Archimedis Opera Omnia cum Commentariis Eutocii, Vol. I (Heiberg 1880)

1.

If a polygon is circumscribed about a circle, the perimeter of the circumscribed polygon is greater than the perimeter of the circle. For let a polygon be circumscribed about the given circle; I say that the perimeter of the polygon is greater than the perimeter of the circle.

Since the combined length of BA and AD is greater than the arc BD, because it encompasses the arc while having the same endpoints, and similarly the sum of DΓ and ΓB is greater than DB, and the sum of ΔK and KΘ is greater than ΔΘ, and the sum of ZHΘ is greater than ZΘ, and further the sum of ΔE and EZ is greater than ΔZ, therefore the entire perimeter of the polygon is greater than the circumference of the circle.

A diagram depicts a circle with a polygon circumscribed around it. Various points along the perimeter are labeled with Greek letters, illustrating the relationship between the polygon's segments and the circle's arcs.

2.

Given two unequal magnitudes, it is possible to find two unequal straight lines such that the ratio of the greater line to the lesser is less than the ratio of the greater magnitude to the lesser.

Let two unequal magnitudes be AB and Δ, and let AB be the greater. I say that it is possible to find two unequal straight lines that satisfy the stated requirement.