354 12. MODERN GEOMETRIESFIGURE 2. Left: AB = 1, so that BE = BCBD. Right: FG = 1,
so that GI = y/GH.the problem that Pappus reported unsolved in his day. It was in this context that
he formulated the idea of using two intersecting lines as a frame of reference, saying
that
since so many lines are confusing, I may simplify matters by consid-
ering one of the given lines and one of those to be drawn... as the
principal lines, to which I shall try to refer all the others. [Smith and
Latham, 1954, p. 29]The idea of using two coordinate lines is psychologically very close to the link-
ages illustrated in Fig. 1. In terms of Fig. 3, Descartes took one of the fixed lines
as a horizontal axis AB, since a line was to be drawn from point C on the locus
making a fixed angle è with AB. He thought of this line as sliding along AB and
intersecting it at point B, and he denoted the variable length AB by x. Then since
C needed to slide along this moving line so as to keep the proportions demanded
by the conditions of the locus problem, he denoted the distance CB by y. All the
lines were fixed except CB, which moved parallel to itself, causing ÷ to vary, while
on it y adjusted to the conditions of the problem. For each of the other fixed lines,
say AR, the angles ö, è, and ø will all be given, ö by the position of the fixed
lines AB and AR, and the other two by the conditions prescribed in the problem.
Since these three angles determine the shape of the triangles ADR and BCD, they
determine the ratios of any pair of sides in these triangles through the law of sines,
and hence all sides can be expressed in terms of constants and the two lengths ÷
and y. If the set of 2n lines is divided into two sets of ç as the 2n-line locus problem
requires, the conditions of the problem can be stated as an equation of the form
p{x,y) = q(x,y),
where ñ and q are of degree at most ç in each variable. The analysis was mostly
"clear and distinct."
Descartes argued that the locus could be considered known if one could locate
as many points on it as desired.^2 He next pointed out that in order to locate points
on the locus one could assign values to either variable ÷ and y, then compute the
value of the other by solving the equation.^3
Everyone who has studied analytic geometry in school must have been struck
at the beginning by how much clearer and easier it was to use than the synthetic
geometry of Euclid. That aspect of the subject is nicely captured in the words
the poet Paul Valery (1871-1945) applied to Descartes' philosophical method in
(^2) The validity of this claim is somewhat less than "clear and distinct."
(^3) This claim also involves a great deal of hope, since equations of degree higher than 4 were
unknown territory in his day.