= Solution
{c}
The figure illustrates <Pierre de Fermat>'s use of <adequality> to construct a <tangent> to a <parabola>. Reading the local exam image is decisive: $D$ is the <vertex of a parabola>, $DC$ lies on its <axis of symmetry>, $B$ lies on the <parabola>, and the proposed <tangent> $BE$ meets the extended axis at $E$. The horizontal through the auxiliary point $O$ meets the axis at $I$; $O$ lies on $BE$, outside the <parabola>. In particular, $O$ must not be treated as a second point of the <parabola>. The target is the <subtangent> $CE$. The diagram and this construction occur in the tangent discussion associated with <Varia opera mathematica by Pierre de Fermat>. The 1679 book was posthumous: its date does not date the invention of the method.<Varia opera mathematica by Pierre de Fermat>[1679 title page and bibliographical record, ETH-Bibliothek]{ref}<Fermat's rule for maxima and minima by the Garden of Archimedes>[Tangent construction, with the same B, C, D, E, I, O labels; the method circulated in the 1630s]{ref}
Write $d=CD>0$, $a=CE>0$, and $e=CI$, with $I$ between $C$ and $D$. These are lengths, and the increment $e$ is distinct from the labelled point $E$. Then $DI=d-e$ and $EI=a-e$. The <parabola>'s characteristic relation makes the square of an ordinate proportional to its distance from the <vertex of a parabola>. At height $I$, the actual ordinate of the <parabola> is shorter than $OI$. Hence
$$
\frac{d}{d-e}>\frac{BC^2}{OI^2}
=\frac{a^2}{(a-e)^2},
$$
where the equality follows from <similar triangles> $BCE$ and $OIE$.
The method of <adequality> replaces the comparison by a relation used to extract the condition of contact:
$$
d(a-e)^2\mathrel{\sim}a^2(d-e).
$$
Expanding and cancelling $da^2$ gives $-2dae+de^2\mathrel{\sim}-a^2e$. Divide by the nonzero increment $e$, then suppress the remaining term containing $e$. The result is $a^2=2da$, so, since $a>0$,
$$
\boxed{CE=2CD.}
$$
Thus $DE=DC$: the intersection with the axis lies as far above the <vertex of a parabola> as $C$ lies below it. This is a concrete construction, not just a rule for computing a numerical slope.
A modern check clarifies why the calculation succeeds without pretending that <Pierre de Fermat> used a modern <limit>. Put $D=(0,0)$, let $y$ increase down the axis, and write the <parabola> as $x^2=ky$, $k>0$. For $B=(b,d)$, with $b^2=kd$, the proposed <tangent> through $E=(0,-d)$ is $y=2bx/k-d$. Substitution gives
$$
x^2-ky=(x-b)^2.
$$
It meets the <parabola> with a <double root> at $B$, and for every other point its $y$ coordinate is less than that of the <parabola>. This also verifies the strict comparison used for $O$.
The significance is \b[a systematic algebraic condition for geometric contact]. Subtracting neighbouring expressions, dividing by an increment, and discarding its remaining terms anticipates <differentiation>. But the historical question is precisely what licenses the last step. The symbol $\sim$ above marks <adequality>, not ordinary equality at a finite displacement. Treating the construction as an already stated <epsilon-delta definition of a limit> would conceal the methodological innovation and its unresolved explanatory burden.
Back to article page
