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 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. The above argument opens Descartes's Géométrie Image Size: 240x320 480x640 960x1280 1440x1920 1920x2560 