Discounting can have a drastic impact on the valuation of future environmental impacts, particularly over long periods of time. Especially profound problems are raised by the use of discounting over multiple generations, leading even some economists and a number of legal scholars to reject its use in the intergenerational context. These problems are indeed troubling, given the possibility that discounting might lead to the nearly complete neglect of the interests of distant generations by the present one. (The classic example is the "Cleopatra" hypothetical, in which it is not worthwhile for Cleopatra to skip a helping of dessert in order to save a child's life today.) Nevertheless, as this paper shows, the logic of discounting is not easily avoided. This paper demonstrates that the standard economic technique of exponential discounting follows from three modest assumptions: (1) environmental benefits have positive value in all future time periods; (2) benefits are additive, so that the benefit during any given time period is equal to the total benefits during its sub-periods; and (3) the preference for permanent solutions over temporary ones does not vary depending with the starting date. Given these modest and seemingly morally acceptable assumptions, exponential discounting is unavoidable. Relaxing these assumptions generates two alternative methods of discounting. If we relax the assumption that benefits are additive, the model allows the assignment of "perpetuity" values - that is, of bonuses for resources with the capacity for indefinite viability. For example, we might attach a higher intrinsic value in each time period to a species which is still viable rather than one that is doomed. This bonus value, which can be thought of as the value of the environmental benefit at infinity, is not subject to discounting. Alternatively, relaxing the third assumption allows the use of hyperbolic discounting. The switch from exponential to hyperbolic discounting is known to have important practical consequences for analysis of problems such as global climate change. Hyperbolic discounting essentially involves a declining discount rate, reducing its impact on far-future benefits. As it turns out, these modified discounting techniques fit naturally with situations involving cooperation across generations, rather than one-shot decisions. These techniques also ameliorate, although they do not completely eliminate, the ethical concerns over discounting across multiple generations.