CONICS I.

Likewise, for two curved lines lying in one plane, I call a transverse diameter a diagonal bisector a straight line which, cutting through both lines, bisects all straight lines drawn in each of the lines parallel to a certain straight line. I call the vertices of the lines the endpoints of the diameter at the curves. I call a rectilinear diameter a central axis one that lies between the two lines and bisects all parallel straight lines drawn to a certain straight line that are intercepted between the lines, and I say that each of the parallels is drawn ordinally parallel to the diameter onto the diameter.

I call conjugate diameters paired bisectors of a curved line, and of two curved lines, the straight lines of which each, being a diameter, bisects the lines parallel to the other.

I call an axis the principal diameter of a curved line, and of two curved lines, the straight line which, being a diameter of the line or lines, cuts the parallels at right angles.

I call conjugate axes paired principal diameters of a curved line, and of two curved lines, the straight lines which, being conjugate diameters, cut each other's parallels at right angles.

I.

Straight lines drawn from the vertex of a conic surface to points on the surface lie within the surface.

Let there be a conic surface whose vertex is point A, and let a point B be taken on the conic surface, and let a straight line AΓB be joined. I say that the straight line AΓB is in the surface.

Critical apparatus at the bottom of the page:
5. πρός] προσ' fortuitous line V. — 6. ὀρθίαν] p; ὀρθείαν V, marginal note by later hand "ὀρθίαn as below". — 9. τέμνει] p, τέμνῃ V. — 11. δύο] omitted by Halley with the Commentary. — 21. α΄] cv, omitted by V.