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If you look at some astronomical instruments of the mid-1700s, you will often 
find that they have been engraved with two scales,
each with its own vernier.

One scale has the familiar 90degrees to the right-angle. The other has 96 
units! Why on earth 96? I hear you ask.

It was to allow really accurate division of the scale. Scales were divided by 
hand in those days, not by machine. The work was done
by an expert artisan, using a beam-compass.

The job was done like this, for a quadrant (quarter of a circle).

It's easy to mark off a 60-degree arc around a circle, by just marking a chord 
across it that's exactly equal to the radius. The
resulting triangle is equilateral, so its angles are exactly 60 degrees. 
School geometry stuff, that many will remember.

Well, if you mark off a 60 degree arc on a circle that's marked with 96 units 
(not 90 degrees) to the right angle, that 60 degree
arc will correspond to exactly 64 units, and there will be an arc of exactly 
32 units left over to make up the rest of the quadrant.

The next thing that it's easy to do precisely is to bisect an angle, by 
striking arcs from either end of a chord. And (of course) 64
and 32 are numbers that simply call out to be bisected, again and again.

In that way, the 96-unit arc could be precisely divided. On the other hand, a 
90-degree arc could be divided into 60 and 30 degrees,
but quickly you find yourself having to divide an angle by three or by five to 
get to degrees; a much more intractable operation.

Once you had that precise scale of 96 units, with its vernier, you could 
easily work out what a scale value, in units, each degree
marking should correspond to, and then transfer it across.

Attempts were made to persuade astronomers to measure, and make predictions, 
measured in those 96ths units, but I don't think it
ever caught on.

There's a good account, that I've borrowed from, of many aspects of scale 
division in- "Dividing the circle" by Allan Chapman.
Second edition was published by Wiley in 1995. It's a really down-to-earth 
account, from a practical viewpoint, which I like. Many
studies of early instruments concern themselves with stylistic matters that 
involve art-history more than technology.

George.

==============

contact George Huxtable at george@huxtable.u-net.com
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

   

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