Reply to Comment on 'Angles in the SI: a detailed proposal for solving the problem' Reply

The comment by Leonard (2022 Metrologia 59 038001) primarily proposes that if angle is treated as a base quantity, with the radian as its base unit, it would be wrong to change the units for torque (from N m to J rad(-1)), angular momentum (from J s to J s rad(-1)) and moment of inertia (from kg m(2) to kg m(2) rad(-2)), as was proposed in the letter being commented on (Quincey 2021 Metrologia 58 053002). This reply clarifies the situation by looking directly at the consequences of the two proposals. Apart from the comfort of retaining the familiar units for these quantities, the benefit of Leonard's proposal would be the preservation of a few favoured equations used in specific situations, while the general relationships between many physical quantities would need to change. The revised units proposed in the letter would leave all the established general relationships unchanged, and are the best option for allowing the longstanding problem of angles being wrongly treated as numbers within the SI to be resolved. This reply includes some historical context, which describes how Euler implicitly introduced the idea that 'the radian is another name for the number one' into the mathematics used for rotating objects, at a time long before anyone had thought about unit systems.

Automated summary

Leonard's comment argues against changing the units for torque, angular momentum, and moment of inertia if angle is considered a base quantity with radians as its base unit. The proposed revised units would maintain established general relationships and solve the problem of angles being treated as numbers within the SI.