3 ms·

At first i thought "aha! yes, how obvious.", but then i thought about how one might measure an angle. i would use a protractor. but how is a protractor calibrat

At first i thought "aha! yes, how obvious.", but then i thought about how one might measure an angle. i would use a protractor. but how is a protractor calibrated? surely the angles determine a relationship between commensurable quantities, and a protractor is calibrated by performing measurements on lengths held at angles and measuring the observed relationship. can one measure an angle directly without performing an experiment on commensurate lengths? a protractor only works because a measurement of an angle can be transported and remain the same (to an approximation). that's a nontrivial fact that i took for granted, but that's really what a protractor does: it is the stored result of a bunch of experiments to determine angles. those experiments must have been done using observations of dimensional quantities. i would like an example of an experiment that directly measures a dimensionless quantity without inferring it from direct measurements of dimensional quantities. it's just a curiosity.

kazinator 11y ago
However, we can use angles in such a way that we never have any estimate that correlates with the size of the rectangle whose aspect ratio we are measuring. Suppose some rectangle is visible at a distance. We can use parallax angles from our vantage point to estimate its aspect ratio. We have no idea how long or wide that rectangle is because we don't know how far away it is. (Indeed, we don't even know whether it is a rectangle, or whether it is some quadrilateral that appears as a rectangle under perspective.)
- danharaj 11y ago
  I think you're saying something a little too strong. I think such a measurement does provide some information about the dimensions of the quadrilateral being measured. I think it would be possible with a bounded number of observations to measure the lengths purely by measuring the angles. You would have to shift your perspective each time, but it shouldn't take too many. So, each such measurement must contribute a non-zero amount of information about the dimensions. Either that, or i'm misinterpreting your first sentence.

kazinator 11y ago

However, we can use angles in such a way that we never have any estimate that correlates with the size of the rectangle whose aspect ratio we are measuring. Suppose some rectangle is visible at a distance. We can use parallax angles from our vantage point to estimate its aspect ratio. We have no idea how long or wide that rectangle is because we don't know how far away it is. (Indeed, we don't even know whether it is a rectangle, or whether it is some quadrilateral that appears as a rectangle under perspective.)

danharaj 11y ago
I think you're saying something a little too strong. I think such a measurement does provide some information about the dimensions of the quadrilateral being measured. I think it would be possible with a bounded number of observations to measure the lengths purely by measuring the angles. You would have to shift your perspective each time, but it shouldn't take too many. So, each such measurement must contribute a non-zero amount of information about the dimensions. Either that, or i'm misinterpreting your first sentence.