Summary: The transverse vibration of a homogeneous Euler-Bernoulli beam of constant depth and lineary varying breadth is investigated in this paper. A direct analytical solution is presented for the mode shape equation. The constraints imposed by the boundary conditions yield the frequency equation. The four ideal boundary conditions considered are clamped, pinned, sliding and free. For a complete beam (sharp ended) it is a necessary condition that the sharp end remain free, and as a result the frequency equations for clamped, pinned, sliding or free boundary conditions at the large end are the determinant of a \(2\times 2\) matrix equated to zero. A truncated beam is considered as part of a hypothetical complete beam. The frequency equation for truncated beam is the determinant of a \(4\times 4\) matrix equated to zero. The roots of the frequency equations are determined by an iterative procedure. The first four natural frequencies of complete beam and the first three frequencies of truncated beams for 16 combinations of the ideal boundary conditions for various truncation factors are presented. The results are benchmarks and are therefore presented in tabular form to preserve the accuracy to six figures.