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    Re: Angles without a protractor
    From: Örjan Sandström
    Date: 2012 Oct 12, 11:41 -0700

    I posted an explanation while ago how to divide a circle into 3 degree segments accurate to level you can measure.
    usually this translates to, at best 0.1 degree accuracy, with a good steel ruler good divider and plenty of time and care. more on posting linked to below.
    http://fer3.com/arc/m2.aspx?i=119943

    the 1° div can be made with a verniere system but it is a bit of a cheat, you need perspex or similar transparent material and scribe lines from centre of that correspondig to 0, 3, 6, 9 and 12 degrees.
    first using 0° as "cardial" line and the others as "divisions" slide the 0° line out on top of and paralell to a "0" line on the scale you want to subdivide until 9° line is touching the next division on the circle you can now drill a hole where the 3 and 6 degree lines of the perspex vernier meet and thus pick out the 1 and 2° ticks, now repeat using 12° as cardinal, the difference should be small enough to require loupe to see if any, error should be shade less than 5% of distance between ticks in theory, but we all know practice and theory.....
    now you have a circle divided into single degrees.

    having done as posting linked and the above you can make a verniere covering 10 degrees slide this out same way until 9 lines of the original scale are covered by 10 lines of the vernier scale using 5° line as centre this way you could get down to half degrees accurately, no sadly not 1/10° as the lines are not paralell they just indicate centre between two div not 1/10.
    if two such verniers are added to a rotating arm (two verniers 180° appart) this arrangement can be used to see errors in individual division ticks so they can be corrected.
    further iterations (second time with scales 181°appart third 183°...) this should improve accuracy if care is taken, hard part is getting below 10' accuracy, for that lines need to be very fine and divisions extremely accurately cut.
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