Frederick II and Mathematics 

Frederick's education was a sort of  correspondence course in science and philosophy with Jews of Spain, Muslims of Egypt, and with his often absent courtiers.  Frederick sought to raise the esteem of his Mediterranean neighbors for him by appearing as a man of learning. For example, he wrote (with the help of his court philosopher Master Theodore)  to the Almohad caliph in Morocco  with a list of philosophical questions which were, in time,  answered by ibn Sabim a prominent man of learning. The questions reveal that Frederick had some familiarity with Aristotle, in a world then dominated by Plato, and a keen interest in learning.

 Michael Scotus was the most famous of Frederick's courtiers, but he also was in contact with Leonardo Fibonacci of Pisa, who even dedicated a book to Frederick.

Castle del Monte,  built late in Frederick's life, is obviously designed to synchronize  with the movements of the heavenly bodies. Frederick's interest in astronomy was well know; not only did he maintain a court astrologer, but  in 1232 the sultan of Egypt sent Frederick a planetarium based on Ptolemy's  zodiacal armillary sphere -it was a clock as well as a map of the heavens. We can easily imagine Frederick seated on the roof of Castle del Monte closely watching Michael Scotus operate the planetarium.

"Frederick, in later days, showed distinguished visitors his priceless planetarium, in which the sun, moon and stars moved in mysterious harmony." - Frederick II by Ernst Kantorowicz (p. 195) (click on image o enlarge)

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God the  Geometer

 

God the Geometer 13th century manuscript illustration. Austrian National Library, Vienna.

The vehicle of this cosmology was Euclidean geometry. Euclid wrote the Elements of Geometry around 300 BC. Husserl, in The Origin of Geometry, supposes that this system arose out of practical activities, such as building. However, the classical conception of space seems to have been based upon visual evidence rather than technique;  the horizon appears to encircle us, and the heavens appear to be vaulted above us. In the West, the primacy of geometry over perception was stressed by St Augustine, who wrote: ‘reason advanced to the province of the eyes ... It found ... that nothing which the eyes beheld, could in any way be compared with what the mind discerned. These distinct and separate realities it also reduced to a branch of learning, and called it geometry.’

Euclid kept  four questions which Aristotle had outlined as the basic questions that must be asked in every science:

1. Does the thing we are trying to study really exist?	For example: Is there really such a thing as a circle? Until we can answer this question there is no use going any further. It is impossible to have  knowledge of something that does not exist, or at least which has not existed at some time.
2. What is its definition?	For example: What is a circle? We must first have the definition of a circle before we can even ask if it exists.
3. What are its properties?	For example: Can a circle be drawn inside a square so as to touch all sides of the square?
4. What is the proper reason that this thing has this property ?	For example: Why can a circle be drawn inside a square?

When we know the answer to Question 4, at the same time we know the answer to Question 3. When we know the proper reason that a property is necessarily connected with its subject, we know that property must exist.

In setting up his Elements, Euclid tries to answer these questions in the correct order. Elements is divided into thirteen books as follows:

The medieval cosmology still holds the interest of modern Professors of Physics
 

"Although it is possible to observe the time on an armillary sphere, it is quite difficult to perpetually mimic the motion of the sun around the earth. The invention of stereography by Hipparchos made the construction of a dynamic representation of the heavens possible through the combination of planispheric projections with the clepsydra. The anaphoric clock and its cousin, the astrolabe, not only helped Ptolemy create the extensive catalogue in the Almagest, but also established the foundation of modern time keeping." (left) Dennis Duke, University of Florida, with a working replica of a zodiacal armillary sphere, as described by Ptolemy in Almagest and Almagest Planetary Model Animations.

 

The astrolabe, a device for locating the position of the stars at any point in time, was  know in Frederick's court.

An Islamic Astrolabe

The astrolabe consists of three major parts:  A fixed disk called a tympanum on which one can measure the position of the stars. The tympanum is an engraved plate, making it easier to use than the wire mesh of the anaphoric clock, but because the position of the horizon differs from place to place, each astrolabe typically contained a number of tympanum. Only one tympanum was used at a time, and the inclusion of several tympanum insured that the astrolabe could be used at a variety of positions on the earth.  A skeletal projection of the stars--called a rete--was fastened over the tympanum.  A simple device for measuring the distance of a star above the horizon--usually a rod attached to the back of the astrolabe. One could produce a map of the sky on any given night by locating a known star, measuring its angular distance above the horizon, and rotating the rete until the representation of the star was aligned with its angular distance on the tympanum

Ogee Arch

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