At first i thought &quot;aha! yes, how obvious.&quot;, 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&#x27;s a nontrivial fact that i took for granted, but that&#x27;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.<p>i would like an example of an experiment that directly measures a dimensionless quantity without inferring it from direct measurements of dimensional quantities. it&#x27;s just a curiosity.