Bisecting (and trisecting) lines

Bisecting (and trisecting) lines

Postby illuminatedwax » Fri May 09, 2014 11:24 pm

In order to make an accurate ruler, map, or blueprint, it will be useful to find the point on a line that is exactly in the middle of two other points. This is called bisection, and you can also trisect a line as well.

First, you'll have to make a drawing compass, which should not be difficult compared to some of the other mechanical tasks in the book. Basically you need two metal pieces, one with a very pointy end, the other with something that can hold a pencil or other marking device. They're connected by a hinge or some other method to allow you to change the distance between the two, and a fastener to allow you to keep it in place:

Image

Then you'll need a straightedge, which you can construct with the help of pulling a piece of thread taut.

Then, let's assume you have a line and two points on that line. You can find the midpoint of the two points by setting your compass to any distance that's longer than the midpoint will be. Then you take it, place the pointy end on one of your endpoints, and draw a circle around it. Then you keep your compass the exact same length, put the pointy side on the other endpoint, and do the same thing. Then you take your straightedge and draw a line between the points where the circles intersect.

Image

(The circles above let you find the midpoint E between points A and B.)

To trisect a line, draw your two circles. We'll call them circles 1 and 2. Then, draw a line between where the circles intersect at the top to the first endpoint (points A and F in that diagram.) Then draw a line where the last line you drew intersects circle 1 below the line. Then draw a line from that point all the way to where circle 2 intersects your original line. Then draw a line from where the circles intersect at the top to the place the last line you drew intersected circle 2. That point on your line is exactly 1/3 of the way from point A to point B.

You can use similar techniques to bisect angles as well. Trisection of angles is impossible except with another tool. To do that, you'll need a neusis -- basically a ruler that allows you to rotate it around a fixed point, e.g. sticking a pin in a paper ruler -- or a string and a cylinder, or a tomahawk.

http://en.wikipedia.org/wiki/Bisection
http://www41.homepage.villanova.edu/rob ... 20segment/
http://en.wikipedia.org/wiki/Angle_trisection
http://en.wikipedia.org/wiki/Neusis_construction

(PS the reason for dividing the hour into 12 parts is partly due to the fact that 12 is divisible by 3 and 4. A lot of ancient number choices are derived from the fact that they are divisible by a lot of things. 60 is great because it's divisible by 2, 3, and 5.)
illuminatedwax
 
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