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/
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