 Newton developed three approaches for his calculus, all of which he regarded as leading to equivalent results, but which varied in their degree of rigor. The first employed infinitesimal quantities which, while not finite, are at the same time not exactly zero. Finding that these eluded precise formulation, Newton focussed instead on their ratio, which is in general a finite number. If this ratio is known, the infinitesimal quantities forming it may be replaced by any suitable finite magnitudes—such as velocities or fluxions—having the same ratio. This is the method of fluxions. Recognizing that this method itself required a foundation, Newton supplied it with one in the form of the doctrine of prime and ultimate ratios, a kinematic form of the theory of limits.