De Sphæra & Cylindro LIB. I.

B. But if a point D is taken on the chord AC, a part of the mixed line DABC, namely ABC, lies towards the side B above CD (connecting the ends D, C), and another part AD is situated along the extension of CD (i.e., in such a way that it coincides with it), but no part is depressed below CD, towards the side contrary to B.

I read "an" (an expletive) instead of "saí".

Or, above it.

II. I call a line concave towards the same parts if, when any two points are taken, the straight lines connecting the points that lie between them either all fall on the same side of the line, or some fall on the same side, and others along the line itself, but none on the opposite side.

Schol. For this somewhat obscure definition to be understood, one should look at and weigh the explanation of the preceding hypothesis; to which I will only add that it is a certain sign of concavity continued towards the same parts if no straight line cuts the line in more than two points.

I read "having" original: "ἔχουσας" instead of "exson".

III. Not unlike these, there are certain bounded surfaces, not indeed in a plane themselves, but having their ends in a plane; and the plane in which they have their ends, either they lie entirely on the same side of it, or have no part on the other.

IV. I call surfaces concave towards the same parts those in which, if two points are taken, the straight lines connecting the points that lie between them either all fall on the same side of the surface, or some on the same side, and others along them, but none on the opposite side.

Schol. Whoever understands the first two will understand these hypotheses without any difficulty.

V. 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.

Fig. 2.

As if BAC were a cone, whose vertex A is the center of the sphere; the figure DAE contained by the conical surface DAE and the spherical surface DE will be a solid sector.

The solid sector DAE is formed by the rotation of the circular sector DAZ around the radius AZ; with the arc DZ equal to ZE. Whence it could be defined otherwise.
Note that when the sector DAE is subtracted, the remainder of the sphere DXE is sometimes katachrestikos metaphorically/improperly called a Spherical Sector, larger than a hemisphere. See the scholastic note 51 of this work.

VI. I call a solid rhombus a solid figure composed of two cones having the same base, whose vertices lie on either side of the plane of the base, such that their axes lie in a straight line.

Fig. 3.

Such is the figure BACD consisting of two cones BAC and BDC, whose common base is the circle BC, and the axis AD passes through the center E.