To continue our “Tools of the Trade” series, we’ll be discussing rules, dividers, compasses, geometry, and their importance to 18th-century construction and design. While today we tend not to think of how geometry and proportion affect our lives, for 18th-century artisans it was the framework of their craft. As Richard Neve explained in his City and Country Purchaser “GEOMETRY [is] indispensably requisite; for it is this that must furnish the Architect with some sure and certain principles, whereon to proceed in the practice of his art.”  But before we learn how to practice this art, we should become acquainted with the tools and terminology required.
First, I’ll be using the term “proposition” to describe any geometrical theory which can be proven by drawing it out. There are really only three tools you need to do this: a rule, a compass, and some form of fine marking tool (I’ll be using pencils–make sure they’re sharp, though!).
The Builder’s Dictionary of 1734 defines a rule as “a simple instrument, ordinarily of hardwood, thin, narrow and strait, serving to draw lines with.” It is essentially a straight edge. The same book describes the compass as a “mathematical instrument used in the describing of circles.” Now that we have an understanding of the terminology, let’s begin looking at the techniques!
Geometry is one of the few parts of 18th-century woodworking which was well-documented within the period. There were a number of authors writing guides on construction and geometry for carpenters and tradesmen. These were intended to teach tradesmen the finer points of geometry as well as to introduce new building techniques, such as the king-and-queen post truss.
Today we’ll use one such book: Francis Price’s British Carpenter of 1733, a treatise which hoped to instruct both carpenters and architects in geometry on a large scale. We’ll work through some of his first plate of geometrical propositions. We’re mostly interested in subdividing a circle into eight parts, as this is how we laid out our waterwheel. I’ve simplified some of the terminology Price used in my explanation of the process so that it makes more sense to a modern reader (this also helped me make sense of what he was saying).
We’ll start with Proposition A, as this teaches us everything we need to know to divide a circle into eighths.
To Erect a perpendicular on a right line given.
Start with line a-b:
Mark a point c on the line. Place the foot of your compass at c:
Open it to your pleasure [i.e., however wide you want]. Mark e and d:
Place the foot at d and mark g.
To divide a circle into eight parts.
Now that we know how to create a perpendicular from a line, we’ll apply these concepts to dividing a circle.
First, draw a line through the center of the circle. Place the foot at point a, and extend your compass beyond the radius of the circle:
Create an arc:
Place the foot at point b. Create an arc:
Draw a line where these two arcs intersect; it should run through the center point as well:
Now your circle is divided into four equal parts. Place the foot of your compass at point b. Extend your compass most of the way to the next intersection:
Make an arc outside of the circle:
Place the compass at the next intersection point. Make an arc to intersect with the first:
Draw a line through that intersection and the center point:
Repeat this process on the opposing side:
That’s eight equal parts!
This concludes our discussion of geometric propositions in the 18th century for today, but be sure to check back here tomorrow for more information on how we applied these same concepts to the layout of our water wheel! As always, stay safe and stay healthy.
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 Neve, Richard, The City and Country Purchaser, London, 1736. Accessed 21 April, 2020 Google e-book
 Bettesworth, A., Hitch, C. and, Austen, S, The Builder’s Dictionary: Or, Gentleman and Architect’s Companion, London, 1734. Accessed 21 April, 2020 Google e-boo
Bettesworth, A., Hitch, C. and, Austen, S, The Builder’s Dictionary: Or, Gentleman and Architect’s Companion, London, 1734. Accessed 21 April, 2020 Google e-book https://books.google.com/books?id=cMQ9AAAAcAAJ&newbks=1&newbks_redir=0&dq=The+Builder%27s+Dictionary+Or+Gentleman+and+Architect%27s+Companion:&source=gbs_navlinks_s
Neve, Richard, The City and Country Purchaser, London, 1736. Accessed 21 April, 2020 Google e-book https://books.google.com/books?id=Hh9hAAAAcAAJ&newbks=1&newbks_redir=0&dq=the+city+and+country+purchaser&source=gbs_navlinks_s
Price, Francis, The British Carpenter, or, A Treatise on Carpentry, London, 1735. Accessed 21 April, 2020 Archive.org https://archive.org/details/britishcarpenter00pric/mode/2up