(Rough draft for an essay on the results of taking plane

sections of a cone) outlined the principles of the new

theory, contained many new results, and offered, for

the first time, a unified theory of the

CONIC SECTIONS

.

His work, however, was largely ignored at the time of

its release, and the important impact of its ideas was

not properly recognized until the 19th century.

Very little is known of Desargues’s personal life.

Born into a family of wealth, Desargues certainly had

access to an excellent formal education and the free-

dom to explore scholarly interests. His first works in

mathematics, although pertaining to practical themes—

the construction of sundials, stone-cutting techniques,

and the use of perspective in art—were highly theoreti-

cal, densely written, and difficult to read, and conse-

quently were of little use to practitioners in the

respective fields. This may explain why the release of

his famous 1639 piece, written in equally obscure lan-

guage, was largely ignored. Only a few copies of the

text were printed, only one of which survives today.

Desargues’s famous theorem, the “perspective the-

orem” that bears his name, was published in 1648 by

French engraver and painter Abraham Bosse (1602–76)

in his treatise Manière universelle de Mr. Desargues

(General methods of Desargues) on the role of perspec-

tive in art. This single result provided a gateway to a

whole new approach to geometric thinking. Desargues

died in Lyon, France, in September 1661. (The exact

date of his death is not known.)

Desargues’s theorem Named after its discoverer,

French mathematician and engineer G

IRARD

D

ESAR

-

GUES

(1591–1661), this theorem states:

Suppose two triangles ABC and A′B′C′are posi-

tioned in two- or three-dimensional space so that

the lines joining the corresponding vertices A

and A′, Band B′, and Cand C′pass through a

common point (that is, so that the two triangles

are in “perspective from a point”). Then, if none

of the pairs of sides AB and A′B′, AC and A′C′,

or BC and B′C′is parallel, then the three points

of intersection of these pairs of sides lie on a

straight line. (That is, the two triangles are in

“perspective from a line.”)

The case of three dimensions is easiest to prove. Noting

that the two triangles cannot lie in parallel planes, one

can show that each point of intersection of a pair of

sides lies on the line of the intersection of the two

planes. The proof of the two-dimensional version of

the theorem is delicate. The converse of Desargues’s

theorem is also true:

If corresponding sides of two triangles have

intersections along the same straight line, then

the lines joining corresponding vertices pass

through a common point.

Desargues’s theorem played a key role in the devel-

opment of

PROJECTIVE GEOMETRY

. As noted, the theo-

rem does not hold if some pair of lines under

consideration turn out to be parallel. But Desargues

observed that this difficulty can be obviated if one were

to adjoin to space additional points, “points at infin-

ity,” where parallel lines do meet. This inspired Desar-

gues to develop a notion of geometry in which each

pair of points determines a unique line and, moreover,

each pair of lines determine (intersect at) a unique

point. In this system, the notions of “point” and “line”

play dual roles, leading to a general principle of duality

in this new projective geometry:

Interchanging the roles of “point” and “line”

in any theorem of projective geometry leads to

another statement that is also true in projective

geometry.

The dual of Desargues’s theorem is its converse.

See also

PERSPECTIVE

.

Descartes, René (1596–1650) French Geometry, Alge-

bra, Philosophy Born on March 31, 1596, in La Haye

(now Descartes), France, philosopher René Descartes is

remembered in mathematics for his 1637 influential

work La géométrie (Geometry), in which he introduced

fundamental principles for incorporating

ALGEBRA

into

the study of

GEOMETRY

, and vice versa. This work

paved the way for developing the notion of a coordinate

system and, although not featured in his work, C

ARTE

-

SIAN COORDINATES

are today named in his honor.

Descartes is also noted for his “rule of signs” for count-

ing the number of solutions to

POLYNOMIAL

equations,

and for promoting the use of symbols in algebraic

work. He advocated the use of letters to represent vari-

ables, suggesting the convention that letters first in the

124 Desargues’s theorem