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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?

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I believe the distinction the first paragraph draws between angles as magnitudes and the measurement of angles is largely artificial.

I would agree that there's a difference if it weren't for the special angle of $90^\circ$. Without that, while you could say that an angle is twice, thrice, half, one-third, ... as large as another angle, you wouldn't be able to measure it, because you wouldn't have a fixed frame of reference available. But once you fix one particular angle as your frame of reference, you can say that one particular angle is twice, half, ... as large as that particular reference angle, and for all pratical intents and purposes your magnitude becomes measurable.

The reference to radians or degrees seems poorly thought out to me. Whether or not those particular units where used at a certain time or not doesn't really matter. As long as there's some unit is available - and $90^\circ$ certainly is one - every comparison of some angle to that particular angle then become a measurement. So unless Euclid never speak of any angle being twice or half or one-third of a rectangular angle, I'd argue that he does measure angles, despit the fact that he doesn't use the units we're used to.

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A measurement is an empirical estimate of a magnitude. Not all magnitudes are empirical. If we declare an angle to be right, then it has a magnitude, $\pi/2$. But that isn't a measurement; it may be an axiom in whatever geometric argument we are making that the angle is right.

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You defined measure in terms of magnitude, can you define magnitude as well? By "Not all magnitudes are empirical", do you mean that not all magnitudes have related empirical estimates? can you give an example? You say that 'right' and 'pi/2' are magnitudes, can you justify that? you also say that 'that isn't a measurement". Why? and what would be a measurement? – bluemoon May 8 '13 at 8:51
A magnitude is a real number, and a measurement is an empirically obtained rational number in the same domain. We can justify that $\pi/2$ radians is a magnitude because it is a real number. A right angle's magnitude isn't a measurement if the right angle is actually a perfect idealization from geometry, and not some approximation to a right angle being constructed or measured. An abstract right angle's magnitude in radians is $\pi/2$ where $\pi$ is the actual transcendental real number, and not a rational approximation like 3.1416. – Kaz May 8 '13 at 19:34
I do not mean to drag this, but I find the subtleties and confusion in this topic both interesting and unsettling. Your conception is at odds with both Joyce's and fgp's. Are you identifying measurement with a physical measurement made by humans then translated to a number using a finite process? Thus constraining it to be rational? Is this a standard definition? – bluemoon May 9 '13 at 8:41

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