Desargues' Brouillon Project and the Conics of ... - J.P. Hogendijk

36 Jan l? Hogendijk
and he does not mention the cone or its apex 0. He proves a converse
of what my introduction may suggest, for he assumes the involution
cp on AF with fixed points X and Q (which will later turn out to be
the projections of the foci) and he proves the existence of the
involution 0 and its centre C. He also proves the non-obvious fact
that the straight line through the centres of CJ and cp passes through
the centre of r. In his proposition r is not a circle but a general
conic section (“section of a roll’), but this difference is inessential. I
now quote his proposition, with some explanatory additions of my
own in parentheses. Most of the botanical terminology in the
quotation can simply be ignored.
Desargues says (Figure 20, adapted from Taton, Fig. 19):
Figure 20.
When in the plane of a general section of a roll 5Y8GH. on the general line AF, (which
is) one of the ordinates of a transversal AV(line f), the transversal point A is paired
with F, the butt of the ordinates, in involution with any other two points X, Q, which
are taken as two double mean knots of the involution (that is, the involution rp), (and
if) each of the pairs of branches springing from this trunk XQ which pass through
one of these pairs of extreme knots, such as FH, AH, and RG, ZG, and are ordinate
to the butts H and G, on the edge of the figure, in such a way that one of the two
touches the figure, such as HA and ZGB, I say that each of such pairs of branches
arranged in this way gives on the transversal VA one of the pairs of knots DA, EB
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