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To: Johannes Pilch who wrote (525254)	1/16/2004 5:20:33 PM
From: AK2004	Respond to of 769667

JP islam science is more famous for it's medicine (abu ali al husain ibn abdala ibn cina). Arabic numeric positional hierarchies predate islam.

To: Johannes Pilch who wrote (525254)	1/16/2004 5:22:00 PM
From: DuckTapeSunroof	Read Replies (2) \| Respond to of 769667

You are correct. They were a trading people, astride the routes between the West and India, and the path to the caravan routes into China. They assimilated much of that Eastern knowledge, and made the West aware of it in due course. (Still, in astronomy and certain other fields, the flame of learning burned rather brightly... long before the West.)

To: Johannes Pilch who wrote (525254)	1/16/2004 5:55:00 PM
From: Lazarus_Long	Respond to of 769667

Hmmm.... Who invented geometry? Greeks? Egyptians? Babylonians? Who? I see your Diophantes (Diophantine equations are a *****, anyway!) and raise you www-groups.dcs.stand.ac.uk These were his treatise on algebra and his treatise on astronomy. The algebra treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. It is the title of this text that gives us the word "algebra" and, in a sense that we shall investigate more fully below, it is the first book to be written on algebra. Rosen's translation of al-Khwarizmi's own words describing the purpose of the book tells us that al-Khwarizmi intended to teach [11] (see also [1]):- ... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned. Now this does not sound like the contents of an algebra text and indeed only the first part of the book is a discussion of what we would today recognise as algebra. However it is important to realise that the book was intended to be highly practical and that algebra was introduced to solve real life problems that were part of everyday life in the Islam empire at that time. Early in the book al-Khwarizmi describes the natural numbers in terms that are almost funny to us who are so familiar with the system, but it is important to understand the new depth of abstraction and understanding here [11]:- When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; ... so forth to the utmost limit of numeration. Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations. His equations are linear or quadratic and are composed of units, roots and squares. For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x2. However, although we shall use the now familiar algebraic notation in this article to help the reader understand the notions, Al-Khwarizmi's mathematics is done entirely in words with no symbols being used. He first reduces an equation (linear or quadratic) to one of six standard forms: 1. Squares equal to roots. 2. Squares equal to numbers. 3. Roots equal to numbers. 4. Squares and roots equal to numbers; e.g. x2 + 10 x = 39. 5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x. 6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2. The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x2 = 40 x - 4 x2 into 5 x2 = 40 x. The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation. For example, two applications of "al-muqabala" reduces 50 + 3 x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal with the numbers and a second to deal with the roots). Al-Khwarizmi then shows how to solve the six standard types of equations. He uses both algebraic methods of solution and geometric methods. More complicated than that, though, as these things always are. ucs.louisiana.edu