from figured, hard maple with a walnut drawer front. The secondary interior wood is clear, Eastern white pine. I’ve finished it with several coats of high quality spar varnish.

If you would like to own this piece, you can contact me at Eric.Wright@planbworkshop.com. The purchase price is $400.00 US for local pickup. If shipping is something you’d like to have arranged, we can discuss that on an individual basis.

Click to view slideshow. ]]>I never got around to posting pictures of my tool chest after completing it last spring. Here they are now.

Click to view slideshow. ]]>I’ve been building a new bench on and off since the spring of 2014. While I still plan to install a shelf inside of the leg stretchers, a deadman, and a planing stop, the bench is now basically complete and functional for the work that is at the top of my queue. I estimate that the bench weighs between 500 and 600 pounds. It’s pretty solid. The wooden screws are both by Lake Erie Toolworks and the holdfasts are by John Switzer of Black Bear Forge.

I’ve also decided to convert my living room to a hand tool woodworking shop. For now this just means my bench and toolchest are the two nicest pieces of furniture in my house and that I have moved the television to a much less used room. My machines still live in my carriage house and I still use them for heavy stock preparation.

Click to view slideshow. ]]>Everybody needs a tea chest to keep their tea organized. I built one from white oak and Douglas fir as a Christmas gift for my parents in 2012. This one is sized to hold three boxes of Tazo tea, but I have found that other brands fit as well. In any case, the design can be modified as needed. I plan to make more. Some might be for sale.

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A few years ago, I built a small display table from a few boards of Khaya (a species of African “mahogany” that isn’t really a true mahogany, but still has a pretty grain). This table is one of the few pieces of furniture that I have built and am still in possession of. It stands about 31 inches high and has a top that measures 20 1/2 inches wide by 10 1/4 inches deep. Currently, it lives in the entryway of my house with a small bowl I turned from purpleheart several years ago and a wooden Buddha that I cannot claim responsibility for.

If you would like to own this piece, you can contact me at Eric.Wright@planbworkshop.com. There is some minor wear on the top of the table. The purchase price is $250.00 US for local pickup. If shipping is something you’d like to have arranged, we can discuss that on an individual basis.

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