School of Athens

1. Pythagoras is depicted writing a book.

2. Socrates, holding up four fingers. These symbolise the four stages one has to pass through to have a philosophical conversation: geometry, astronomy, arithmetic and stereometry. The four steps of the staircase also refer to this.

3. Plato

4. Aristotle

A single gesture symbolises the core of their philosophy: Plato points upwards, Aristotle stretches his arm in front of him, with his palm facing the earth. Plato symbolises speculative philosophy by pointing upwards, towards the spiritual, the supernatural. Aristotle, on the other hand, points to the earth, as the source of all scientific knowledge. The books they hold in their hands also refer to this. Plato is holding his Timaios, in which he explains his thoughts about the origin and creator of the cosmos, while Aristotle is holding his Ethics, which is about the right behaviour of people.

5. Euclid, pictured, stooping to draw the terms of a principle with a compass.

Pythagoras – c. 570-500 BC.

Socrates – c. 469-399 BC.

Plato – c. 427-347 BC.

Theaetetus of Athens – c. 417-369 BC.

Eudoxus of Cnidus – c. 410-355 BC.

Aristotle – c. 384-322 BC.

Archimedes – c. 287-212 BC.

Euclid of Alexandra – c. 265-200 BC.

“There is geometry in the humming of the strings.

There is music in the spacing of the spheres.”

– Pythagoras

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PYTHAGORAS (c. 570-500 BC.)

One of the Presocratic philosophers, was known as a philosophical and religious reformer. Researchers emphasise both Pythagoras’ mathematics and philosophy, and his shamanistic allure.

He learned numerology from the Phoenicians, geometry and astronomy from the Egyptians and Chaldeans and music and other sciences from the Persian magisters.

In Egypt, he was initiated into the ‘divine mysteries’.

In Croton, around 530 BC., he founded a brotherhood around religious, mystical and philosophical doctrines. Women were also admitted.

The aim of the school of Pythagoras was ‘to free man’s mind from the many shackles and bonds that have bound it from childhood’. For the mind ‘sees everything and hears everything, but everything else is blind and deaf’. The school was a ‘mystery school’, where the hidden spiritual identity could be developed.

According to the world view of Pythagoras and Plato, music was a reflection of the cosmic spheres of stars and planets.

Here we see the Tetragrammaton in a triangle, ascending to the number 72, which in esoteric Jewish tradition is considered to represent God numerically.

Tetractys

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“Geometry will draw the soul towards truth, and create the spirit of philosophy.”

– Plato

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PLATO (c. 427-347 BC.)

Greek philosopher and writer, pupil of Socrates and teacher of Aristotle. With his Theory of Ideas he became the patriarch of metaphysical realism. The Timaeus, one of his written dialogues, is a description of the nature of the physical world and cosmogony in the form of a myth. The Timaeus can be regarded as the ‘theory of everything’, starting from the creation of the universe to human anatomy.

For Plato, the demiurge is a deity who shapes the material world from pre-existing chaos. Using the Forms (ideas) as a model, he transforms the chaos into the best possible image of these eternal and unchanging archetypes. The demiurge forms a harmonious universe in the image of the eternal Forms.

Plato assumes that the smallest particle of each element has a well-defined geometric form: tetrahedron (fire), octahedron (air), icosahedron (water), and cube (earth). Furthermore, Plato postulated the existence of a fifth element, the dodecahedron or quintessence, of which the cosmos itself would be made.

He travelled to Egypt where he learned geometry and astronomy.

“Numbers rules the universe.”

– Pythagoras

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EUDOXUS of CNIDUS (c. 410-355 BC.)

was a student of Plato and a versatile scientist who excelled in mathematics, geography and astronomy. All his works have been lost in the course of time. What we know about him comes from secondary sources, for example from the works of Archimedes.

In mathematics, he was the forerunner of Euclid and established several axioms. Euclid adopted many proofs from Eudoxus. Eudoxus’ main interests included the golden section, the intersection of curves and the ‘Delic problem’ (= doubling of a cube).

The Kampyle of Eudoxus, an algebraic curve, is named after him:

a2x4 = b4(x2+y2)

Kampyle of Eudoxus © Paul Bourke

“And the whole [is] greater than the part.”

– Euclides

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EUCLIDES (c. 265-200 BC.)

Hellenistic mathematician, who worked in the library of Alexandria. Euclid is often called the ‘father of geometry’.

His most important work is called ‘The Elements’. In 13 books, it presents the oldest systematic treatise on geometry. It is the most successful manual and one of the most influential works in the history of mathematics. It has been, after the Bible, the book most produced and studied in the Western world. The first edition of the Elements dates back to 1482, preceded by many handwritten copies.

In this work, Euclid presented geometry in an ideal axiomatic form that came to be known as Euclidean geometry. The treatise is not, as is sometimes thought, a compendium of all that the Hellenistic mathematicians knew at that time about geometry, but rather an elementary introduction to the mathematics of that time, which corresponds to a large extent to what we call geometry today.

1410 AD.

Excerpt from the Encyclopedia of Sultan Iskandar, by Nāşir al-Kātib.

British Library

17th c. AD.

Fragment from Euclid Elements.

Manuscript Frans van Schooten

This image shows the sixteenth element of Euclid.

University of Glasgow Library

1817-28 AD.

Diagrams to Samuel Cunn’s Euclid’s Elements of Geometry by Joseph Mallord William Turner

1847 AD.

Euclid by Oliver Byrne in collaboration with William Pickering, London

NON-EUCLIDEAN GEOMETRY.

For two millennia, Euclid’s geometry of points, lines and bodies was thought to be the only possible one and to contain the ultimate truth. However, in the course of the 18-19th centuries, Gauss, Lobachevski and Riemann developed another geometry, the so-called non-Euclidean geometry. Riemann’s geometry made it possible for Einstein to formulate his theory of relativity, and it lies at the basis of modern cosmology.

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“I wanna know Gods thoughts in a mathematical way.”

– Einstein

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ARCHIMEDES of SYRACUSE (c. 287–212 BC.)

Greek mathematician, physicist, engineer, inventor and astronomer. Archimedes is widely regarded as the greatest mathematician of antiquity and one of the greatest mathematicians of all time. He gave a remarkably accurate approximation of pi and also defined the spiral that bears his name.

Archimedean solids

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“Geometry is one and eternal shining in the mind of God.”

– Johannes Kepler

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14th c. AD.

Illustration in Euclid’s Elementa, translation attributed to Adelard of Bath, England.

Detail of a scene in the letter ‘P’ of a woman; she is using a caliper (compass) to measure distances in a diagram. In her left hand, she holds a measurement angle to draw right angles. She is observed by a group of students. In the Middle Ages, it was unusual to see women as teachers, especially when the students appeared to be monks. She is most likely the personification of Geometry, based on Martianus Capella’s famous book, the Nuptiis Philologiae et Mercurii, 5th c. AD., a standard source for allegorical imagery of the seven liberal arts.

13th c. AD.

Anonymous

Science, especially geometry and astronomy/astrology, was for most medieval scholars directly connected to the divine. The compass in this 13th century manuscript is a symbol of God’s creative act. God created the universe according to geometric and harmonic principles, therefore the search for these principles was the search and worship of God.

15th c. AD.

Portrait of Luca Pacioli,

by Jacopo de’ Barbari.

It shows Pacioli drawing a construction on a board whose edge bears the name Euclid. His left hand rests on an open book (Summa de Arithmetica, Geometria, Proportioni et Proportionalità or a copy of Euclid). On the table rest the instruments of a mathematician: a protractor and a compass. In the right-hand corner of the table, a dodecahedron rests on a book with Pacioli’s initials. To the left of the painting, there is a rhombicuboctahedron (a convex solid composed of 18 squares and 8 triangles).

“Geometry is the underlying organizing principle and key to unifying and understanding

interactions between the macro and microcosmic.”

– Nassim Haramein

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Ancient Babylonian clay tablet with geometric diagrams accompanied by problem statements.

19-17th c. BC.

2 tablets from the Old Babylonian period illustrating the Pythagorean theorem.

(Photo © Yale Babylonian Collection)

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