The works translated here--The Books at the Sphere and Cylinder--were a resource of serious satisfaction for Archimedes, the best scientist of antiquity. Accompanying this translation is the 1st clinical variation of the diagrams, which includes new details from the lately came upon Archimedes Palimpsest. the quantity additionally comprises the 1st English translation of Eutocius' statement. Reviel Netz's remark analyzes Archimedes' paintings from modern study views reminiscent of clinical sort and the cognitive background of mathematical texts.
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Additional info for The Works of Archimedes, The Two Books On the Sphere and the Cylinder: Translation and Commentary
For instance, one may envisage the circle as (i) composed of all its radii; or (ii) of all the chords parallel to a given diameter. When this is translated from the language of inﬁnitely many line-segments into the language of indeﬁnitely many slices, the circle is then bounded by (i) sectors, in the ﬁrst case, or by (ii) rectangles, in the second case. An approach analogous to (ii), bounding the circle by rectangles, was taken by Archimedes in CS. An approach analogous to (i), bounding the circle by sectors, was taken by Archimedes in SL.
That is, to say that the property of Deﬁnition 2 is meant to apply only to the family singled out in Deﬁnition 1 is an empty claim: the property can apply to no other lines. It seems to me that the clause of Deﬁnition 1 is meant to introduce the main idea of Deﬁnition 2 with a simple case – which is what I did above. In other words, the function of Deﬁnition 1 may be pedagogic in nature. Postulates 1–2: about what? The wording of the translation of Postulate 1 gives rise to a question of translation of signiﬁcant logical consequences.
To ﬁnd a sphere equal to a given cone or cylinder (Proposition 1). To cut a sphere so that the surfaces of the segments have to each other a given ratio (Proposition 3). To cut a sphere so that the segments have to each other a given ratio (Proposition 4). To ﬁnd a segment of a sphere similar to a given segment, and equal to another given segment (Proposition 5). 25 26 i n t ro d uc t i on 6 7 8 To ﬁnd a segment of a sphere similar to a given segment, its surface equal to a surface of a given segment (Proposition 6).