 Why does Archytas think that logistic is superior to the other sciences? In B4, he particularly compares it to geometry, arguing that logistic (1) “deals with what it wishes more vividly than geometry” and (2) “completes demonstrations” where geometry cannot, even “if there is any investigation concerning shapes.” This last remark is surprising, since the study of shapes would appear to be the proper domain of geometry. The most common way of explaining Archytas’ remark is to suppose that he is arguing that logistic is mathematically superior to geometry, in that certain proofs can only be completed by an appeal to logistic. Burkert sees this as a reason for doubting the authenticity of the fragment, since the exact opposite seems to be true. Archytas could determine the cube root of two geometrically, through his solution to the duplication of the cube, but could not do so arithmetically, since the cube root of two is an irrational number (1972a, 220 n. 14). Other scholars have pointed out, however, that certain proofs in geometry do require an appeal to logistic (Knorr 1975, 311; Mueller 1992b, 90 n. 12), e.g., logistic is required to recognize the incomensurability of the diagonal with the side of the square, since incommensurability arises when two magnitudes “have not to one another the ratio which number has to number” (Euclid X 7). These suggestions show that logistic can be superior to geometry in certain cases, but they do not explain Archytas’ more general assertion that logistic deals with whatever problems it wants more clearly than geometry.