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

1.

Previously, I sent you the things I had examined at that time, having written them down with their proofs: that every segment contained by a straight line and a parabola is one-third greater than a triangle having the same base as the segment and equal height. Subsequently, however, when some theorems remained unproven, I worked out their proofs. They are as follows: first, that the surface of any sphere is four times the greatest circle; second, that for any segment of a sphere, there is a circle equal in surface, whose radius is equal to the straight line drawn from the vertex of the segment to the circumference of the circle which is the base of the segment; to...

Critical apparatus containing variant readings from manuscripts B, F, V, A, D and the Latin translation by Commandino (Cr).

1. greeting] well-doing B. 2. I have sent] VAD; I sent F; I have sent... with the other words: to you — of them in line 3 omitted B; "I sent" Cr. to that time] at that time F. examined] theorized F. 3. the] om. F. of them] om. F with a gap left. 4. a straight line and] B; om. F; "from a straight line and" Cr. 5. Between "sur-" and "-third" a gap exists in F. triangle] om. F; triangle having B. 6. the same base] base the same B; this base F. having] om. B. 7. after these things] gap B. falling upon] falling from F; falling B; "now however of some occurring" Cr. of some unproven] opposing F, gap B; "which seem to be proven by effect" Cr. 8. I have worked out] Rivaltus; having worked out then after F; worked out... with a gap left B. of them] of them F; them — of every line 9 omitted B with gap left; "we have written the demonstrations" Cr. 9. as follows] if as follows F; "of this kind" Cr.