April 9, 1997

"Islam and the Arab World", Edited by Bernard Lewis
American heritage Publishing Co., Inc. New York.

P.200

Discussion on the different styles of Copernicus and Euclid.

What did you observe about the styles of Copernicus and Euclid? How are their arguments constructed?

Note that there is a paragraph of "setting up" the problem, identifying what's to be computed, compared, etc. Auxiliary lines, labels and so forth are added into the figure. Then, there is think D a formal argument that the claimed relationship holds, framed in terms of theorems of Euclidean geometry.

Break into groups and identify these parts of the argument from Des Revolution, Chapter 6, and Euclid, prop 44.

2

These two parts of the argument were formal requirements. The "setting up" is called zetetics and the proof is called poristics. Equal care had to be taken with both pieces of the argument. All parts of the proof came from axioms or previously demonstrated theorems of Euclidean geometry.

3

Here is how this form of argument would work in an algebra problem of the time:

Problem: Given the sum of the roots and the sum of their cubes, to derive the roots.

Zetetics: Let B be the sum of the roots and D the sum of the cubes. The roots are to be derived.

Poristics: Four times the sum of the cubes minus the cube of the sum of the roots divided by three times the sum of the roots yields the square of the difference between the roots. Given the sum and difference between the roots, the roots are determined.

in modern language: 


           (((4D - B3) / 3B)1/2 + B)2/2
        
and 
 
           (((4D - B3) / 3B)1/2 - B)2/2

check this answer. What is lost in this formalism?

4

How did Renaissance scholars come to possess Euclid's elements?

During the early middle ages,the Arabic world produced capable mathematicians, who translated Greece mathematics into Arabic. The first copies of Euclid were translated into Latin from Arabic. For a thousand years, the Islamic world had access to these documents and were studying and building upon them, especially in the area of computing roots of difficult equations, al-jabr (to restore). When europeans got their hands on the ancient greek texts, they were in Arabic. Along with them came mathematical works of a new sort by Arabic mathematicians, including an algebra text by Al-Kwarizmi (one of several spellings). This contained material known to Diophantus, new work on solutions of certain cubics and so on. It was translated into latin by late medieval Christian scholars who were also concerned with solutions to particular equations. These were called "cossists" by others, because they wanted to restore the value of the thing (cosa)rto be found. But they never translated the last part of Al-Kwarizmi's book into Latin, probably because they couldn't figure out what it meant, and in part because it just wasn't Greek. Lluyl (?) Catalan monk with the Institute for learning arabic. The Dark Ages weren't dark in the Arab speaking world, nor in Africa.

J. L. Berggren: "Episodes in the Mathematics of Medieval Islam"
1986, Springer-Verlag New York Inc.

Plate 6.4, P.178

5

Status: The high status of Greek culture in the West over Arabic culture expressed itself in a Euro-Christian preference for geometry over algebra. This trend was also triggered by a question of genuine comprehensibility. Why is the algebraic proof above harder to understand than a geometry proof? The algebraic formalism does not allow us to see how the formula was derived. If we change the problem slightly, you won't know how to solve it. You can hear the teacher saying; show your work. In this formalism, there is no way to show your work. You might say algebra was in a kind of preverbal state.

6

Organize vs. Disorganized: Algebra was a large collection of individual problems and special cases. There was no statement the equivalent of: "In all isoceles triangles the base angles are congruent". The category "isoceles triangle" covers a lot of cases, and the equivalent organization of thought in algebra had not occured yet. How did the 15th century mathematical mind react to this?

7

Many believed Diophantus and other greeks actually knew all the Arabic algebra, and of course had it organized in a better way. This corroborated with the popular idea of "lost wisdom of the ancients". The Aarabic texts were really Greek ideas that had been "defiled by barbarians" in some way so as to become muddled and unclear.

8

What happened: Geometry was studied by astronomers and nearly every educated person. Algebra was studied by scholars, and for some reason, French lawyers. Language in mathematics improved for stating general cases and a new form of proof was introduced, called "exegetics".

J. L. Berggren: "Episodes in the Mathematics of Medieval Islam"
1986, Springer-Verlag New York Inc.

Plate 6.1, P.160

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