Polytropy

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

y² = ℓx(2d − x)/2d

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 − 2by/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.