July 17th, 2017, 11:35 PM  #11  
Senior Member Joined: Oct 2009 Posts: 428 Thanks: 144  Quote:
Example, Hartshorne (authority on modern geometry): Quote:
 
July 18th, 2017, 12:26 AM  #12  
Senior Member Joined: Jun 2015 From: England Posts: 830 Thanks: 244  Quote:
I did indeed mean to refer to common notion 1 in the second part of my post. Secondly whilst I agree with you that Euclid's lists were not exhausitve, I was disappointed with the casual dismissal of my first comment from someone whose posts are normally highly perceptive and a pleasure to read. Criticism, yes. But constructive criticism please. I can only assume you have not understood my point, which is particularly ironic considering your username. When I first read Heath's translation I thought the lists so much fluff and nonsense, although a valiant attempt to define his working materials. You cannot develop theorems about points, lines and so on until you have at least a working definition. It was later I realised just how far reaching these definitions are. "That which hath no part" goes right to the heart of the application of the (Newton's) Calculus to Physics. Please define density for me.  
July 18th, 2017, 02:39 AM  #13  
Senior Member Joined: Oct 2009 Posts: 428 Thanks: 144  Quote:
Also, more on point. I am not saying (or should not have said) that definition I is useless. It is a fine heuristic and it puts a very good idea in the head of the reader of what a point is. But you have to agree it is not like a modern mathematics definition. Nowadays, we wouldn't call this a definition, but rather an explanation or something else. That is all I meant with my comment. I don't really understand the density thing. Do you mean it like "rationals are dense in the reals" or in another way? Maybe I'm being dense (pun intended). Also, I do agree that most of Heath is fluff and only interesting for historians and not mathematicians. However, it is I think the most authorative English translation of Euclid nevertheless.  
July 18th, 2017, 03:20 AM  #14  
Senior Member Joined: Jun 2015 From: England Posts: 830 Thanks: 244  Quote:
My library does not run to multiple direct translations of Euclid. Heath is all I have. But what I originally thought as fluff, I later came to think that Euclid was talking about dimensions in the first few definitions. So a line has no width or height but has length. Now imagine passing a line through a point that has dimensions. The questions arise Which part of the point does the line pass through? If two lines pass through the same point can they not actually touch (meet) by passing through differnt parts of that point? "That which hath no part" ie zero dimension in all directions avoids this. So such apparently banal statements are necessary to develop the theory. On to my remark about density  nothing personal intended, glad you could take it that way. So much of Physics is founded on continuum mechanics which is about distributed or extensive properties, such as mass, charge, electric fields, stress etc. Today we gaily integrate density over some region to obtain the total mass, but what is the density at a point which has zero volume, since density is mass/volume? We must look to the theory of limits to answer this question. Today if I want fire I have all sorts of sophisticated means of obtaining it. I don't have to resort to rubbing two boy scouts together to start one.  
July 21st, 2017, 07:30 AM  #15 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
AFAIK , the 5th postulate was not called 'The Parallel Postulate' by Euclid , but reading commentaries by others about it one gets the impression he called it so. 

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