The theory of stress-induced cavitation is applied here to the problem of cavitation of a viscous liquid in the streaming flow past a stationary sphere. This theory is a revision of the pressure theory which states that a flowing liquid will cavitate when and where the pressure drops below a cavitation threshold, or breaking strength, of the liquid. In the theory of stress-induced cavitation the liquid will cavitate when and where the maximum tensile stress exceeds the breaking strength of the liquid. For example, liquids at atmospheric pressure which cannot withstand tension will cavitate when and where additive tensile stresses due to motion exceed one atmosphere. A cavity will open in the direction of the maximum tensile stress which is 45o from the plane of shearing in pure shear of a Newtonian fluid. This maximum tension criterion is applied here to analyze the onset of cavitation for the irrotational motion of a viscous fluid, the special case imposed by the limit of very low Reynolds numbers and the fluid flow obtained from the numerical solution of the Navier--Stokes equations. The analysis leads to a dimensionless expression for the maximum tensile stress as a function of position which depends on the cavitation and Reynolds numbers. The main conclusion is that at a fixed cavitation number the extent of the region of flow at risk to cavitation increases as the Reynolds number decreases. This prediction that more viscous liquids at a fixed cavitation number are at greater risk to cavitation seems not to be addressed, affirmed or denied, in the cavitation literature known to us.