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A fascinating question from Andy Marczewski and an even more fascinating explanation by Jane Rees. 

Andy was shown this folding ruler, thoughAndt to belong to a carpenter. It has a conventional inch scale on one edge, but on the other side has two different scales.  They are labelled "8 SQUARE"

.

and "LINE". 

Andy asked if Can anyone  could explain the mysterious scales.  

As ever Jane Rees came up with an explanation: 

Dear Andy

Your enquiry about the 8 Square line has been passed to me. These are indeed lines used by a woodworker and your rule is what is known as a carpenters’ rule.

The 8 square line, which is actually two lines is also known as the E & M line or the Mastmaker’s scales. These give the dimensions needed to set out an octagon onto a square timber. Without these aids, trigonometry is required – or just trial and error!

The one marked 8 Square on your rule is the edge (E) line (needed to set out the octagon from the edge of a square, and the one marked Line on your rule is the middle (M) line, ended to set out the octagon from the centre of a square. The lines are not complete in themselves but must be placed on the rule to allow the answer to be read off an adjacent line of measure. They are direct reading, i.e. by looking across the rule from the size of square timber on the E or M line, the layout of the octagon, either in from the edge, or out from the middle, can be read off the line of measure.

The attached diagram and photograph show use of the E & M line is used and how you read it off a rule.

Jane  

Andy has replied:

Andy Marczewski 2017-09-16 13:33

Andy M