Tag Archives: Apollonius

Three more mathematical documents in the Neues Museum

Photography by Carlos Dorce

The Berlin Papyrus 6619 s not the only mathematical “paper” of the Neues Museum of Berlin because we can see two more documents on exhibition. The first of them is a Greek papyrus (139 AD) with some geometrical problems and their solutions (first picture).
The second is a table with Greek fractions from the Byzantine epoch (7th century):

Photography by Carlos Dorce

Finally, we find this ceramic piece which is part of a more complete catalogue composed of pieces P 11999, P 12000, P 12002, P 12007, P 12008, P 12609 and P 12611 from the 3rd and 2nd centuries BC, all of them found in Elefantina.

Photography by Carlos Dorce

These pieces contain one of the most difficult problems of the Greek mathematics: the construction of a regular icosahedron. This P12609 was translated and analised by Jürgen Mau and Wolfgang Müller (`Mathematische Ostraka aus der Berliner Sammlung’, Archiv für Papyrusforschung XVII (1962), 1-10.), and we find some words which help us to understand the text. For example, word σφαιρας suggests that we are studying a tridimensional figure, τριγωνων πλευρον refers to equilateral triangles and δεκαγων is used by Euclid in some propositions of the Elements.
We can think about the transmission of the Greek science from Alexandria to other Greek cities because of these pieces were found in Elefantina and not in Alexandria. After Euclid’s Elements, the only reference to a construction of a regular icosahedron is found in the work of Hypsicles (c. 190 BC-c.120 BC) who explains that his father and Basilides of Tyrus discussed in Alexandria about Apollonius’ construction of the regular dodecahedron and icosahedron.
LocationNeues Museum (map)
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Omar Khayyâm’s doodle

Omar Khayyam was born in Nishapur on May 18, 1048. He went to school and two of his best friends there were Nizam al-Mulk and Hassan al-Sabbah. The three boys agreed that the first of them who could get a rich position will help his friends in a future. Al-Mulk became vizier in Alp Arslam’s palace and al-Sabbah and Khayyam also benefited of his nrew position although Khayyam got a job which let him to study Astronomy, Literature and Mathematics. Khayyam’s life wasn’t easy but his astronomical studies and his participation in the reform of the calendar were so decisive that he always had a city or a place where being able to life and work. According to al-Arudî al-Samarcandî, Khayyâm died on December 4, 1131.

Stamp issued by Dubai

Khayyâm studied Euclid’s Elements and Data, Apollonius’ Conics and al-Khwârizmî’s Algebra and wrote his major work on Algebra around 1074 whre was able to solve geometrically the cubic equation. The treatise begins with three introductory lemmas:

  1. To find two segments x and y which a/x = x/y = y/b (Khayyâm finds a point (x,y) which is the crossing point of the parabolas x2 = ay and y2 = bx).
  2. To determine the height of a parallelepiped with known squared base b if we know that its volume must be the same as another parallelepiped with squared base a and height (Khayyâm determines a line k such that a/b =b/m and the searched height K is k/a = h/K).
  3. To determine the side of the base of the second parallelepiped.

Khayyâm solves fourteen canonic cubic equations (he didn’t know the negative numbers!) from these three geometric lemmas. For example:

Lemma 1. From lemma 1, we can find x and y such that 1/x = x/y = y/c and this point (x,y) satisfies x3 = c.

The other thirteen cases are solved crossing parabolas, circles and hiperbolas.

In 1857 E.Fitzgerald discovered Khayyam’s Rubâyyât in the British Museum and trabslated some verses from this “new” manuscript. The translation to English was so popular since 1861 and the Khayyâm’s name was very famous in the literary circles. The Rubâyyât contains over 400 quatrains written in Persian and were translated to English again in 1970s by Robert Graves:

Wake! for Morning in the Bowl of Night

Has flung the Stone that puts the Stars to Flight:

And Lo! the Hunter of the East has caught

The Sultán’s Turret in a Noose of Light

Stamp issued by the Federated States of Micronesia

Stamp issued by Guyana

Dreaming when Dawn’s Left Hand was in the Sky

I heard a Voice within the Tavern cry,

Awake, my Little ones, and fill the Cup

Before Life’s Liquor in its Cup be dry.

The mathematical doodle was published by Google in the Arabic countries two years ago to commemorate Khayyâm’s birthday.