After Descartes gave geometry the power of algebra in 1637, a purely geometrical theorem of Apollonius that is both useful and beautiful was forgotten. This is what I conclude from looking at texts from the seventeenth century on.

Naturally there are versions of the theorem for the parabola and hyperbola; but when a point *P* moves along an ellipse, as in the animation above (which you can probably enlarge by clicking; or if there is a problem, as one person experienced, try the still version below), the theorem is that the blue triangle (which has rising rules from left to right) is always equal either to the red trapezoid (with falling rules), when it *is* a trapezoid, or to the red triangle less the green triangle (cross-hatched). In letters, the triangle *XPY* is always equal to the trapezoid *VXY*E**, whose sides may cross one another, but which can always be understood as the triangle *VKE** less the triangle *KY*X*. In brief,

*XPY* = *VXY*E**,

though our sign of equality dates only from Robert Recorde’s *Whetstone of Witte* of 1557. Descartes uses a different symbol for equality; Apollonius, none.

Apollonius applies the foregoing theorem as follows. By adding quadrilateral *XYX*Y** to both sides of the equation, and using the equality of triangles *VFE* and *E*FD*, we obtain

*Y*PX** = *EYX*D*,

as suggested in the animation below. In algebraic or “analytic” terms, we have effected a change of coordinates and shown that, with respect to the new coordinates, the ellipse has the same equation as before. The original coordinate axes were *VK* and *VE**; the new, *DK* and *DE*.

Descartes’s 1637 *Géométrie*

- established the convention of naming lengths with single minuscule letters and
- justified making algebraic computations with these letters.

In *De Sectionibus Conicis, Novo Methodo Expositis, Tractatus* of 1655, John Wallis used Cartesian methods to establish the change-of-basis theorem that we have just proved geometrically. According to Morris Kline in *Mathematical Thought from Ancient to Modern Times* (1972), Wallis “was probably the first to use equations to prove properties of the conics.” In my judgment, Wallis’s argument concerning change of coordinates is tedious and opaque. It is easier to follow the argument of Jan de Witt in *Elementa Curvarum Linearum, Liber Primus* of 1659, though it helps that an English translation, by Albert W. Grootendorst, was published in 2000. I have looked at Ian Bruce’s translation of Euler’s *Introductio in Analysin Infinitorum* of 1748; at the 1773 English translation (*Geometrical Treatise of the Conic Sections*) of Hugh Hamilton’s *De Sectionibus Conicis, Tractatus Geometricus* of 1758; and at more recent books. Nobody uses *geometry*—areas—as Apollonius does.

The best algebraic proof of change of coordinates in an ellipse is by means of the affine transformation called an elliptical rotation. In his commentary on Apollonius, Boris Rosenfeld suggests that this rotation (he calls it a “turn”) is what Apollonius’s theorem amounts to. In a sense this is true, but it ignores how radically Descartes has changed our way of thinking about mathematics.

We can translate the first equation above into the modern form

*y*^{2} = *ℓx*(2*d* − *x*)/2*d*

for some *ℓ* (the *latus rectum* or upright side), where *d* is the length of *VK*. Then we can verify that, if satisfied by (*a*, *b*), the equation is fixed by the transformation

*x*′ = *cx*/*d* − 2*by*/*ℓ* + *a*,

*y*′ = −*bx*/*d* − *cy*/*d* + *b*,

where *c* = *d* − *a*. However, these computations have nothing obvious to do with our earlier computations with areas.

I return to Apollonian modes of thinking, to fill in the details of why the earlier geometric equations hold. Our ellipse has center *K*, and *VW* is a diameter. As a diameter, *VW* bisects all chords that are parallel to the tangent *VE**. In particular, *PX* is half of such a chord and is thus an ordinate with respect to the given diameter. This ordinate cuts off from the diameter two abscissas, *VX* and *XW*. Also *DM* is an ordinate, and *KV* is extended to *E* so that *KE* is a third proportional to *KM* and *KV* (that is, as *KM* is to *KV*, so is the latter to *KE*). Then *PY* is drawn parallel to *DE*, which will turn out to be tangent to the ellipse.

The properties of the ellipse are derived initially from the cone from which it is cut, as in the animation above (or the still version below), with apex *A* and base the circle with diameter *BC*. The chord *DD′* is at right angles to, and is therefore bisected by, the diameter. Hence the square on the ordinate *DM* is equal to the rectangle bounded by the abscissas *BM* and *MC*. The same is true in a section of the cone parallel to the base; thus the square on *PX* is equal to the rectangle bounded by *QX* and *XR*. By the similarity of triangle *QXV* to *BMV* and *XRW* to *MCW*, the ratio of the squares on *PX* and *DM* is the product of the ratios of *VX* to *VM* and of *XW* to *MW*.

Now we are in the plane of the ellipse. The square on *PX* varies jointly as the abscissas *VX* and *XW*. But the square also varies as the triangle *XPY*, and the trapezoid *VXY*E** varies jointly as the abscissas *VX* and *MX*. In case *P* coincides with *D*, the triangle is equal to the trapezoid. Therefore it is always equal.

The algebraic equation involving the *latus rectum* is Proposition 13 of Book I of the *Conics* of Apollonius of Perga; the consequence concerning ratios, Proposition 21. The equation of triangle and trapezoid is Proposition 43; the consequence concerning change of basis, Proposition 50. The equation of triangles used to derive this consequence is later isolated as Proposition 1 of Book III.

For the record, I made the animations as follows. I drew the still diagrams in `LaTeX` with the `pstricks`, `pst-3dplot`, and `pst-eucl` packages. I included a parameter for an angle. Then, with the `\multido` command, I could at once make a `pdf` file of 360 diagrams, one for each angle, each diagram on a new page. In `The Gimp` program, I imported the `pdf` file as layers, then exported as a gif file. (The procedure seems to have changed a bit since “Self-similarity.”)

When I first posted this article, *VE** in the 2-d animations was accidentally equal to what is labelled as *KL* in the 3-d animation. Since one should not assume that this will always be so, I have made it not so.

Still versions of the animations added July 31, 2019.