Abstract: The device has a base mounting a support which constantly contacts the surface of an object being measured, the object having a nominally circular cross-section. The support is a plurality of essentially self-adjusting balance beams arranged in steps or in a staggered order. The swing axles of the balance beams of a first step are secured to the base, while the swing axles of the balance beams of a subsequent step are secured to the ends of the balance beams of a preceding step. The balance beams of a last-in-succession step contact the profile of the cross-section of the object being measured. The central angles at which the balance beams are mounted are chosen on the basis of a condition of maximal stabilization of the position of the center of the profile of the cross-section of the object being measured relative to the base in the direction of the disposition of a sensing element of a displacement transducer secured to the base.