How does a "Line of chords" work?

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Request for Question Clarification by efn-ga on 04 Nov 2005 00:06 PST

Could you explain your question more?  Are you asking about the math
that makes the tool known as a "line of chords rule" work to calculate
the size of an angle, or are you asking how to use such a tool for
this purpose?  Do you know any trigonometry?

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Clarification of Question by kabaj-ga on 04 Nov 2005 19:50 PST

To clarify my question I wanted to know exactly how the "line of
chords" is used to measure angles. In other words what is the function
of Line of chords?
Thanks Derek Johnson

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Request for Question Clarification by efn-ga on 04 Nov 2005 22:32 PST

Sorry, I'm still not sure I understand your question.  Are you asking
about the use of a line of chords rule, a device with two ruler-like
arms connected by a pivot at one end?  I did a search and most of the
references to "line of chords" I found that were not in a musical
context were to a device like this, but I am not sure that this is
what you are asking about.

I've found a picture and also some directions from an 1870 book for
creating a certain angle. I think that these can be reversed to
measure an angle.

The site

http://www.uvm.edu/~dahammon/museum/WoodSector.html

has pictures of a sector, which can be used to create or measure
angles. "After 300 years of use, the sector fell out of favor in the
mid 1800's. The sector is engraved with scales of various mathematical
functions and provides similar information as that of a slide rule."
One of the scales is a "line of chords."

The site

http://www.sliderules.clara.net/a-to-z/sector/sector.htm

gives directions for creating a specific angle less than 60 degrees.
Scroll down to the section titled "Line of chords." (For angles
greater than 60 degrees, two or three smaller angles are created, that
is, to get an angle of 138 degrees, first an angle of 46 degrees is
created, then a second 46 degrees measured off from the first, then a
third 46 degree angle added. Other directions are provided for angles
less than 5 degrees.)

The directions state: "make the transverse distance of 60 and 60 equal
to the length of the radius of the circle, and with that opening
describe the arc BC." Without having the instrument, I have to guess
at the meaning of this. (The transverse distance is defined just above
the first figure on the left near the top of the site.) I visualize
the two legs of the sector are opened until the transverse distance is
at the two 60 marks is also 60, but I don't really know if that is
true. This transverse distance is then used to as the radius to create
an arc. After an arc is generated this way (shown at the left in the
"Line of chords" section), a distance (B to C) is marked on the arc.
This distance was chosen by a distance corresponding to 46 (to get an
angle of 46 degrees). Lastly, radii AB and AC are drawn in.

To measure an angle, the distance corresponding to the "60 and 60"
would be marked off, then the distance between those two marks (as
shown on the "Line of chords" scale) would give the angle.

If you have a Line of Chords instrument, you can probably figure out
the "60 and 60" meaning better than my estimation.

A Line of Chords instrument is still sold today, but I didn't find
directions for it. It is shown at

http://www.tooled-up.com/Product.asp?PID=10241

but there is not enough detail for me to know exactly how to get a "60 and 60."