trivia - trisecting an angle & gradians

I'd long believed that it was impossible to trisect an angle. Started googling it recently and found that thats not quite true. You can do it if you have marks on the straight edge, the original "rules of the game" were pencil paper compass and unmarked straight edge.

http://en.wikipedia.org/wiki/Angle_trisection see 1/2 way down the page or attached snippett below

Am curious - why did 360 degrees survive decimalisation ? I have seen web pages advocating a decimal degree system but it never caught on. ie 400 units defining a full circle each quadrant being 100 units and abolishing minutes and seconds just have decimal places.

I've read on the internet that old European theodolites can be marked up in 400 Gradians and that the French Military used Gradians

https://books.google.com.au/books?id...circle&f=false

Bill

I'll put my hand up

at various times I've used all 5 of the main systems of measuring angles

Degrees - minutes - seconds

Degrees - minutes and decimal minutes

radians -- as in 90° = pi/2 radians

Gradians -- 400 gradians to a circle

Mils -- where 1 mill = 1 metre at a range of 1000m


I think the 360° system has survived because of it's "simplicity"

15° = 1 hour
0°15' = 1 minute
0°1' = 1 nautical mile, which makes working out distance relatively easy.

If you moved to any other system, these relationships break down.

Trisecting an angle with pencil, compass and straight edge.

I'm way too old to remember anything from 10th year geometry. But logic prevails.

Assumption is that the angle is drawn on a piece of paper and can be extended to the edges of the paper.

1 ~ Draw a line. Use a compass to mark off three equal sections on the line.

2 ~ Stretch the compass to the full distance of the three sections. Use the compass to mark on the angle lines where the distance fits, touching both lines of the angle. Draw a line between those two points.

3 ~ Stretch the compass to the length of one of the three equal sections of the drawn line in step one.

4 ~ Use the compass to mark transfer the length of the equal sections on the line drawn between the sides of the angle.

5 ~ Use the straight edge to draw lines from the point of the angle to the marks on the line in step 4.

The angle is now trisected.

Quote:

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Originally Posted by rrich

Trisecting an angle with pencil, compass and straight edge.

I'm way too old to remember anything from 10th year geometry. But logic prevails.

Assumption is that the angle is drawn on a piece of paper and can be extended to the edges of the paper.

1 ~ Draw a line. Use a compass to mark off three equal sections on the line.

2 ~ Stretch the compass to the full distance of the three sections. Use the compass to mark on the angle lines where the distance fits, touching both lines of the angle. Draw a line between those two points.

3 ~ Stretch the compass to the length of one of the three equal sections of the drawn line in step one.

4 ~ Use the compass to mark transfer the length of the equal sections on the line drawn between the sides of the angle.

5 ~ Use the straight edge to draw lines from the point of the angle to the marks on the line in step 4.

The angle is now trisected.

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rrich,

I think I have misinterpreted what you have written, is a sketch possible ?

Bill

I don't know if this will work as I drew it using Visio 2000. (15 years out of date)

It won't so I'll have to do it with pencil and paper and scan it in.

The real beauty of 360 is the large quantity of numbers it is exactly divisible by;
2,3,4,5,6,8,9,10,12 to name but a few.

Just to confuse things...in oil and gas drilling they use a decimalized foot. That is 10 inches to a foot. A drill pipe would be 40.7ft long.

Quote:

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Originally Posted by ian

I think the 360° system has survived because of it's "simplicity"

15° = 1 hour
0°15' = 1 minute
0°1' = 1 nautical mile, which makes working out distance relatively easy.

If you moved to any other system, these relationships break down.

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the relationship to nautical miles is only true for longitude at the equator

Quote:

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Originally Posted by Dion N

the relationship to nautical miles is only true for longitude at the equator

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true, but I'm sure there's a [relatively simple] function -- which I can't recall -- that adjusts for latitude

Quote:

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Originally Posted by ian

I'll put my hand up

at various times I've used all 5 of the main systems of measuring angles

Degrees - minutes - seconds

Degrees - minutes and decimal minutes

radians -- as in 90° = pi/2 radians

Gradians -- 400 gradians to a circle

Mils -- where 1 mill = 1 metre at a range of 1000m


I think the 360° system has survived because of it's "simplicity"

15° = 1 hour
0°15' = 1 minute
0°1' = 1 nautical mile, which makes working out distance relatively easy.

If you moved to any other system, these relationships break down.

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Correct - the direct relationship between time and angle measurement is why degrees have survived decimalization. Celestial navigation, geodesy, astronomy, spherical trigonometry and computations are far simpler using the derived unit / accepted unit of degree. The wikipedia article is also a little misleading as the degree is not only a measure of "plane angles" it is also a measurement of "spherical angles". To add a little more confusion the sum of the spherical angles in a spherical triangle always exceed 180 degrees.

Here is the sketch.

To me it looks trisected. I can't remember the theorem from Geometry but that was way back and long before JC was a choir boy.

Attachment 346044

if I read your diagram correctly, the line you have trisected forms a right angle with one side of the angle.

From trigonometry, the tan of an angle is the opposite over adjacent, so for the sketch you provided, and assuming (for convenience) that the angle to be trisected is 30°, then
Tan(30°) should equal 3 x tan (10°) = 3 x 0.1763 = 0.5289, but tan(30°) = 0.5774


The only way I can see to do a trisection, is by trial and error
Draw an arc across the angle, then by repeated trials find the compass setting that divides the arc into three

Quote:

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Originally Posted by rrich

Here is the sketch.

To me it looks trisected. I can't remember the theorem from Geometry but that was way back and long before JC was a choir boy.

Attachment 346044

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Rrich,

Thanks for taking the time to post the sketch explaining your method.

Bill

Shown here are a few methods for trisecting a line, and they are reasonably neat:
http://www41.homepage.villanova.edu/...ing%20segment/
(press the "play" button on each of the diagrams)

The last one has a very oblique angle of intersection between the circle and the straight line, and so too much error would be introduced (even just the pencil lead thickness changing from drawing the first few lines would introduce error).

But that won't allow us to trisect an angle (I don't think). To trisect an angle you'd have to trisect an arc of a circle first.

Quote:

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Originally Posted by FenceFurniture

To trisect an angle you'd have to trisect an arc of a circle first.

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:whs: