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We used a scanning tunneling microscope to probe the interactions between spins in individual atomic-scale magnetic structures. Linear chains of 1 to 10 manganese atoms were assembled one atom at a time on a thin insulating layer, and the spin excitation spectra of these structures were measured with inelastic electron tunneling spectroscopy. We observed excitations of the coupled atomic spins that can change both the total spin and its orientation. Comparison with a model spin-interaction Hamiltonian yielded the collective spin configuration and the strength of the coupling between the atomic spins.