The horospherical Cauchy transform on Riemannian symmetric spaces and curved Radon's inversion formula

Following to Gelfand's conception of integral geometry the Plancherel formula on symmetric spaces is equivalent to the inversion of the horospherical transform. We modify the horospherical transform and show that there is its inversion coinciding with the inversion formula for its flat model. The last one can be obtain using the classical Fourier transform.
The basic tool is a "curving" of flat inversion formulas starting of usual Radon inversion formula on the plane. Our final Plancherel formula on Riemannian symmetric spaces differs from Harish-Chandra's formula.