At that time the use of radians was still uncommon in physics.
The frequency of periodic events was measured at that time in cycles per second. The unit of frequency "cycle per second" has been renamed "hertz" much later. The renaming was first proposed in 1935, but it became adopted in SI only in 1960.
Because the use of cycles per second (now hertz) is entrenched, when the radian has been promoted as the preferred unit for plane angle, that has caused a split in the SI system of physical quantities, between the frequency of periodic phenomena measured in cycles per second and the angular velocity measured in radians per second.
This split is a big cause of inconsistency in SI, because "frequency of periodic phenomena" and "angular velocity" are just 2 names for the same physical quantity and the "hertz"/"cycle per second" and the "radian per second" do not belong in the same consistent system of units, the former corresponds to the choice of the cycle as the unit of plane angle, while the latter corresponds to the choice of the radian as the unit of plane angle.
Thus the SI system of units is a mixture of units from 2 distinct systems of consistent units. There are a number of physical quantities in SI for which 2 units are used, one derived from the cycle and one derived from the radian. In some cases distinct names are used to show the intended unit, like "frequency" and "angular velocity", while in other cases there are no distinct names. The non-SI unit of angular velocity, "rpm", i.e. "rotations per minute", is another name for "cycles per minute".
All the units of physical quantities related to rotations, including the angular momentum, depend on the unit chosen for the plane angle, so they must be multiplied or divided by conversion factors (i.e. 2 times pi for conversion between cycle and radian), whenever the unit for plane angle is changed.
The use of radian is a cause of confusion for those who learn about the physics of rotation, because they learn for instance that the angular momentum and the torque are proportional with the radius to the center of rotation.
This is false and it appears to be true only when the radian is chosen as the unit of plane angle. In reality, the factor of conversion between the physical quantities that refer to linear motion and those that refer to rotations is not the radius, but the ratio between the arc length and the central angle corresponding to the arc. This ratio happens to be equal to the radius only when the central angle happens to be measured in radians.
The correct conversion factors must be used to convert the values of a physical quantity like angular momentum between systems with different units for the plane angle.
While in the system with plane angles measured in radians the angular momentum is the linear momentum multiplied by radius, in the system with plane angles measured in cycles the angular momentum is the linear momentum multiplied by perimeter. The same for torque.
When you divide a kinetic energy to a frequency expressed in cycles per second, you get an angular momentum that is equal to the product of linear momentum by perimeter, corresponding to the cycle as the unit of angle.
When you divide a kinetic energy to a frequency expressed in radians per second, you get an angular momentum that is equal to the product of linear momentum by radius, corresponding to the radian as the unit of angle.
These 2 variants correspond to the alternative "h" and "h bar" values of the Planck constant. The same quantum of angular momentum corresponds to both choices (i.e. the double of the spin of the electron), it is just expressed in different units.
The frequency of periodic events was measured at that time in cycles per second. The unit of frequency "cycle per second" has been renamed "hertz" much later. The renaming was first proposed in 1935, but it became adopted in SI only in 1960.
Because the use of cycles per second (now hertz) is entrenched, when the radian has been promoted as the preferred unit for plane angle, that has caused a split in the SI system of physical quantities, between the frequency of periodic phenomena measured in cycles per second and the angular velocity measured in radians per second.
This split is a big cause of inconsistency in SI, because "frequency of periodic phenomena" and "angular velocity" are just 2 names for the same physical quantity and the "hertz"/"cycle per second" and the "radian per second" do not belong in the same consistent system of units, the former corresponds to the choice of the cycle as the unit of plane angle, while the latter corresponds to the choice of the radian as the unit of plane angle.
Thus the SI system of units is a mixture of units from 2 distinct systems of consistent units. There are a number of physical quantities in SI for which 2 units are used, one derived from the cycle and one derived from the radian. In some cases distinct names are used to show the intended unit, like "frequency" and "angular velocity", while in other cases there are no distinct names. The non-SI unit of angular velocity, "rpm", i.e. "rotations per minute", is another name for "cycles per minute".
All the units of physical quantities related to rotations, including the angular momentum, depend on the unit chosen for the plane angle, so they must be multiplied or divided by conversion factors (i.e. 2 times pi for conversion between cycle and radian), whenever the unit for plane angle is changed.
The use of radian is a cause of confusion for those who learn about the physics of rotation, because they learn for instance that the angular momentum and the torque are proportional with the radius to the center of rotation.
This is false and it appears to be true only when the radian is chosen as the unit of plane angle. In reality, the factor of conversion between the physical quantities that refer to linear motion and those that refer to rotations is not the radius, but the ratio between the arc length and the central angle corresponding to the arc. This ratio happens to be equal to the radius only when the central angle happens to be measured in radians.
The correct conversion factors must be used to convert the values of a physical quantity like angular momentum between systems with different units for the plane angle.
While in the system with plane angles measured in radians the angular momentum is the linear momentum multiplied by radius, in the system with plane angles measured in cycles the angular momentum is the linear momentum multiplied by perimeter. The same for torque.
When you divide a kinetic energy to a frequency expressed in cycles per second, you get an angular momentum that is equal to the product of linear momentum by perimeter, corresponding to the cycle as the unit of angle.
When you divide a kinetic energy to a frequency expressed in radians per second, you get an angular momentum that is equal to the product of linear momentum by radius, corresponding to the radian as the unit of angle.
These 2 variants correspond to the alternative "h" and "h bar" values of the Planck constant. The same quantum of angular momentum corresponds to both choices (i.e. the double of the spin of the electron), it is just expressed in different units.