1. Geometry of the Ellipse

Menaechmus (Μέναιχμος, − 375 to − 325), was a Greek mathematician and member of Plato’s Academy. He is credited by Eratosthenes with the discovery of the conic sections, objects which arose in the so-called Delian problem, also known as doubling the cube (given a cube of volume a ³, construct another of volume 2a ³). Menaechmus sought two numbers x and y such that \(a/x = x/y = y/2a\), which leads to x ³ = 2a ³, by finding the intersections of a parabola (\(y = {x}^{2}/a\)) and a hyperbola (\(y = 2{a}^{2}/x\)).

The word “conic” comes from the Greek kônikos, an adjective derived from kônos (ὁ κω̃νος, ου), meaning “pine cone”, the fruit of certain conifers.

A direct application of this definition can be used to draw an ellipse in what is sometimes known as the gardener’s method. Two stakes are stuck in the ground, some distance d apart. A non-stretch string of length ℓ > d is attached with one end at each stake. If we then hold a third stake against the string and move it in such a way that the string is always taut, this stake will trace out an ellipse with focal points at the two fixed stakes. The major axis is equal to ℓ and the eccentricity is d∕ℓ.

The term was introduced into scientific Latin (New Latin) by Kepler in 1603, from the Latin word focus, i, which means “a place where fire is made”. A pencil of light rays passing through one focus will converge at the other, and could start a fire there. This is indeed the explanation for the name “focus” (see Fig. 1.11).

The word “center” comes from Old French centre, which comes from the Latin centrum, i, meaning “compass point”, “center of a circle”, or “midpoint of an ensemble”. The word derives from the Greek kentron, τὸ κέντρον, ου, a “goad”, from the verb κεντέω, meaning “to prick or sting”. This is a rather rare example of a technical country word which became scientific and later assumed a very general meaning. The Latin word spread to all the Latin languages and many non-Latin ones, too.

Appolonius used the following three terms to characterise conic sections, inspired by the language of the Pythagoreans:

The first term comes from εκ, meaning “in”, and the verb λείπειν, meaning “to leave or neglect”. The last two derive from the verb βάλλειν, meaning “to throw”. The word “parabola” should not be associated with the idea of throwing something in the sense of throwing a projectile. This link between “parabola” and “throw” was described by Galileo, but was unknown to the Ancient Greeks. It is interesting to note that, taken in this order, the three terms “lacking”, “comparable”, and “in excess” associated with the conic sections correspond to the values of the eccentricity as compared to unity. In fact, Apollonius introduced the length p, the parameter of the ellipse, and calculated the areas of squares and rectangles based on this length. With modern notation, taking the focal axis as the axis Ox, with Oy perpendicular to it and the origin O at the apex of the conic section, we obtain the following relation for the parabola: which means that the area of a square of side y is equal to that of a rectangle of dimensions 2p × x. For the two other conic sections, we have:

In comparison with the parabola, it is thus the quantity (p∕a)x ² which lacks for the ellipse and which is in excess for the hyperbola. The name for the conic sections was introduced by Kepler, Desargues, and Descartes, first in Latin, then in the modern European languages.

This theorem, proven in 1822, is also called the Dandelin–Quételet theorem, after the two Belgian mathematicians Germinal Dandelin (1794–1847) and Adolphe Quételet (1796–1841).

To illustrate this property, V.I. Arnold suggests the following experiment in one of his books: Drop a droplet of tea close to the center of the cup. The waves will come together at the symmetric point. This is due to the definition of the foci of the ellipse, which implies that waves coming from one focus of the ellipse will converge at the other.

This angular parameter u corresponds to the parametric (or reduced) latitude u in geodesy or Kepler’s eccentric anomaly E in astronomy, as we shall see later on.

Such integrals are classified into three kinds. Those of the second kind have the form with parameter k such that 0 < k ² < 1. These elliptic integrals are said to be incomplete when ϕ is arbitrary. When \(\phi =\pi /2\), one speaks of complete elliptic integrals.

Srinivâsa Aiyangâr Râmânujan (1887–1920) was a highly original Indian mathematician, considered as a genius in his field. He taught himself mathematics using a compendium of 6,000 theorems, most of which were given without proof. From the age of 17, and in particular during his time in England between 1913 and 1919, he established hundreds of formulas which he noted down without proof in his notebooks. His intuition and memory were astonishing. He worked on number theory, elliptic integrals, Bernoulli numbers, and so on. Ramanujan established the values of π and e using continued fractions, series, or extremely concise formulas.