Wednesday, January 8, 2020

Practical Geometry - Drawing a Square with a Compass, Part 2

 4 more ways to draw a square with a compass.

For Part 1 see:

How to draw a square with a compass  #3
Peter Nicholson wrote about Practical Geometry in 1793.  His first plates are introductions to the first rules of geometry: using a compass to bisect a line,

My blog post about him is :
It includes images of Plate 2 and Plate 3.

Here I have copied just the image of a square. Nicholson includes instructions for finding the square 'abcd' by dividing the arc a-e (the black spot) in half then adding that half to a-e and b-e find d and c.

Asher Benjamin and Owen Biddle in their pattern books copy Nicholson.
They do change the order of the letters which makes the steps easier to follow: a and b are 2 corners of the square. The arcs of a and b create c. Half of arc a-c is d. Add the length b-c to the arcs of a-c and b-c to find e and f: the square has its 4 corners.

How to draw a square with a compass, #4

A 3/4/5 triangle always has a right angle (90*) where the lengths 3 and 4 meet.
2 3/4/5 triangles are a rectangle which is 3/4 of a square.
I have drawn this on graph paper for clarity.

When carpenter squares became widely available and accurate, the square corners were easy to establish. The compass was only needed to lay out the length.

Before that - before about 1830 - the carpenter could have laid out his square like this:

His length is laid out in 4 units.
He knows approximately where the 2 sides will be. He does not know if his angle is 90*.

Here I have drawn the arc of the length of 4 units - on the right side. Then the arc of 5 units with its center at 3 units  on the left side. where they meet will be the 3/4/5 triangle.

The carpenter did not need to layout the full arcs as I have drawn them.
If he held his Line at the right lengths he could have marked a bit of both arcs where he thought they cross, and then placed a peg where they did cross. He would have checked his square by matching diagonals.

The relationship between the 3/4/5 triangle and the square is good to recognize. However, the 3/4/5 triangle is usually the only geometry. Layout by a carpenter square, widely available in the 1840's, was simpler and took less training than using a compass.

This small, simple house, built c. 1840 for a cobbler, was probably laid out using a carpenter square. I've tried other geometries which almost fit. The 3/4/5 triangle does.

I wrote the original post in 2014. It's time to revisit and review.
Here's the link to the post:

How to draw a square with a compass, #5

 Lay out a perpendicular through a line. Draw a circle with its center where the lines cross.
Draw lines - here dash/dot lines - between the points where the circle crosses the lines.

This square, as a diamond, was often used by finish carpenters because it easily evolves into more complex layouts. 

Below is the entrance porch for Gunston Hall, designed by William Buckland, c. 1761. The rotated squares determine the size of the porch. They also locate the floor, the pediment, the roof pitch, the size of the arch, the height of the rail.
 My post on Gunston Hall is:

Here the glass facade of
 the Mass. Ave. entrance to MIT. For more, see:

 How to draw a square with a compass, #6

 On a line select a length - see the dots .
Using the length as the radius draw a circle using one dot as the center.
Now there are 3 dots. Draw 3 circles using all 3 dots as centers.
Drop a perpendicular line at the first circle's center.
Now there are 2 new dots for centers of more circles.
Connect the petals where the 4 circles cross.
A square.

This modest farm house, c. 1840, used the square crossed as the squares above are for the Gunston Hall porch.

One last note: the circle to square diagram #6 can also become the diagram for #5. 

Each master builder probably had his preferred way of using his compass, even when he practiced within a tradition.
Still, just as a 3/4/5 triangle is part of a square, these diagrams are also simply different choices, different perceptions of the same geometry.