Precision in Measurement: Perfect Protractor?

Date: 10/16/2001 at 07:52:57
From: Ivan McDonagh
Subject: Marking my own protractor

Hi,

In the reply to a question regarding constructing an angle of 
one degree it was stated that an angle of one degree cannot be 
constructed using just a straight edge and a compass because 
the sine and cosine of one degree both require cube roots and 
only square roots can be constructed 
(http://mathforum.org/dr.math/problems/callanta.5.25.00.html   ).

My understanding is that only construction using "perfectly" reliable 
instruments will give "perfectly" accurate results. If we have to 
resort to measuring and calculating then there has to be (as I see it) 
a certain amount of uncertainty regarding the result.

Given that protractors are expected to be accurate to the degree and 
in some instances the minute or second how are such angles accurately 
constructed and marked?

On the subject of precision, would you please tell me how to "60-sect" 
a one-degree angle so as to generate the markings for the minutes?

Thanks for your help. Unfortunately I just don't have the maths to have 
even started on this. Once I got past bisection and trisection I 
referred to this resource and discovered that construction just can't 
be done (as I read it).

Regards,
Ivan McDonagh
(Perth, Western Australia)

Date: 10/16/2001 at 09:01:57
From: Doctor Peterson
Subject: Re: Marking my own protractor

Hi, Ivan.

You are right that it is impossible to construct all the angles you 
need with compass and straightedge. But that isn't really necessary.

First, there are other tools available to do constructions; the 
restriction to compass and straightedge is just part of the  ancient 
Greek game of geometry, due to their desire to reduce all truth to the 
fewest possible starting points, or axioms. There are many ways to 
trisect an angle, starting with Archimedes' construction using a 
MARKED straightedge. See the Dr. Math FAQ:

   Impossible Constructions
   http://mathforum.org/dr.math/faq/faq.impossible.construct.html   

Second, a perfect construction is not necessary, or even useful, in 
making a protractor. The kind of perfection we talk about here is only 
a theoretical perfection: if we had a perfect straightedge, a pencil 
that actually draws LINES (with no width!), and so on, then we could 
prove that the result would be exact. But often such constructions, 
done with real tools, are actually LESS accurate than just measuring 
and estimating. That's because the construction can get very 
complicated and introduce errors repeatedly whenever we have to draw a 
line through a non-quite-perfectly defined "point."

One very reasonable way to mark a protractor would be to make a 
special tool with a 60:1 gear ratio, so that for every 60 degree turn 
of the handle, the marker would move one degree. Having marked 
degrees, if you are making a protractor large enough to show minutes, 
you could use the same tool to divide each degree into 60 parts. But 
usually we don't get that kind of precision. The marks on a typical 
protractor are not much less than a degree wide, so greater precision 
would be useless!

In summary: outside of a mathematician's mind, there are no "perfectly 
reliable instruments," nor is there a need for (or the possibility of) 
"perfectly accurate results." There is always uncertainty in the real 
world. So use whatever kind of measuring or calculating you wish. One 
reasonable way would be to use basic trigonometry to construct 
triangles with, say, 1- and 10-degree angles. (The larger one would 
allow you to construct the major divisions without having to build up 
large angles from many small ones, accumulating errors as you go.) You 
might construct 30- and 60-degree angles with a compass, since they 
are very simple and reliable, then fill in every 10 degrees, and then 
fill in the degrees between.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/