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DESCARTES, RENÉ DU PERRON (b. La Haye,
Touraine, France, 31 March 1596; d. Stockholm,
Sweden, 11 February 1650), natural philosophy, scientific
method, mathematics, optics, mechanics, physiology.
DESCARTES: Mathematics and Physics.
Mathematics.
other than that art which they call by the barbarous
name of “algebra,” if only it could be so disentangled
from the multiple numbers and inexplicable figures that
overwhelm it that it no longer would lack the clarity
and simplicity that we suppose should obtain in a true
mathematics.5
Descartes expressed his second programmatic goal
in a letter to Beeckman in 1619; at the time it appeared
to him to be unattainable by one man alone.
He envisaged “an entirely new science,”
... by which all questions can be resolved that can be
proposed for any sort of quantity, either continuous or
discrete. Yet each problem will be solved according to
its own nature, as, for example, in arithmetic some
questions are resolved by rational numbers, others only
by irrational numbers, and others finally can be imagined
but not solved. So also I hope to show for continuous
quantities that some problems can be solved by
straight lines and circles alone; others only by other
curved lines, which, however, result from a single motion
and can therefore be drawn with new forms of
compasses, which are no less exact and geometrical, I
think, than the common ones used to draw circles; and
finally others that can be solved only by curved lines
generated by diverse motions not subordinated to one
another, which curves are certainly only imaginary (e.g.,
the rather well-known quadratrix). I cannot imagine
anything that could not be solved by such lines at least,
though I hope to show which questions can be solved
in this or that way and not any other, so that almost
nothing will remain to be found in geometry.6
Descartes sought, then, from the beginning of his
research a symbolic algebra of pure quantity by which
problems of any sort could be analyzed and classified
in terms of the constructive techniques required for
their most efficient solution. He took a large step
toward his goal in the Rules and achieved it finally
in the Géométrie.
Descartes began his task of “purifying” algebra by
separating its patterns of reasoning from the particular
subject matter to which it might be applied.
Whereas cossist algebra was basically a technique for
solving numerical problems and its symbols therefore
denoted numbers, Descartes conceived of his “true
mathematics” as the science of magnitude, or quantity,
per se. He replaced the old cossist symbols with
letters of the alphabet, using at first (in the Rules) the
capital letters to denote known quantities and the
lowercase letters to denote unknowns, and later (in
the Géométrie) shifting to the a,b,c; x,y,z
notation
still in use today. In a more radical step, he then
removed the last vestiges of verbal expression (and
the conceptualization that accompanied it) by replacing
the words “square,” “cube,” etc., by numerical
superscripts. These superscripts, he argued (in rule
XVI), resolved the serious conceptual difficulty posed
by the dimensional connotations of the words they
replaced. For the square of a magnitude did not differ
from it in kind, as a geometrical square differs from
a line; rather, the square, the cube, and all powers
differed from the base quantity only in the number
of “relations” separating them respectively from a
common unit quantity. That is, since
1:x = x:x2 =
x2:x3 = ...
(and, by Euclid V, ratios obtain only among homogeneous
quantities), x3 was linked to the unit magnitude
by three “relations,” while x was linked by only one.
The numerical superscript expressed the number of
“relations.”
While all numbers are homogeneous, the application
of algebra to geometry (Descartes's main goal
in the Géométrie) required the definition of the six
basic algebraic operations (addition, subtraction,
multiplication, division, raising to a power, and extracting
a root) for the realm of geometry in such a
way as to preserve the homogeneity of the products.
Although the Greek mathematicians had established
the correspondence between addition and the geometrical
operation of laying line lengths end to end
in the same straight line, they had been unable to
conceive of multiplication in any way other than that
of constructing a rectangle out of multiplier and multiplicand,
with the result that the product differed in
kind from the elements multiplied. Descartes's concept
of “relation” provided his answer to the problem:
one chooses a unit length to which all other lengths
are referred (if it is not given by the data of the
problem, it may be chosen arbitrarily).
FIGURE 1
Then, since
1:a = a:ab, the product of two lines a and
b is constructed
by drawing a triangle with sides 1 and a; in
a similar triangle, of which the side corresponding to
1 is b, the other side will be ab, a line length. Division
and the remaining operations are defined analogously.
As Descartes emphasized, these operations do not
make arithmetic of geometry, but rather make possible
an algebra of geometrical line segments.
