The analogy between certain properties of the co-ordinates of elliptic and hyperbolic sectors, which forms the subject of this memoir, was observed by the early writers on the fluxional or differential calculus, and employed by them in its improvement. But, independently of this important application, the propositions in question are some of the most elegant theorems in geometry, and highly interesting as abstract truths.

Maclaurin, in the third chapter of the second book of his Treatise of Fluxions, has proved the truth of this theorem: “Supposing n to be any number, let E and n × E be elliptic sectors, which stand on arcs that begin at a vertex of either axis, and H and n × H hyperbolic sectors, which stand on arcs that begin at the extremity of the transverse axis: the algebraic equation which expresses the relation between x and z, ordinates drawn to the other axis from the extremities of the arcs, must be the very same in the two curves.”