...judging it well to communicate them to those who are devoted to mathematics, I have sent to you the proofs, having written them out, concerning those matters which you requested should be examined by those much occupied in mathematics. Farewell.

OF ARCHIMEDES ON THE SPHERE

AND CYLINDER, BOOK I.

First, there are written these postulates, and the things assumed for their proofs. There are in a plane certain finite curved lines which either have their extremities joined by straight lines, or are entirely on the same side [of the line joining the extremities], or have no part on the other side. I call such a line concave on the same side, in which, if any two points are taken, the straight lines between the points either all fall on the same side of the line, or some on the same side and some upon the line itself, but none on the other side. Similarly, there are certain finite surfaces, themselves in a plane, which have their boundaries in a plane. I call those on the same side which have their boundaries in a plane, and either will be entirely on the same side, or have no part on the other side. And I call those surfaces concave on the same side, in which, if two points are taken, the straight lines between the points either all fall on the same side of the surface, or some on the same side and some upon the surfaces themselves, but none on the other side. I call a solid sector the figure contained by the surface of a cone and the surface of a sphere within the cone, when a cone cuts a sphere, having its vertex at the center of the sphere. I call a solid rhomb the solid figure composed of two cones which, having the same base, have their vertices on opposite sides of the plane of the base, such that their axes lie on a straight line. And I assume these things: that of lines having the same extremities, the straight line is the least. Of other lines, if they are in a plane and have the same extremities, such lines are unequal when they are both concave on the same side, and either one is entirely enclosed by the other and the straight line having the same extremities as it, or it is partly enclosed and has some parts in common; and the enclosed one is the lesser. Similarly, of surfaces having the same boundaries, if they have their boundaries in a plane, the plane surface is the least. Of other surfaces having the same boundaries, if the boundaries are in a plane, such surfaces are unequal when they are both concave on the same side, and either one is entirely enclosed by the other surface and the plane having the same boundaries as it, or it is partly enclosed and has some parts in common; and the enclosed one is the lesser. Further, of unequal lines, unequal surfaces, and unequal solids, the difference by which the lesser is exceeded is such that, when added to itself, it is capable of exceeding any proposed magnitude of those which are compared to one another. These things being assumed, if a polygon be inscribed in a circle, it is manifest that the perimeter of the inscribed polygon is less than the circumference of the circle. For each of the sides of the polygon is less than the arc of the circle cut off by it.

BIBLIOTECA NAZ.
VITT. EMANUELE
ROMA

IF a polygon be circumscribed about a circle, the perimeter of the circumscribed polygon is greater than the perimeter of the circle. For let the underlying figure be circumscribed about a circle. I say that the perimeter of the polygon is greater than the perimeter of the circle. For since the sum of BA, AL is greater than the arc BL, because [the lines] enclosing the arc have the same extremities; similarly also the sum of AD, DC, and CB, CD... and the sum of LK, KH, HI... and the sum of LK, ZH, H... and further the sum of ED, E, EZ, ZD... therefore the whole perimeter of the polygon is greater than the circumference of the circle.

A geometric diagram illustrating a circle inscribed within a pentagonal polygon. The polygon's vertices and various intersection points are labeled with Greek letters including Α, Β, Γ, Δ, Ε, Ζ, Η, Θ, Ι, Κ, Λ. Lines connect the center of the circle to the vertices of the polygon.

GIVEN two unequal magnitudes, it is possible to find two unequal straight lines, such that the greater straight line has to the lesser a ratio less than the greater magnitude has to the lesser. Let there be two unequal magnitudes, A, B...