362 12. MODERN GEOMETRIES
DA C
ÂFIGURE 6. Menelaus' theorem for a plane triangle.Although Desargues' terminology is very difficult to follow, his Rough Draft
contained some elegant theorems about points on conies. Two significant results
are the following:^13
First: If four lines in a plane intersect two at a time, and the points of inter-
section on the first line are A, B, and C, with  between A and C, and the lines
through A and  intersect in the point D, those through A and C in Šand those
through  and C in F, then
The situation here was described by Pappus, and the result is also known as
Menelaus' theorem. The proof is easily achieved by drawing the line through Å
parallel to AB, meeting BD in a point G, then using the similarity of triangles
EGF and CBF and of triangles DEG and DAB, as in Fig. 6. From Eq. 1 it is
easy to deduce that BD • AE • CF = BF • AD • CE. Klein (1926, p. 80) attributes
this form of the theorem to Lazare Carnot (1753-1823).
Second: The converse of this statement is also true, and can be interpreted as
stating that three points lie on a line. That is, if ADB is a triangle, and Å and F
are points on AD and BD respectively such that AD : AE < BD : BF, then the
line through Šand F meets the extension of AB on the side of  in a point C,
which is characterized as the only point on the line EF satisfying Eq. 1.
In 1648 the engraver Abraham Bosse (1602-1676), who was an enthusiastic
supporter of Desargues' new ideas, published La Perspective de Mr Desargues, in
which he reworked these ideas in detail. Near the end of the book he published the
theorem that is now known as Desargues' theorem. Like Desargues' work, Bosse's
statement of the theorem is a tangled mess involving ten points denoted by four
uppercase letters and six lowercase letters. The points lie on nine different lines.
When suitably clarified, the theorem states that if the lines joining the three pairs
of vertices from two different triangles intersect in a common point, the pairs of
lines containing the corresponding sides of these triangles meet in three points all
on the same line. This result is easy to establish if the triangles lie in different
planes, since the three points must lie on the line of intersection of the two planes
containing the triangles, as shown in Fig. 7.
For two triangles in the same plane, the theorem, illustrated in Fig. 8, was
proved by Bosse by applying Menelaus' theorem to the three sets of collinear points
(^13) To keep the reader's eye from getting too tangled up, we shall use standard letters in the
statement and figure rather than Desargues' weird mixture of uppercase and lowercase letters and
numbers, which almost seems to anticipate the finest principles of computer password selection.
(i)
BD _ AD CE
BF~AE'CF'