Taming the Unknown by Parshall Karen Hunger Katz Victor J

Author:Parshall, Karen Hunger, Katz, Victor J.
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2014-03-14T16:00:00+00:00

Figure 10.3.

He thus began with a single axis NZM and a linear equation (figure 10.3). (Note that he adhered closely to Viète’s notation: vowels for unknowns; consonants for knowns; “in” to represent “times”; and no symbol for “equals.”) To show that his equation, written in modern notation as dx = by, represents a straight line, he used basic geometry:

Indeed, we have B to D as A to E. Therefore, the ratio A to E is given, as is also the angle at Z. So the triangle NIZ is given in species and the angle NIZ is also given. But the point N and the line NZ are given in position. Therefore the line NI is given in position. The synthesis is easy.

This was “easy” in the sense that it was straightforward to complete the argument by showing that any point T on NI determines a triangle TWN with NW : TW = B : D.

Although the basic notions of modern analytic geometry are apparent in Fermat’s description, his ideas differ somewhat from those now current. First, Fermat used only one axis. He thought of a curve not as made up of points plotted with respect to two axes but as generated by the motion of the endpoint I of the variable line segment ZI as Z moves along the given axis. Fermat often took the angle between ZI and ZN as a right angle, although nothing compelled that choice. Second, for Fermat as for Viète and even Harriot (at least most of the time), the only proper solutions of algebraic equations were positive. Thus, Fermat’s “coordinates” ZN and ZI—solutions to his equation D in A equals B in E—represented positive numbers, and he drew only the ray emanating from the origin into the first quadrant.

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