In an introduction to Euclid's Elements , D.E Joyce writes:
Treating angles as magnitudes should not be confused with measuring angles. The angles themselves are the magnitudes. The only measurement of angles in the Elements is in terms of right angles (defined in the next definition). Degree measurement and radian measurement were not used until later. In terms of degrees a right angle is 90°, while in terms of radians a right angle is pi/2 radians.
I think I do appreciate the difference between magnitude and measure and understand what he means, but it still feels quite fuzzy to me.
How would you explain the difference? are my thoughts below at all right? would you please improve on them?
What is a magnitude? It is a thing that we can perform arithmetic on, and compare (think ordered field). A measure is a more physical concept, the relation between a standard magnitude (of measure one per example), and another. A magnitude is more abstract, while a measure is more concrete; we measure centimeters, grams, etc... and our standard measure is totally arbitrary, but consistent with what is imposed by abstract magnitude. When it does not, the quantity involved probably stops being seen as measurable. I find it difficult to find things that could not be called measurable. Is beauty measurable? If not, is it because firstly, there is no standard magnitude like centimeter? In the sense that beauty is subjective? and secondly because even if it was, and we decided on an algorithm to measure beauty, beauty would still not add up? Per example, overlaying two paintings of beauty measure five can produce a painting of measure one, but also of measure six depending on the paintings? Is there a more subtle example than beauty?