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René Descartes (1596-1650) |
René
Descartes has been dubbed the "Father of Modern Philosophy", but he
was also one of the key figures in the Scientific Revolution of the 17th
Century, and is sometimes considered the first of the modern school of
mathematics.
As
a young man, he found employment for a time as a soldier (essentially as a
mercenary in the pay of various forces, both Catholic and Protestant). But,
after a series of dreams or visions, and after meeting the Dutch philosopher
and scientist Isaac Beeckman, who sparked his interest in mathematics and the
New Physics, he concluded that his real path in life was the pursuit of true
wisdom and science.
Back
in France, the young Descartes soon came to the conclusion that the key to
philosophy, with all its uncertainties and ambiguity, was to build it on the
indisputable facts of mathematics. To pursue his rather heretical ideas
further, though, he moved from the restrictions of Catholic France to the more
liberal environment of the Netherlands, where he spent most of his adult life,
and where he worked on his dream of merging algebra and geometry.
In
1637, he published his ground-breaking philosophical and mathematical treatise
"Discours de la méthode" (the “Discourse on Method”), and one of its
appendices in particular, "La Géométrie", is now considered a
landmark in the history of mathematics. Following on from early movements
towards the use of symbolic expressions in mathematics by Diophantus, Al-Khwarizmi and François Viète, "La
Géométrie" introduced what has become known as the standard algebraic
notation, using lowercase a, b and c for known quantities and x, y and z for unknown
quantities. It was perhaps the first book to look like a modern mathematics
textbook, full of a's and b's, x2's, etc.
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Cartesian Coordinates |
It
was in "La Géométrie" that Descartes first proposed that each point
in two dimensions can be described by two numbers on a plane, one giving the
point’s horizontal location and the other the vertical location, which have
come to be known as Cartesian coordinates. He used perpendicular lines (or
axes), crossing at a point called the origin, to measure the horizontal (x) and vertical (y) locations, both
positive and negative, thus effectively dividing the plane up into four
quadrants.
Any
equation can be represented on the plane by plotting on it the solution set of
the equation. For example, the simple equation y = x yields a straight line linking together
the points (0,0), (1,1), (2,2), (3,3), etc. The equation y = 2x yields a straight line linking together
the points (0,0), (1,2), (2,4), (3,6), etc. More complex equations involving x2, x3, etc, plot various
types of curves on the plane.
As
a point moves along a curve, then, its coordinates change, but an equation can
be written to describe the change in the value of the coordinates at any point
in the figure. Using this novel approach, it soon became clear that an equation
like x2 + y2 = 4, for example, describes a circle; y2 - 16x a curve called a parabola; x2⁄a2 + y2⁄b2 = 1 an ellipse; x2⁄a2 - y2⁄b2 = 1 a hyperbola;
etc.
Descartes’
ground-breaking work, usually referred to as analytic geometry or Cartesian
geometry, had the effect of allowing the conversion of geometry into algebra
(and vice versa). Thus, a pair of simultaneous equations could now be solved
either algebraically or graphically (at the intersection of two lines). It
allowed the development of Newton’s and Leibniz’s subsequent discoveries of calculus.
It also unlocked the possibility of navigating geometries of higher dimensions,
impossible to physically visualize - a concept which was to become central to
modern technology and physics - thus transforming mathematics forever.
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Descartes' Rule of Signs |
Although
analytic geometry was far and away Descartes’ most important contribution to
mathematics, he also: developed a “rule of signs” technique for determining the
number of positive or negative real roots of a polynomial; "invented"
(or at least popularized) the superscript notation for showing powers or
exponents (e.g. 24 to show 2 x 2 x 2 x 2); and re-discovered
Thabit ibn Qurra's general formula for amicable numbers, as well as the
amicable pair 9,363,584 and 9,437,056 (which had also been discovered by
another Islamic mathematician, Yazdi, almost a
century earlier).
For
all his importance in the development of modern mathematics, though, Descartes
is perhaps best known today as a philosopher who espoused rationalism and
dualism. His philosophy consisted of a method of doubting everything, then
rebuilding knowledge from the ground, and he is particularly known for the
often-quoted statement “Cogito ergo sum”(“I think, therefore I am”).
He
also had an influential rôle in the development of modern physics, a rôle which
has been, until quite recently, generally under-appreciated and
under-investigated. He provided the first distinctly modern formulation of laws
of nature and a conservation principle of motion, made numerous advances in
optics and the study of the reflection and refraction of light, and constructed
what would become the most popular theory of planetary motion of the late 17th
Century. His commitment to the scientific method was met with strident
opposition by the church officials of the day.
His
revolutionary ideas made him a centre of controversy in his day, and he died in
1650 far from home in Stockholm, Sweden. 13 years later, his works were placed
on the Catholic Church's "Index of Prohibited Books".
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