On the origin of the word “ellipse”

Jan P. Hogendijk

Department of Mathematics, University of Utrecht, Netherlands

2017 Ancient Greek mathematics.

Period: 550 BC - ca. 250 CE Area: Greece, Southern Italy, Western Turkey, Egypt. Ancient Greek words that we still use.

“Mathematics” = ta math`emata,the things that can be learned

axiom, theorem, lemma

asymptote, hyperbola, ellipse, parabola The basic work: the Elements by Euclid of Alexandria (ca 300 BC). Axioms, definitions, proofs. High theoretical level. Written originally on papyrus, medieval copies on parchment. Activity of Greek citizens in their free time (“schole”). Elements, 10th century CE manuscript of parchment (> 10 sheep necessary). Euclid’s theory of ratios (which also works for irrational ratios) in “Book 5” of the Elements.

Book 5, Definition 4 Logon echein pros all`elamegeth`elegetai ha dunantai pollaplasiazomena all`el¯onhuperechein.

Book 5, Definition 5 En t¯oiaut¯oilog¯oimegeth`elegetai einai pr¯otonpros deuteron kai triton pros tetarton hotan ta tou pr¯otou kai tritou isakis pollaplasia t¯ontou deuterou kai tetartou isakis pollaplasi¯onkath’ hopoionoun pollaplasiasmon hekateron hekateron `ehama huperech`ei`ehama isa `ehama elleip`eil`efthentakatall`ela Euclid’s Greek text (tr. Heath)

Def. 4 “Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.”

Def. 5 “Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if equimultiples be taken of the first and third, and equimultiples of the second and fourth, whatever equimultiples they are, then the former equimultiples alike exceed, are alike equal to, or alike fall short of the latter equimultiples respectively taken in corresponding order.” Def. 5 A : B = C : D if for any integers m, n mA > nB → mC > nD mA = nB → mC = nD and mA < nB → mC < nD

Theorem: if A : B = C : D and C : D = E : F then A : B = E : F . (Proposition 11 of Book 5).

In modern notation

In modern terms: Def. 4. A and B can have a ratio if there are integers m, n such that mA > B and nB > A. In modern notation

In modern terms: Def. 4. A and B can have a ratio if there are integers m, n such that mA > B and nB > A.

Def. 5 A : B = C : D if for any integers m, n mA > nB → mC > nD mA = nB → mC = nD and mA < nB → mC < nD

Theorem: if A : B = C : D and C : D = E : F then A : B = E : F . (Proposition 11 of Book 5). The ellipse

Term ellipse introduced by Apollonius of Perga (Western Turkey, 200 BC) in his work Conics

The Conics consists of 8 Books, of which 4 have been preserved in Greek, books 5-7 in Arabic only. Book 8 is lost.

It is assumed that the reader knows (most of) Euclid’s Elements.

The exposition follows a strict ritual. Example of text and translation: Conics I:13, Greek

A

E Λ P Π M O ∆ ΘΞ

N

Z B Γ H K Example of text and translation: Conics I:13, Greek. If a cone is cut by a plane through the axis, and by another plane meeting each side of the axial triangle, being neither parallel to the base nor subcontrary, and let the plane containing the base of the cone meet the cutting plane in a straight line perpendicular either to the base of the axial triangle or to its rectilinear extension; then any parallel line drawn from the section of the cone (parallel) to the common section of the planes as far as the diameter of the section will be equal in square to a certain area applied to a certain straight line, such that the diameter of the section will bear to it the same ratio as the square on the line drawn from the vertex of the cone parallel to the diameter of the section as far as the base of the triangle bears to the rectangle contained by the intercepts made by it on the sides of the triangle; the (area) having (as its) breadth the intercept made by it on the diameter in the direction of the vertex of the section, and being deficient (Greek: elleipon) by a figure similar and similarly situated to the rectangle contained by the diameter and the line along which they are equal in square; and let such a section be called an ellipse. Step 2: the same in notation (see handout p. 319, taken from Greek Mathematical Works, tr. Ivor Thomas, vol. 2).

To understand this, note:

I conic section is intersection of oblique cone with a plane. An oblique cone has a circular base and a vertex not in the base plane.

I A “diameter” of a conic section with the correspondiung “ordinates” are defined in the definitions, see handout p. 287. (This is related to, but not the same as, a modern coordinate system)

I every conic section has (at least) one diameter, proved in Prop. 7, handout pp. 289-295.

I “subcontrary” section is the intersection of a cone with a plane not parallel to the base, which section nevertheless produces a circle. Name “ellipse” is related to its “equation”:

T1T2 diameter of the ellipse, QR an arbitrary ordinate. Put T1T2 = d, T1R = x, QR = y. There is a constant segment p = T1U1 perpendicular to T1T2 such that 2 p 2 y = px − d x = T1RVW , the rectangle along p = T1U1 with side x = T1R, which falls short by a rectangle WVSU1 similar to T1T2U2U1. Exercise can be of two levels.

Level 1: Try to understand Conics book 1, proposition 13 (Ellipse). Write a clear account of what Apollonius tries to do, you can use modern notation.

Level 2: Try to understand the transition to a new diameter, Conics book 1, proposition 50 (handout pp. 328 and further), but only for the ellipse. Write a clear account of what Apollonius tries to do, you can use modern notation.

You can use any literature you like! Level 2 is only for those of you who want hard math. This presentation can be downloaded at http://www.jphogendijk.nl/talks/ellipse.pdf