Harte *et al*. (1) assumed the probability rule: if a species occurs in an area*A*
_{0}, then the probability that it occurs in half of that area is a constant, *a*, independent of area*A*
_{0}, satisfying 0.5 ≤ *a* ≤ 1. From this rule, Harte *et al*. (1) give a mathematical proof of the power law form of the species-area curve: if *S*is the number of species in *A*, then *S* =*cA ^{z}
*, where (0.5)

^{z}=

*a*, 0 ≤

*z*≤ 1, and

*c*is constant.

Harte *et al*. (1) do not justify the last step in their proof. Their final equation, *S _{i}
* =

*cA*

_{i}

^{z}, is equivalent to

*S*

_{i}= (2^{−i})

^{z}

*S*

_{0}. To complete their proof, it would be necessary to prove, for example, that the number of species occurring in an area (0.75)

*A*

_{0}is (0.75)

^{z}

*S*

_{0}.

The distribution of a given species in a habitat can be thought of as a random point distribution over that habitat. Harte *et al*. (1) need to provide at least one nontrivial example of such a distribution that satisfies their probability rule. As far as we can see, no random point distribution satisfies the rule unless*a* = 0.5, in which case the only example known to us is the uniform distribution of a single, randomly chosen, point.

We shall demonstrate from the probability rule of Harte *et al*. (1) that *a* = 1 or *a* = 0.5 or*c* = 0. The power law happens to hold for these values, but in all other cases the rule and the power-law are in conflict.

Let x|o mean “species occurs only in right half,”
The probability rule of Harte *et al*. (1) yields
Applying 6. to 1. and 1. to 6. gives
The first equality is derived in (2). Notice thatCancelling equal probabilities, we are left withThe probability rule also yieldsandhence,* a* = 1 or *a* = 0.5.

Another way of arriving at this conclusion is to apply the probability rule and the power law to an area consisting of three of the quarters created by subdividing the habitat into four quarters.*E*
_{2} is the number of species found only in one of the quarters. The rule implies, by Eq. 7. of Harte *et al*. (1), *E*
_{2}= (1 − a )^{2}
*S*
_{0}. Therefore, the number of species in the remaining three quarters is
By the power law,Therefore,which holds if *S*
_{0} = 0 =*c*. If *S*
_{0} ≠ 0, multiplying both sides by 2^{2z}/*S*
_{0} produceswhich implies *z* = 0 or *z* = 1, hence, *a* is either 0.5 or 1.

The rule proposed by Harte *et al*. (1) implies that species are distributed in one of three trivial ways. In general, the equivalence of the probability rule with the power law is invalid, as are all conclusions that rely upon it, such as the “endemics-area relationship” (1, 2).

*Response*: In our derivation of the familiar species-area relationship (SAR), of a new endemics-area relationship, and of a new abundance distribution from self-similarity, we explicitly made use of successive shape-preserving bisections of a biome that is taken initially to be a golden rectangle with length-to-width ratio of ** a**, can take on only particular values. There is plenty of evidence cited in our report that shape does matter, and that the specific prediction made by our self-similarity theory about the dependence of species richness on patch shape is reasonable.

Maddux and Athreya also refer to the example of a random placement model, in which the parameter *a* is indeed restricted to the value 1/2. In our theory, the fraction of the species in a rectangle that is also found in a particular half of that rectangle is not given by random placement but rather is governed by the parameter,*a*, which is independent of scale and not restricted to 1/2. Whereas the assumption of independence is necessary to our findings, our statement that there averages 1 species per unit square was unnecessary and overly restrictive; if *S _{m}
*differs from 1, the distribution plotted on a ln(

*n*) scale is displaced horizontally but its shape is unchanged.