The Complete Works of Archimedes (1544) - Page 11

1 point. And let there be inscribed in the cone a pyramid, having as its base the equilateral triangle ABC; and let DA, DC, DB be joined. I say that the triangles ADB, ADC, BDC are [together] equal to a triangle whose base is equal to the perimeter of triangle ABC, and whose altitude from the vertex to the base is equal to the altitude drawn from D to the [side of] ABC. For let the perpendiculars DK, DL, DM be drawn; these are therefore equal to one another. And let there be set out a triangle EZH having the base EZ equal to the perimeter of triangle ABC, and the altitude [H] equal to DL. Since, therefore, the rectangle under BC, DL is double the triangle BDC; and the rectangle under AB, DK is double the triangle ABD; and the rectangle under AC, DM is double the triangle ADC; therefore the rectangle under the perimeter of triangle ABC—that is, EZ—and DL—that is, KH—is double the triangles ADB, BDC, ADC. The rectangle under EZ, KH is also double the triangle EZK; therefore triangle EZK is equal to triangles ADB, BDC, ADC.

If a pyramid be circumscribed about an isosceles cone, the surface of the pyramid, 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 side of the cone. Let there be a cone whose base is the circle ABC, and let a pyramid be circumscribed so that its base—that is, the polygon DEZ—is circumscribed about the circle ABC. I say that the surface of the pyramid, excluding the base, is equal to the aforesaid triangle. For since the axis of the cone is perpendicular to the base, that is, to the circle ABC, and the straight lines joined from the center of the circle to the points of contact are perpendicular to the tangents; therefore the lines joined from the vertex of the cone to the points of contact will also be perpendicular to DE, EZ, ZD, [namely] KA, KB, KC. Therefore the aforesaid perpendiculars are equal to one another; for they are sides of the cone. Let there then be set out the triangle TKL, having TK equal to the perimeter of the polygon DEZ, and the altitude LM equal to KA. Since, then, the rectangle under DE, AK is double the triangle EDK; and the rectangle under EZ, KB is double the triangle EZK; and the rectangle under ZD, CK is double the triangle ZDK; therefore the rectangle under TK and AK—that is, ML—is double the triangles EDK, EZK, ZDK. And the rectangle under TK, LM is also double the triangle LKT. For these reasons, then, the surface of the pyramid, excluding the base, is equal to a triangle having a base equal to the perimeter of DEZ, and height equal to the side of the cone.

Three geometric diagrams illustrating the relationship between cones and pyramids. 1\. Top Diagram: A cone with vertex δ and a triangular pyramid inscribed within it. The vertices of the pyramid's base are labeled α, β, and γ. Lines connect the vertex δ to the points α, β, and γ. 2\. Middle Diagram: A cone with vertex κ. A triangle δϵζ is circumscribed about its circular base, touching the circle at points α, β, and γ. Lines connect vertex κ to the vertices of the large triangle (δ, ϵ, ζ) and to the points of tangency (α, β, γ). 3\. Bottom Diagram: A standalone triangle θκλ with an internal altitude line λμ. Below this triangle is another representation of a circular base with an inscribed triangle.