Three early translations of it, however, into Latin, Æthiopic, and Arabic, are extant, all of which have obviously been made from the Greek.

The dual isogeny of an isomorphism is its inverse. The dual of a composition of isogenies is the composition of the duals in the reverse order. If 𝜑 is an isogeny and ̂𝜑 is its dual then the formulas 𝜑(z+𝛬)=mz+𝛬', 𝜑(z'+𝛬')=( operatorname 𝜑/m)z'+𝛬 show that also 𝜑∘̂𝜑=[ operatorname deg(𝜑)]=[ operatorname deg(̂𝜑)], so that 𝜑 is in turn the dual isogeny of its dual ̂𝜑. Isogeny of complex tori, rather than isomorphism, will turn out to be the appropriate equivalence relation in the context of modular forms.

Their only regret was, that Mademoiselle Carrara would taste none of the conserves and the pastry they were so busily concocting.

In his field journal, he estimated that since his preliminary dig in 1870, he and Sophia had removed more than a quarter of a million cubic meters of earth by wheelbarrow, mancart, and oxcart.