Background: The oxygen fl ow in humans and other higher animals depends on the erythrocyte-to-blood volume ratio, the hematocrit. Since it is physiologically favourable when the fl ow of oxygen transport is maximum it can be assumed that this situation has been achieved during evolution. If the hematocrit was too low, too few eryth- rocytes could transport oxygen. If it was too high, the blood would be very viscous, so that oxygen supply would again be reduced. Methods: The theoretical optimal hematocrit can be calculated by considering the dependence of blood viscosity on the hematocrit. Different approaches to expressing this dependence have been proposed in the literature. Here,wediscussearlyapproachesinhydrodynamicsproposedbyEinsteinandArrheniusandshowthatespecial- ly the Arrhenius equation is very appropriate for this purpose. Results & conclusions: We show that despite considerablesimpli fi cationssuch asneglecting thedeformation,ori- entation and aggregation of erythrocytes, realistic hematocrit values of about 40% can be derived based on opti- mality considerations. Also the prediction that the ratiobetween the viscosities ofthe blood and blood plasma at highshearratesnearlyequalsEuler'sconstant(2.718)isingoodagreementwithobservedvalues.Finally,wedis- cusspossibleextensionsofthetheory.Forexample,wederivethetheoreticaloptimalhematocritforpersevering divers among marine mammals to be 65%, in excellent agreement with the values observed in several species. General signi fi cance: These considerations are very important for human and animal physiology since oxygen transport is an important factor for medicine and physical performance.