1. In the Cambridge and Dublin Mathematical Journal, vol. v., 1859, De Morgan gives the definition of the “area contained within a circuit” as the area swept out by a radius vector which has one end (the pole) fixed and the other describing the circuit (in a determinate mode), on the supposition that each element of area is positive or negative, according as the radius is revolving positively or negatively. He remarks that the definition satisfies existing notions, that it provides the necessary extension of the meaning of the word area, and proceeds to show that it gives to every circuit the same area, whatever point the pole may be. The object of this paper is to give an Area-Theory beginning with the triangle and going on to circuits bounded by straight or curved lines. The fundamental proposition is derived from Analysis, and the geometry of the applications is therefore an Analytical Geometry; indeed, one of the objects of the paper is to emphasise the advantage of keeping Analysis and Geometry in close correspondence. As evidence of the difficulty of pursuing an Area-Theory in Geometry, without the aid of Analysis, it may be noticed that Townsend in his Modern Geometry (1863), §83, lays down Salmon's Theorem in this form: “If A, B, C, D be any four points on a circle taken in the order of their disposition, and P any fifth point, without, within, or upon the circle, but not at infinity, then always area BCD.AP2 − area CDA.BP2 + area DAB.CP2 − area ABC.DP2 = 0, regard being had only to the absolute magnitudes of the several areas which from their disposition are incapable of being compared in sign.”