alphabet should refer to known quantities and letters in
the latter part of the alphabet unknown quantities. (For
instance, we today, without question, interpret the
equation ax+b=0 as one containing a variable x, with a
and bassumed to be known quantities.) Descartes also
developed the index notation for
EXPONENTS
: x2, x3,
and the like. As a philosopher, Descartes was interested
in exploring the deepest underlying principles of all of
scientific knowledge. He felt that mathematics lay at the
base of all understanding.
After receiving a law degree from the University of
Poitiers in Paris in 1616, Descartes traveled to Holland
to enlist in the military school at Breda to study mathe-
matics and mechanics. Duty with the army took him
across Europe for a number of years, but in 1628
Descartes returned to Holland and began a comprehen-
sive treatise on physics, optics, and celestial mechanics.
He decided, however, not to publish this work after
hearing that G
ALILEO
G
ALILEI
(1564–1642) was con-
demned to house arrest for espousing modern scientific
thought. Descartes later modified the piece and pub-
lished a treatise on science in 1637 in which La
géométrie appeared as an appendix.
In 1644 Descartes published Principia philo-
sophiae, a comprehensive tome exploring all aspects in
scientific investigation and knowledge. Divided into
four parts—The Principles of Human Knowledge, The
Principles of Material Things, Of the Visible World,
and The Earth—Descartes’s work argued that mathe-
matics lies at the foundation of all thinking and that
all studies of nature and of the universe can be
reduced to the mathematical principles of mechanics.
This work was extremely influential, but the specific
details of some of the theories Descartes outlined in it
were problematic. For instance, Descartes believed
that forces, such as gravitational forces, could not be
transmitted without some kind of ambient medium.
Thus, he was forced to conclude that vacuums do not
exist, and that the entire universe is filled with matter.
He developed a “vortex theory” in which he argued
that the planets and the Sun are carried through space
by a system of vortices in the ambient medium. This
theory of vortices was accepted for nearly 100 years
in France until S
IR
I
SAAC
N
EWTON
(1642–1727)
demonstrated mathematically that such a dynamical
system is impossible.
In 1649 Descartes moved to Stockholm to accept a
position to tutor Queen Christina of Sweden in mathe-
matics. The cold climate did not suit Descartes well,
and he died a few months later on February 11, 1650,
of pneumonia.
Descartes instigated a philosophical shift as to how
mathematics and analytical thought are utilized in the
role of scientific investigation. Rather than shape scien-
tific theories around what is observed experimentally,
Descartes argued to identify fundamental “self-evident”
principles first and to use logical reasoning to under-
stand the causes behind experimental phenomena. His
desire to carry the topic of geometry into physics as a
part of this process is still felt today: most every branch
of modern physics is described in geometric terms.
Descartes’s rule of signs Discovered by philosopher
and mathematician R
ENÉ
D
ESCARTES
(1596–1650), the
“rule of signs” gives a bound on the maximum number
Descartes’s rule of signs 125
René Descartes, an eminent scholar of the 17th century, was the
first to apply methods from algebra to solve problems in geometry.
The field of analytic geometry was born. (Photo courtesy of
Topham/The Image Works)