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The First Appearance of the Plane
Keren Zaks

The main concern in ``La géométrie" is the construction of points that were solutions of geometric problems(p439). The first appearance of the ``plane" is in the first book of ``La géométrie, on page 310, with a diagram on the previous page.

It is stated: ``First, I suppose . . . and since so many lines are confusing, I may simplify matters by considering on of the given lines and one of those to be drawn, for example AB, & BC, as the principal lines, to which I shall try to refer all the others. Call the segment of the line AB between A and B, x, and call BC, y. Produce all the other given lines to meet these two . . .

``Now since all the angles of the triangle ARB are known, the ratio between the sides AB and BR is known. If we let the AB:BR=z:b, since AB=x, we have ; and since B lies between C and R we have . (When R lies between C and B, CR is equal to , and when C lies between B and R, CR is equal to ). . . the ratio between the sides CR and CD is determined. Calling this ratio z:c, since , we have . . . the distance from A to E is known. If we call this distance k, then EB=k+x; although EB=k-x when B lies between E and A, and . . . the ration of BE to BS is known. We call this ratio z:d. Then and . . . the ratio of CS and CF [is] z:e. Therefore, . Like wise . . . BG=l-x. . . z:f is known . . . Thus you see these lines can always be expressed by three terms, one of which consosts of the unknown quantity y multiplied by some known quantity; another consisting of the unknown quantity x multiplied or divided by some other known quantity; the third consisting of some known quantity.(Descartes, p309-312)."

(For the sake of simplicity, I shall exclude the next two pages of equations. They are in the text which is provided, and one need not have reading knowledge of French to understand them.)

What I have demonstrated is the way in which Descartes relates algebra to geometry, as well as how he uses his ``plane" to accomplish the task. It is ``plane as the nose on his face".