Measuring Lengths and Angles: Derived and Direct Physical Measurements

In the previous posting I argued that I was queasy about the idea of claiming that the radian measure was unitless. The claim that the radian measure was unitless is based on the definition of an angle as:

Theta= S/R

where S is the arc distance associated with the given central angle of a circle of radius R. R is the radius of the circle.

The claim is that arc distance and radial distance can be measured in the same physical units. My counterargument is that there are fundamental ontological differences between curved and straight lines and these differences merit utilizing distinct dimensional units.

I would argue that the dimensional units appropriate to arc length measures derive directly from definitional equation:

S = theta x R

S = Radian x Meters


So I would measure arc length in radian-meters, rather than just meters.

Thus to the extent we are interested in measuring arc lengths (and we are not usually interested in this measure) we should utilize a derived unit measure. The distinction between derived and direct measurement bears greater scrutiny.

Speculations on Direct and Derived Measurements

In physics there are only two types of measurement: direct and derived. I suspect that there are only two direct possible physical measurements and that all other measurements are ultimately derived from these two. Under proper conditions we can construct physical devises that allow the direct measurement of length and of angles. All other measurements are ultimately derivative from these two. All measurements are also based on fundamental beliefs (axioms) about the nature of the physical world. In other words measurements are theory laden. But, almost none of these fundamental beliefs are very controversial. I will have more to say on these points in later posts. For now I want to look very closely at the physical construction of the straight edge and the protractor.

The Straight Edge

Any straight object such as a taut string or strip of cloth can be used to fashion a more durable meter or yard stick. The unit length is a matter of convention (but a very important convention). There is no natural unit of gradation. To deploy a meter or yard stick usually entails an alignment process which in turn requires approximation because precise measures with a yard stick require straight edges on the object to be measured. For objects with highly irregular shapes calipers can be used. Measurements of area and volume derive directly from the measures of length. Maintaining bars representing conventional units of length and assuring the reproduction of faithful copies has been a matter of great commercial and scientific importance.

The Protractor

Any semi-circle disc can serve as a measure of angles. A circle or a semi circle can in turn be drawn and hence used to construct a protractor using only a compass, which in turn can be a device as rudimentary as a taut string with a marker on the end. So it does not take much to construct a protractor.

Further, because we can bisect any angle with just a straight edge and a compass we can create equal gradients on a compass without resorting to the heroic efforts needed to replicate and define our conventional units of linear length such as the meter or yard. In effect the circle or semi circle is naturally divisible into equal units. For example, dividing the straight line (a.k.a. straight angle) in two, gives us 90 degrees or pi/2 radians. Bisecting this angle again gives us 45 degrees or pi/4. And we can go on bisecting to create more and more refined angle measures with simply a straight edge and a compass, neither of which needs any unit gradations of any sort.

Finally, we do not need to specify any specific sized protractor because all circles are proportional and the angle gradients constructed on one sized protractor will yield equivalent angle measures on any other sized protractor. Of course the arc lengths between gradients on one sized protractor will differ on different sized protractors. But as noted above we normally are not interested in measuring arc lengths but rather angles.