The trend in the development and adoption of neural training regimes for supervised learning in feedforward neural networks has been to follow gradient approaches already successful in numerical analysis. However, weak direction can be seen to be common to the neural implementation of such approaches which in large measure is due to vector summation. We provide a training method, Tangent Hyperplanes, which avoids such summation. The method is designed to be more strongly directed and easier to use successfully than the standard gradient regimes. Solutions are found to problems for single layer nets with linear activation functions in a single iteration. For multi-layer nets with non-linear activation functions, weight transitions regularly form smooth solution sequences which contain relatively low numbers of iterations. A number of benchmark problems including well-known ones have been tested to compare Tangent Hyperplanes with the standard gradient regime of Back-Propagation for penetration to solution and robustness. Training parameters for the new method were given universal values across all the problems and achieved 99-100 % success rates. Tangent Hyperplanes also managed to solve the 2-spirals problem with a single hidden layer architecture for which standard gradient regimes have reports of failure.