Taking the circumference as intermediate between the perimeters of the inscribed and the circumscribed regular n-**gons**, he showed that, the radius of the circle being given and the perimeter of some particular circumscribed regular polygon obtainable, the perimeter of the circumscribed regular polygon of double the number of sides could be calculated; that the like was true of the inscribed polygons; and that consequently a means was thus afforded of approximating to the circumference of the circle.

The general theorems which enabled him to do this, after a start had been made, are A2n = 11A„A ' n (Snell's Cyclom.), P 2A„A' n - 2A' „AZ, Gre o A 2 ” - A n +A2n or A' n +A2„ (g r1') where A „, A'„ are the areas of the inscribed and the circumscribed regular n-**gons** respectively.