the historical development - for many reasons. Among them - the best sense I have of the history of civilization is through some understanding of a history

of geometry's development. I would go further and say that in many ways

Art <asiegel@eico.com> wrote: that is

precisely the importance of geometry study - and the same was true in Euclid's time. Euclid was a summing up, and a history lesson in its time -

the hmm... Not wishing to open any previous debates, but here are my 'coordinates' in this fascinating topic. Hope it'll help us work better together... I agree history is essential [my father was historian] And history of science no less. But I am not sure first introduction to subjects should begin with history or follow our present versions of that history. :::maybe, maybe not::: Any subject, especially 'history' itself, cannot be understood without some personal experiences to relate to. For example, history as taught in schools, is hard to grasp even with a good teacher, mostly because as young creatures, we have not lived much ourselves. On the day we are born, one day is a lifetime [in free air minus womb] At the end of our first week our sense of time has already shifted radically.. After six months our sleep wake eat rhythms have transformed from rhythm to noise http://www.nomadicsltd.com/timelines/sleep_wake.html By ten years old we are very conscious of 'years' and somewhat of seasons. But historical time is still so hard to relate to.. We are taught about this or that events in human history and 'important' people or developments, and we really can't scale to them, nor in most cases to the emotions, logic or conditions behind them. Novels and films are sometimes easier to relate because they personalize, albeit at the expense of detail or correct sequence.. which of course makes historians crazy, but can provoke us to want to know more.. When we ourselves or someone near us is dying [at whatever age] then we can say. "oh that's what a lifetime is.." The most readable historians I know who embody some of this are Simon Schama and Fernand Braudel. A History of Britain : At the Edge of the World, 3500 B.C.-1603 A.D by Simon Schama (October 2000) Citizens : A Chronicle of the French Revolution by Simon Schama (March 1990) The Embarrassment of Riches : An Interpretation of Dutch Culture in the Golden Age The Perspective of the World : Civilization and Capitalism 15th-18th Century by Fernand Braudel The Structures of Everyday Life : The Limits of the Possible The Wheels of Commerce In the case of Science, Math and Geometry, I believe much the same applies. We have not yet much experience and we need help relating that to the formal. The 'history' of geometry is not only Euclid. One must go much further back and ask how did Euclid come to this strange juncture. We should also ask what form his source code took. Our modern Euclid passed through Arab, Renaissance, Enlightenment and Victorian interpretations before it reached , Dover, McGraw-Hill or <insert your favorite publisher here> Since the Library of Alexandria was destroyed we shall probably never know what we are missing. But we can imagine by asking from the earliest times what did these people do: Navigators, sailors and travelers, constructors of ancient monuments, merchants, farmers, fishermen, weavers, potters, artists, Geometers themselves [the ones who measured the earth]. This is why I say start with string and sticks, pen and papers, balls and cubes... as they say ...'think Egyptian.. String and sticks in the sand will allow a huge amount of interactive geometry to emerge. The basics of symbolic algebra follow 'naturally' from building things. If you want to cut a piece of marble or wood to fit against another, the most accurate way is hold up the first and make a mark. But it may be too heavy to move and already carefully postioned. The most portable way is use a stick [or string if you don't stretch it] and make a mark or knot. The use of measuring units only enters the picture when you get very ambitious or start worrying about cost, time, having enough material etc. Since we don't even know by whom, or how Stonehenge and the Pyramids were built, we are clearly missing a big chunk of the history of geometry. Not too mention other parts of Africa and Asia. As a sometime carpenter in my early 20s I was so brainwashed by school and rulers [and love of them], that it took me a couple of years of hands-on finish work, to learn when NOT to use a tape measure - thus getting more accurate results and faster.. The same applies for moving pianos and any large heavy object like lengths of lumber.. carrying them precisely by the center of gravity etc.. My main argument is if you are going to the history do it deeply and question the history we have been taught to find the real history. I think in this way the lessons of the past become the experiences of today. The computer allows to crate virtual simulators for all of this. But we must not ignore the value of direct manual contact and our own senses. Thus Dig for the experience of fundamentals and then reach for the brilliant formalists [Euclid etc] Other examples are to do with counting and relationship of number and geometry For example how to count a lot of small objects, how to measure and compare things These are the things humans have needed to do for perhaps 50,000 years. A fabulous book which covers this from the earliest roots is: The Universal History of Numbers : From Prehistory to the Invention of the Computer by Georges Ifrah Hardcover - 663 pages (November 19, 1999) John Wiley & Sons; ISBN: 0471375683 Other Editions: Paperback What is brilliant about Ifrah's book is its trans-culturalism. Not just derived from classical Greece-Renaissance Europe, he really goes all over the world and looks at fascinating comparative history of how humans have used number. Especially in the world today and USA perhaps more than anywhere because of its melting pot, we must look at these things in global historical context. I know an experienced wise old Chemist who is of the firm opinion that the best way to _start_ teaching chemistry is through cooking. Then start asking questions and providing paradigm guidance and method. She insists this if done right will form the right foundation. Among her arguments is that when the children go home, the chemistry lesson will be waiting for them on the dinner table and throughout the rest of their lives. She is a firm believer in scientific method. Returning to the classical Greeks - we must ask how did they do their geometry? What tools did they use? Likewise what their peers in China or elsewhere were doing. Consider the multiple approaches to Pythagoras. Paper and scissors are a great tool here. Take a pile of stones as you sit on beach by a fire watching the stars.. at any time in history Does Euclid pop out ... or do tetrahedral numbers? Take a some string, some shells some sticks.. start drawing circles in the sand and marking interesting points made by your marks, the sticks and the fire light. Does Bucky pop out or does Euclid? I think what we need in addition to spending some time together on a real beach, is a virtual beach, string, sticks, shells.. a computer could do nicely :-) - Jason ________________________________________________________________ Jason CUNLIFFE = NOMADICS['Interactive Art and Technology'].DesignDirector