On the origin of the word ellipse

Transcription

1 On the origin of the word ellipse Jan P. Hogendijk Department of Mathematics, University of Utrecht, Netherlands 2017

2 Ancient Greek mathematics. Period: 550 BC - ca. 250 CE Area: Greece, Southern Italy, Western Turkey, Egypt.

3 Ancient Greek words that we still use. Mathematics = ta mathèmata, the things that can be learned axiom, theorem, lemma asymptote, hyperbola, ellipse, parabola

4 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 ).

5 Elements, 10th century CE manuscript of parchment (> 10 sheep necessary).

6 Euclid s theory of ratios (which also works for irrational ratios) in Book 5 of the Elements. Book 5, Definition 4 Logon echein pros allèla megethè legetai ha dunantai pollaplasiazomena allèlōn huperechein. Book 5, Definition 5 En tōi autōi logōi megethè legetai einai prōton pros deuteron kai triton pros tetarton hotan ta tou prōtou kai tritou isakis pollaplasia tōn tou deuterou kai tetartou isakis pollaplasiōn kath hopoionoun pollaplasiasmon hekateron hekateron è hama huperechèi è hama isa è hama elleipèi lèfthenta katallèla

7 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.

8 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.

9 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).

10 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.

11 Example of text and translation: Conics I:13, Greek A O E Π M Λ P Θ Ξ N B Γ H Z K

12 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.

13 Step 2: the same in notation (see handout p. 319, taken from Greek Mathematical Works, tr. Ivor Thomas, vol. 2). To understand this, note: 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. A diameter of a conic section with the correspondiung ordinates are defined in the definitions, see handout p (This is related to, but not the same as, a modern coordinate system) every conic section has (at least) one diameter, proved in Prop. 7, handout pp subcontrary section is the intersection of a cone with a plane not parallel to the base, which section nevertheless produces a circle.

14 Name ellipse is related to its equation : T 1 T 2 diameter of the ellipse, QR an arbitrary ordinate. Put T 1 T 2 = d, T 1 R = x, QR = y. There is a constant segment p = T 1 U 1 perpendicular to T 1 T 2 such that y 2 = px p d x 2 = T 1 RVW, the rectangle along p = T 1 U 1 with side x = T 1 R, which falls short by a rectangle WVSU 1 similar to T 1 T 2 U 2 U 1.

15 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.

16 This presentation can be downloaded at