Ophthalmological device including an applanation tonometer tip having a bi-curved cornea-contacting surface structured to minimize the intracorneal stress, and method of using such device for measurement of intraocular pressure. The cornea-contacting surface includes a first central portion and a second portion that encircles and adjoins the first central portion. The curvatures of the first and second portions have opposite signs. In one case, the first central portion can be rotationally-symmetric. In a related case, the first portion has a curvature with a sign opposite to that of a curvature of a typical cornea, while the curvature of the second portion has a sign equal to that of the curvature of the cornea. Method for using the device to procure values IOP with increased ac curacy as compared with the use of a conventional flat-surface tonometer tip.