Descartes: A Biography

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In Search of a Career (–) 

Descartes seems to have spent the early months ofin studying

mathematics and, especially, in reflecting on ways to open up the dis-

cipline to new techniques. Descartes’ mathematical training was proba-

blyelementary, and was based for the most part on the publications of

Christopher Clavius (–), which included a well-known edition

of Euclid’sElements.Despite the recent applications of algebraic meth-

ods to geometrical problems, due mostly to Franc ̧ois Viete (` –),

geometrical methods of construction were still limited to what could be

realized with only straight lines and circles.However, there were already

some intimations of the Cartesian project in a startling claim at the con-

clusion of Vi`ete’sIntroduction to the Analytic Art(). ‘The analytic

art...claims for itself the greatest problem of all, which is to leave no

problem unsolved.’Descartes wrote to Beeckman with similarly extrav-

agant ambitions in March.

Indeed, to tell you openly what I plan, I do not want to construct a LullianBrief Artbut

a completely new science by which all questions that can be raised about any kind of

quantity, either continuous or discrete, may be solved by a general method. However,

each one must be solved in accordance with its own nature. In arithmetic, for example,

some questions can be resolved using rational numbers, some only by surd numbers,

whereas others can be imagined but cannot be solved. Likewise, I hope to prove, in the

case of continuous quantity, some problems that can be solved by using only straight

lines or circles; other problems can be solved only by using other curved lines which,

however, result from a single motion and can therefore be traced by the new compasses,

which I think are no less certain and geometrical than the common compass by which

circles are traced. Finally, there are problems that can be solved only by curved lines

traced by separate motions that are not subordinate to one another; such curves are

certainly merely imaginary, such as the relatively well-known quadratrix. I think that it

is impossible to imagine anything that cannot be solved by at least those lines. However,

I hope to prove in due course which questions can be solved in one way or another,

or not at all, so that there would be almost nothing left to be discovered in geometry.

That is indeed an infinite task, and not for a single person. It is incredible rather than

ambitious, but I have seen some light through the boundless obscurity of this science

bywhich I think I can dispel the most dense clouds. (x.–)

Here Descartes tries to provide a complete classification of all mathe-

matical problems, borrowing evidently from a tradition that bequeathed

unresolved questions from one generation to another. In the case of arith-

metic, he divided problems into equations whose solutions were ()ration-

al numbers, () surd numbers, and ()complex numbers (i.e., those that

involve the square root of a negative number, such as

√

−). In the case of