Note the circles inside this bookcase which offers geometric clues to the proportions

Pre-industrial design books frequently employed squares, circles, and simple rectangles to convey the basic proportions in a design. Often these drawings show circles surrounded by squares and rectangles to help the reader quickly grasp the composition. A circle conveys that the space is equal in width and height (essentially a square) and combinations of overlapping circles easily convey a square expanding into a rectangle. The beauty of using these simple overlapping circles is that it’s easy to depict rectangles which have harmonic width to height ratios. Draw two circles where the diameters just touch the focal points and the surrounding rectangle has a ratio of 2 parts high to 3 parts wide ( 2:3 is a fifth in music).

This design uses a simple rectangle with overlapping circles to reveal a rectangle that is 2: 3

I often encourage students to draw these simple rectangles to help them visualize harmonic shapes, rectangles with ratios of 1:2, 2:3, 3:4, 3:5, and 4:5.

Let’s say you want to draw a rectangle that is 4:5 or four parts high by five wide. Historically this was called a square and one quarter square. Begin by drawing a circle then scribe a horizontal line through the center and extend it in the direction you want to expand. Then use dividers to step off the line into four equal parts inside the circle. Go back to your compass and draw an overlapping circle so the circumference of your second circle overlaps all but one quarter of the first. Surround both with a rectangle and you have a nice harmonic shape to use for the opening on a fireplace or the outline of an end table.

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Thanks for this. Glad for something I can do on the couch when my mind wants to be in the shop.

I loved this exercise when I went through it in the book. It was interesting to see how accurate “stepping off” sections (quarters, thirds, etc) of a shape can be.

If you have any real confidence in what you write and teach, my earlier comment shouldn’t be especially threatening.

I’m currently working my way through By hand and eye, and I want to say great work! I’ve been amazed at how much sense it makes and how easy it has been to incorporate in the workshop.

But I’m wondering if you could explain Figure 3.2.7 in the book, page 126 “Erecting a 45 degree angle from a point on a line”. I’ve tried it several times, with the information given there (which is kind of sparse) but I can’t get it to work accurately. I might be missing something. Also I’ve measured the angle in the figure in the book and it isn’t really 45 degrees.