364 12. MODERN GEOMETRIESHaving received a copy of this work from Marin Mersenne, Descartes took the
word draft literally and regarded it as a proposal to write a treatise—which it may
have been—such as a modern author would address to a publisher, and a publisher
would send to an expert for review. He wrote to Desargues to express his opinion
of "what I can conjecture of the Treatise on Conic Sections, of which [Mersenne]
sent me the Draft." Descartes' "review" of the work contained the kind of advice
reviewers still give: that the author should decide more definitely who the intended
audience was. As he said, if Desargues was aiming to present new ideas to scholars,
there was no need to invent new terms for familiar concepts. On the other hand,
if the book was aimed at the general public, it would need to be very thick, since
everything would have to be explained in great detail (Field and Gray, 1987, p.
176).2.4. Blaise Pascal. Desargues' work was read by a teenage boy named Blaise
Pascal (1623-1662), who was to become famous for his mathematical work and
renowned for his Pensees (Meditations), which are still read by many people today
for inspiration. He began working on the project of writing his own treatise on
conies. Being very young, he was humble and merely sketched what he planned
to do, saying that his mistrust of his own abilities inclined him to submit the
proposal to experts, and "if someone thinks the subject worth pursuing, we shall
try to carry it out to the extent that God gives us the strength." Pascal admired
Desargues' work very much, saying that he owed "what little I have discovered to
his writings" and would imitate Desargues' methods, which he considered especially
important because they treated conic sections without introducing the extraneous
axial section of the cone. He did indeed use much of Desargues' notation for points
and lines, including the word order for a family of concurrent lines. His work, like
that of Desargues, remained only a draft, although Struik (1986, p. 165) reports
that Pascal did work on this project and that Leibniz saw a manuscript of it—
not the rough draft, apparently- in 1676. All that has been preserved, however,
is the rough draft. That draft contains several results in the spirit of Desargues,
one of which, called by Pascal a "third lemma," is still known as Pascal's theorem.
Referring to Fig. 9, in which four lines MK, MV, SK, and SV are drawn and then
a conic is passed through Ê and V meeting these four lines in four other points P,
Ï, N, and Q respectively, Pascal asserted that the lines PQ, NO, and MS would
be concurrent (belong to the same order).
2.5. Newton's degree-preserving mappings. Newton also made contributions
to projective geometry, in a way that related it to Descartes' analytic geometry and
to algebraic geometry. He described the mapping shown in Fig. 10 (Whiteside, 1967,
Vol. VI, p. 269). In that figure the parallel lines BL and AO and the points A,
B, and Ï are fixed from the outset, and the angle è is specified in advance. Thus
the distances h and Ä and the angles ø and è are given before the mapping is
defined. Then, to map the figure GHI to its image ghi, first project each point G
parallel to BL so as to meet the extension of A Â at a point D. Next, draw the
line OD meeting BL in point d. Finally, from d along the line making angle è with
BL, choose the image point g so that gd : Od :: GD : OD. The original point,
according to Newton, had coordinates (BD, DG) and its image the coordinates
(Bd,dg). Thus, if we let ÷ = BD and y = DG, the coordinate transformation in