*Cogito ergo sum* drawn on the Cartesian plane.
Cogito ergo sum drawn on the Cartesian plane.Euler’s Eyes Collection

Here at Euler’s Eyes we like to read “I think, therefore I am” as a little machine. The thought hears something (in Latin, audire) and speaks another (fabulari). The rule that binds what What you hear is the function.

An example: “say what you hear twice”. Empty parentheses act as a waiting ear a number:

$$f = 2\,(\;\;) \qquad\text{hoje escrevemos}\qquad f(x) = 2x.$$
The parentheses ( ) as an ear: the function expects to hear a number.
The parentheses ( ) as an ear: the function expects to hear a number.Euler’s Eyes Collection
*Dominus*, *functus* and *graphiko's*: the domain, the function and the trace it leaves.
Dominus, functus and graphiko’s: the domain, the function and the trace it leaves.Euler’s Eyes Collection

Try the machine with three rules: $f = 2(\quad)$, $f = (\quad) - 2$ and $f = [(\quad) - 1]\cdot[(\quad) - 3]$. Each number heard becomes a point $(x;\ f(x))$, and the set of points forms the graph, the graphiko’s of the rule:

Feed the machine with numbers. The line f = ( ) − 2 cuts the axis at 2; the parabola of [( ) − 1]·[( ) − 3] cuts at 1 and 3.
*Mathema graphiko's*: the trace of f = ( ) − 2, a straight line that cuts the axis at 2.
Mathema graphiko’s: the trace of f = ( ) − 2, a straight line that cuts the axis at 2.Euler’s Eyes Collection
The trace of f = [( ) − 1]·[( ) − 3], a parabola that cuts the axis at 1 and 3.
The trace of f = [( ) − 1]·[( ) − 3], a parabola that cuts the axis at 1 and 3.Euler’s Eyes Collection

All topics by René Descartes · ← Descartes and the idea of ​​function · Descartes and the problem of cannons →

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