Reading "Courant's – What is Mathematics", where a definition has just been introduced stating "[a] conic is the locus of intersection of corresponding lines in two projectively related pencils". A little later "If the pencil O and O' are congruent we obtain a circle. If angles are equal but with opposite sense, the conic is an equilateral hyperbola (see Fig 99 [me: below])".
The sentence in italics is basically the only explanation given as to how the hyperbolas had been obtained, and I have a hard time understanding it; for example I can't understand why and how we use points of intersection here (I know about cross-ratio, but not how it relates to the construction)... Could someone explain?
I believe that it's saying this: take the (vertical) line from $O$ to $O';$ draw a line $15$ degrees clockwise from this, going through $O'$ (the line going from $O'$ to the vertex called "7". Draw line $15$ degrees counterclockwise from this, going through $O$ (the line from $O$ to the vertex called "21"); find their intersection and plot it. Now repeat for 30, 45, 60, and so on degrees.
The problem is that this isn't what's shown in the drawing. The line $O-21$ is there, but the one marked to intersect with it is $O'-5$. The $O-20$ intersects $O'-4$, and so on. The resulting little circles, when connected, form a hyperbola.
The other side of this story is how the circle got there in the first place: for each line through $O$, you find a "corresponding" line through $O'$ (and "corresponding" here seems to mean "rotated by 90 degrees"), and you find their intersections. The result is the set of all points of the circle.
How is such a correspondence defined? Well, it looks as if Courant is using angles, but my preferred version is one where you define a projectivity as any sequence of perspectivities, as Hartshorne does in his little book on projective geometry. Doubtless in the real projective plane, the two things end up amounting to the same thing.