Number Analysis
MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

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Euclid - Part 2
Diophantus

 

This paper will explore the difference between Euclid’s methods and the methods of Diophantus, paying particular attention to the concept of "reverse-engineering" (assume an answer, x, and clear away the clutter until the x is shown equal to a number –- basically algebra). This method was well-developed by Diophantus but only vaguely understood in Euclid’s time.

 

Euclid

He evidently flourished about 295 B.C. His Elements have been such a profound influence on our society that the word "geometry", in common usage, has become practically synonomous with Euclidean geometry. Euclid compiled and edited many mathematical publications from his time, and he developed many arguments and innovations of his own. He probably developed an improved proof of the pythagorean theorem, and defined parallel lines as lines that do not meet when extended indefinitely, rather than two lines equidistant from each other at all points, see The Parallel Problem. [Calinger]

Elements, Book VII and IX

Book VII and IX deal only indirectly with what we consider geometry. The focus is the defining qualities and characteristics of numbers. Geometry is used to have a familiar vehicle of examining the definitions of, and relationships between, numbers. However, the very fact that Euclid used geometric figures to express each operation suggests that he felt physical intuition was sufficient to explore these operations, and that therefore operations which couldn’t be visualized in this manner either didn’t exist, or were unimportant. 

General ideas

Euclid accepted only positive integers as numbers. The unit was indivisible, and fractions had no meaning.

Many of his arguments become more clear when put in more familiar, algebraic terms. However, it should be noted that algebra was not even known at the time, which makes a study of his methods and results, without relying on our general algebraic techniques, a challenging and revealing undertaking. It should also be noted that reducing his arguments to algebraic notation will often obscure the non-algebraic essence of his methods.

The following is a list of some of Euclid’s definitions, added here to introduce some basic concepts he put forth. After the definitions, I discuss his methods in some detail, and then compare them to the methods of Diophantus. [Heath]

Select Definitions, Book VII

"1. An unit is that by virtue of which each of the things that exist is called one." [Notice this neatly side-steps what one actually is, it just simply defines unit as "one-ness", whatever it is]
"2. A number is a multitude composed of units." [notice 1 is not a multitude, and hence not a number]
"3. A number is a part of a number, the less of the greater, when it measures the greater." [in our current language, the less is a factor of the greater]
"4. but parts when it does not measure it." [or, not a factor of it]
"5. The greater number is a multiple of the less when it is measured by the less."
"6. An even number is that which is divisible into two equal [integer] parts."
"7. An odd number is that which is not divisible into two equal [integer] parts, or that which differs by an unit from an even number." [note the second part is provable from the first, or the first from the second]

The next several definitions deal with such things as, even times odd numbers (defining them as, in effect, repeated addition of the even, a number of times equal to the odd), and other similar notions. As these are now unused concepts, we will move on to...

"11. A prime number is that which is measured by an unit alone." [this is still a common definition]

 

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