The Complete Works of Archimedes

...sides of the [circumscribed] polygon to the sides of the inscribed [polygon] have a lesser ratio than that of [the line] to K, as we have learned. For this reason indeed, the duplicate ratio is also less than the duplicate ratio. And as the ratio of side to side is, the ratio of polygon to polygon is the duplicate; for they are similar. And as G is to D, so is G to D. Therefore the circumscribed polygon also has to the inscribed a lesser ratio than [G] has to D. Much more then the ratio of the circumscribed to the inscribed has a lesser ratio than E has to Z. Similarly, we shall show that, two unequal magnitudes being given, and a sector, it is possible to circumscribe a polygon about the sector, and to inscribe another similar to it, so that the circumscribed has to the inscribed a lesser ratio than the greater magnitude has to the lesser. And this also is manifest: that if a circle or a sector and a certain area be given, it is possible, by inscribing equilateral polygons into the circle or the sector, and ever into the remaining segments, to leave certain segments of the circle or sector which are less than the proposed area. For these things have been handed down in the Elements.

6 It must now be shown that, a circle or a sector and an area being given, it is possible to circumscribe a polygon about the circle or sector, so that the remaining segments of the circumscription are less than the given area. For let these things be shown for the circle, and the same reasoning may be transferred to the sector. Let the circle be A and the area B; it is indeed possible to circumscribe a polygon about the circle so that the segments left over between the circle and the polygon are less than the area B. For indeed, there being two unequal magnitudes, the greater being the sum of the area and the circle, and the lesser being the circle itself, let a polygon be circumscribed about the circle, and let another be inscribed, so that the circumscribed has to the inscribed a lesser ratio than the aforementioned greater magnitude has to the lesser. This circumscribed polygon shall be that for which the remainders will be less than the proposed area B. For if the circumscribed has to the inscribed a lesser ratio than the sum of the circle and the area B has to the circle itself; and the circle is greater than the inscribed [polygon], much more does the circumscribed [polygon] have to the circle a lesser ratio than the sum of the circle and the area B has to the circle itself. And by subtraction, therefore, the remainders of the circumscribed polygon have to the circle a lesser ratio than the area B has to the circle. Therefore the remainders of the circumscribed polygon are less than the area B. Or thus: since the circumscribed has to the circle a lesser ratio than the sum of the circle and the area has to the circle, for this reason the circumscribed is less than the sum of both. So that the other remainders will also be less than the area B. And similarly for the sector.

A geometric diagram showing a circle with both an inscribed hexagon and a circumscribed hexagon. A central point is marked, with radial lines extending to the vertices of the hexagons.

7 If a pyramid be inscribed in an isosceles cone, having an equilateral base, its surface excluding the base is equal to a triangle having a base equal to the perimeter of the base [of the pyramid], and height equal to the perpendicular drawn from the vertex to one side of the base. Let there be an isosceles cone, whose base is the circle ABG, and let there be inscribed in it a pyramid having an equilateral triangle ABG as its base. I say that its surface excluding the base is equal to the said triangle. For since the cone is isosceles, and the base of the pyramid is equilateral, the heights of the triangles comprising the pyramid are equal to one another. And the triangles have as bases AB, BG, and GA, and as height the aforementioned [perpendicular]. So that the triangles are equal to a triangle having a base equal to AB, BG, and GA, and the aforementioned height; that is, the surface of the pyramid excluding the triangle ABG.

A geometric diagram showing a circle (base of a cone) with an inscribed equilateral triangle labeled with Greek letters α, β, and γ at the vertices. A central point is marked with lines radiating to the triangle's vertices.

Let there be an isosceles cone, whose base is the circle ABG, and vertex the...