On The Construction of Parabolas

Question

I am not aware of any other geometry other than Euclidean. Correction: I am aware of other geometries, however I am not aware of their axioms, theorems, or applications. I only know that they disregard the 5th, parallel postulate; that is all. I have, however, considered the construction of the parabola using only a rule and compass (Euclidean postulates). I soon found it quite the cumbersome activity. Is seems impossible to construct a parabola, for reasoning coordinate geometrically: a parabola is a figure that has the property that it's $y$ coordinate continually increases quadratically while it's $x$ coordinate increases linearly. Such a property doesn't seem constructible using a rule (which could construct loci with both coordinates increasing linearly) and a compass (which could construct loci with both coordinates increasing quadratically). Does this mean that the parabola doesn't exist in a Euclidean plane? Same goes for the hyperbola, or other loci where one coordinate increases at the rate $= n$ and the other coordinate at the rate $= m ≠ n$. Thank you in advance.

Answer 1

In Conics, I, 11-13 Apollonius constructs and defines the parabola, hyperbola, and ellipse. They are constructed by cutting a cone with a plane. A cone is generated (Definitions 1-2) by moving one end of a line segment around the circumference of a circle while the other end of the segment, not in the plane of the circle, remains fixed. This may seem un-Euclidean, but then Euclid generates a sphere (Elements, XI, Def. 14) by rotating a semicircle once around its fixed diameter, so Apollonius seems to be within that tradition. As for cutting a cone with a plane, however, it does seem Apollonius requires a postulate not found in Euclid—a “knife” in addition to his rule and compass.

Addendum: Even confining ourselves to a plane, however, it seems possible to construct a portion of a conic section, continuously and not just point by point.

In his 1637 La Geometrie (tr. D.E. Smith & M.L. Latham, Dover, 1954), Rene Descartes explains the instrument and figure below: "Suppose the curve $EC$ to be described by the intersection of the ruler $GL$ and the rectilinear figure $CNKL$, whose side $KN$ is produced indefinitely in the direction of $C$, and which, being moved in the same plane in such a way that its side $KL$ always coincides with some part of the line $BA$ (produced in both directions), imparts to the ruler $GL$ a rotary motion about $G$ (the ruler being hinged to the figure $CNKL$ at $L$. If I wish to find out to what class this curve belongs, I choose a straight line, as $AB$, to which to refer all its points, and in $AB$ I choose a point $A$ at which to begin the investigation....Then I take an arbitrary point, as $C$, at which we will suppose the instrument applied to describe the curve. Then I draw through $C$ the line $CB$ parallel to $GA$. Since $CB$ and $BA$ are unknown and indeterminate quantities, I shall call one of them $y$ and the other $x$. To find the relation between these quantities I must consider also the known quantities which determine the description of the curve, as $GA$, which I shall call $a$; $KL$, which I shall call $b$; and $NL$ parallel to $GA$. which I shall call $c$."

From this, in which we can see the beginnings of coordinate geometry, Descartes derives, by similar triangles, the proportion$$\frac{y}{\frac{b}{c}y-b}=\frac{a}{x+\frac{b}{c}y-b}$$Cross-multiplication gives$$\frac{ab}{c}y-ab=xy+\frac{b}{c}y^2-by$$or$$y^2=cy-\frac{cx}{b}y+ay-ac$$Descartes says, though he does not explain, "From this equation we see that the curve $EC$ belongs to the first class, it being, in fact, a hyperbola."

Descartes' instrument here seems neither ruler nor compass but a sort of combination of the two. He argues, however, (pp. 40-44) that the true distinction between geometric and merely "mechanical" curves is that the former are "described by a continuous motion or by several successive motions, each motion being completely determined by those which precede," while the latter, e.g. the spiral or quadratrix, "must be conceived as described by two separate movements whose relation does not admit of exact determination." Moreover, the former can be expressed by an algebraic equation in two unknowns while the latter cannot.