In this work, Grinnellian niche theory (a body of theory about geographic distributions of species in terms of noninteracting niche variables) is used to demonstrate that species-area relationships emerge with both size of environmental space and size of geographic area. As environmental space increases, more species' fundamental niches are included, thus increasing the number of species capable of living in the corresponding region. This idea is made operational by proposing a size measure for multidimensional environmental space and approximating fundamental niches with minimum volume ellipsoids. This framework allows estimating a presence-absence matrix based on the distribution of fundamental niches in environmental space, from which many biodiversity measures can be calculated, such as beta diversity. I establish that Whittaker's equation for beta diversity is equivalent to MacArthur's formula relating species numbers and niche breadth; this latter equation provides a mechanism for the species-niche space relationship. I illustrate the theoretical results via exploration of niches of the terrestrial mammals of North America (north of Panama). Each world region has a unique structure of its environmental space, and the position of fundamental niches in niche space is different for different clades; therefore, species-area relationships depend on the clades involved and the region of focus, mostly as a function of MacArthur's niche beta diversity. Analyzing species-area relationships from the perspective of niche position in environmental space is novel, shifting emphasis from demographic processes to historical, geographic, and climatic factors; moreover, the Grinnellian approach is based on available data and is computationally feasible.