1. If PQRS be an Hyperbola, OE, OF its asymptotes, P, Q, R, S any points on it such that the sectorial area OPQ = sectorial area ORS; and if PA, QB, RC, SD be ordinates to one asymptote and parallel to the other, it is known that
Hence if A, B, C … be taken so that OA, OB, OC … are in continued proportion, the areas OPQ, OQR, ORS … are all equal, and since the number of points can be made as large as we please, the sum of the sectorial areas can be made as large as we please. ∴ The area between an asymptote, the curve, and any radius vector is infinite.