A Designer’s Guide to Euclid’s Elements

Introduction

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.

Recipes from Pre-Industrial Design

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.

Euclid’s Elements through a Designer’s Lens

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.
  • 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.

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