euclidian geometry vs. Cartesean coordinates

[gib65]

In terms of the reference frame used in geometrical modeling, is there a difference between "Euclidian Geometry" and "Cartesean Coordinate Systems"? One thing I'm wondering is, how is Euclidian Geometry possible without a Cartesean Coordinate System? Yet, Descartes is credited with inventing the Cartesean system 2 millenium after Euclid invented his geometrical system. The Cartesean system cannot have been new in the 17th century - maybe given a new name, but not new.

[DV8 2XL]

The Cartesian system is Euclidean Geometry with coordinates - that was the innovation - the Cartesian Coordinate System allowed the unification of algebra and geometry in one system - analytical geometry

[Sisyphus]

It most certainly was new. Descartes solved a lot of ancient problems that no one else could by algebrizing geometry, and thus by finding a way to express geometrical lines as the loci of points in a specifically varying ratio of distances between two other given lines, which eventually evolved into the Cartesian coordinate system that everyone learns in school. Euclid had nothing whatsoever like this, in fact keeping any concept of number away from geometry, and certainly not having a coordinate system.

 

You should find a copy of Euclid's Elements to see how it works. Everyone should, in fact, as it is arguably the most influential book ever written in the course of Western thought. (Some others say the Bible.)

[cosine]

Yeah, basically Cartestian co-ordinates are an application of the abstract concepts discussed in the Elements.

[DV8 2XL]

You should find a copy of Euclid's Elements[/i'] to see how it works. Everyone should, in fact, as it is arguably the most influential book ever written in the course of Western thought. (Some others say the Bible.)

 

[bitter Old Man's opinion] The decline of education in the West and the rise of faith/mysticism over logic/rationalism, began when Euclidean Geometry was removed from the General Syllabus. [/bitter Old Man's opinion]

[Sisyphus]

[bitter Old Man's opinion'] The decline of education in the West and the rise of faith/mysticism over logic/rationalism, began when Euclidean Geometry was removed from the General Syllabus. [/bitter Old Man's opinion]

 

I agree it was a major blow, and everyone should be made to study Euclid if they want to call themselves educated. It has only been in the last hundred years or so that any text aimed at an educated audience stopped assuming a familiarity with the Elements, and I agree that's a terrible thing.

 

On the other hand, I don't think there's ever actually been a point in human history where faith/mysticism didn't have the upper hand...

[jr_metaphysicist]

So you have inspired me to get a copy of Euclid's Elements, thank you for that. But I still do not see what Descartes added to Euclid's spatial models of X and Y coordinates, and in turn as I understand it a third dimension Z. Was it simply the algebra that defined the nature of lines, curves and shapes within this three dimensional space? I've done a bunch of on line research here actually, and believe I have a good understanding of Einstein's introduction of the fourth dimension yielding spacetime of course, but it's not clear at all what Descartes actually added to Euclid's model, despite my understanding of what Cartesian coordinate systems are.

[imatfaal]

http://aleph0.clarku.edu/~djoyce/java/elements/toc.html and other web-based versions are a great start - or if you can splash the cash Byrne's book in facsimile is not only a great (the greatest?) work of maths, but also very intuitive and a work of art

 

http://www.amazon.co.uk/Byrne-Six-Books-Euclid-Facsimile/dp/3836517752/ref=sr_1_6?ie=UTF8&qid=1333532719&sr=8-6

[D H]

[bitter Old Man's opinion] The decline of education in the West and the rise of faith/mysticism over logic/rationalism, began when Euclidean Geometry was removed from the General Syllabus. [/bitter Old Man's opinion]

Yes, yes, yes, and students should also be forced to walk ten miles to school like they did in the old days, trudging uphill both ways, oftentimes encountering deep snow drifts in blazingly hot 120 degree temperatures (Fahrenheit).

 

 

This notion that nobody has come up with a better way of teaching geometry in the 2300 years that have passed since Euclid is downright ridiculous. The notion that Euclid's Elements is the bible with regard to geometry is even more ridiculous. These notions hearken back to the pre-scientific notion that the best way to learn is to read from the great works, replete with monks chanting as background music.

 

From the perspective of history of science and mathematics, Euclid's Elements and Newton's Principia are incredibly important works. From the perspective of how best to teach science and mathematics, there are better ways to teach geometry and introductory physics. Newton's Principia book is wall o' text page upon page of geometric reasoning, largely devoid of algebra and calculus. In fact, nobody teaches physics via Newton's Principia. So why this urge to teach geometry from the 2300 year old Euclid's Elements? Some of the proofs are flawed, others are rather roundabout, and the order of teaching does not match up with how kids think and progress.

[imatfaal]

DH - are any proofs flawed beyond the parallel postulate? I agree with you on the value of teaching - there are better ways and more direct routes; I love the book as a piece of history, but there are just better ways to do things.

[D H]

There is no proof of the parallel postulate in Euclidean geometry. It's a postulate.

 

Regarding flaws in the proofs, the flaws start from the very onset with proposition 1 in Euclid's book #1. There are many others. One is his proof of the side-angle-side proposition. Many books make SAS an axiom because of this.

[imatfaal]

Thanks DH. I meant beyond proof that used the parallel postulate, but when I read up David Joyce's online Euclid he completely bears out what you have said - on 1:1 he comments at the end "What needs to be shown (or assumed as a postulate) is that two infinitely extended straight lines can meet in at most one point."

[The Observer]

So you have inspired me to get a copy of Euclid's Elements, thank you for that. But I still do not see what Descartes added to Euclid's spatial models of X and Y coordinates, and in turn as I understand it a third dimension Z. Was it simply the algebra that defined the nature of lines, curves and shapes within this three dimensional space? I've done a bunch of on line research here actually, and believe I have a good understanding of Einstein's introduction of the fourth dimension yielding spacetime of course, but it's not clear at all what Descartes actually added to Euclid's model, despite my understanding of what Cartesian coordinate systems are.

 

The entire point was the notion of x and y coordinates were not known to Euclid. He talks primarily about proportions of lines and such, he did not yet have a notion of (x,y) points on a grid labelled with numbers.

 

A space is called Euclidean if it assumes what they are talking about, the parallel postulate. ie. that parallel lines never cross. Euclid never considered spaces where this postulate did not hold, so we have named this "flat" space Euclidean space in honour of him. Note that he did not ever use such a term himself.

Edited April 4, 2012 by The Observer

[vthompson12000]

Math folks commonly refer to Euclidean vector spaces. In view of all the properties and operations applicable to vectors, isn't the Cartesian contribution more relevant than the Euclidean? Would it be more accurate to refer to Cartesian vector spaces?

[mathematic]

Old fashioned plane and solid geometry are based on Euclid. Euclidean spaces generalize these notions to arbitrary finite dimensions. Next step up is Hilbert space.