That he might ascertain whether any of the cloths of ancient Egypt were made of hemp, M. Dutrochet has examined with the microscope the weavable filaments of this last vegetable.

The back of the used book shop was one of her favorite nooks; she could read for hours and no one would bother her or pester her to buy.

A common bee will sting and kill another common bee, for cause, but when it is necessary to kill the queen other ways are employed.

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.