The importance if parallelograms as the general form for products was missed by Justus. He after all was trying to logically found arithmetic, and that is based on the rectangular parallelogram, and particularly the square.
However this is a misconception carried over by Islamic cultures from Aristotelian analysis.
The Sumerians had a greater respect for factors, factoring , the rectangle and the parallelogram inscribed in a circle.
Recent work by Norman Wildberger and his colleague is uncovering the richness of their sexagesimal system.
<![CDATA[<a href="http://www.youtube.com/v/J5Ug3Cr8RUE&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/J5Ug3Cr8RUE&rel=1&fs=1&hd=1</a>]]>http://m.youtube.com/watch?v=J5Ug3Cr8RUEFactors become the bedrock for apprehending scale, proportions( Analogos) squares( duoplassoi) cube ( triplassioi) and other logoi( dynamic ratios) relationships found in book 5 of the Stoikeia. These ratios are all based on the intersection of chords and secants associated with the circle, in the most general propositions of books 1 to 4
Book 2 is the clue that is overlooked its propositions are based on the general parallelogram with a diameter, the general half parallelogram. And the special rectangular parallelogram, and it's half.
Tables have since Sumer been associated with quick and handy calculation. The process using the tables was therefore key to being competent.
In a real sense algebra( symbolic arithmetic) is really an expression of the general process for using a set of tables! Justus understood that his ring theoretical approach had to be based on this dynamic geometry of the Greeks, but he was not clear how multiplication old be logically founded. There was no proposition for multiplication, only the rectangular form.
Un fact Hermann his son realised that the proposition for multiplication is based on the dynamic parallelogram. Not the static rectangular form. Like all propositions every type of multiplication had to be constructed to demonstrate its validity.
Several products can be constructed geometrically. Books 1 and 2 deal with those as foundational concepts. Pythagoras theorem is a product, intersecting chords is a product, gnomon, parallelograms and rectangles are products.
Where multiplication enters the discussion is in the notion of factoring. Factoring is breaking into parts, and it is this notion of breaking into regular or irregular parts that is emphasised over and over. Those parts that are equal are called artios or perfect, those that are not equal are called perisos or approximate( near to some perfect part) .
The process of multiplication is founded on duplication. The number of times a form is duplicated( isaskis) is punted and this product is called a polyplassios or multiple form .
The discourse in book 6 is about dynamic ratios . Skesis is a held stage in that dynamic process( kinesis) . The ratios are thus studied in proportions, that is in some relationship of 2 or more ratios. . The first relationship is called Analogos, and involvesc2 ratios . Book 6 goes on to define squared, cubed, inverse and other relationships rarely mentioned.
All of these have a foundation in a geometrical form inscribed in or exterior to a citcle.
Measuring the circular arc was done by using the chord bow.. Remarkably the radius of a citcle is a chord that cuts the circle precisely 6 times by definition! From a proportion perspective one might think twice the radius( diameter) should cut it 3 times, but in fact it cuts the cicle in equal semicircles. What cuts it into 3 is the diagonal of a parallelogram called a rhombus .
As you study chords more you fing these rhomboids appealing regularly in the dissection of chords. So it is most diagonal length that is important, but what rhomboid form is it that bisects, trisecting quadrasects and quintasects a chord bow.
Proportions in a circle are based on area not length .
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