> > As I said, there is NO missing premise. The link (aleph0) I provided also states something along those lines, but the author David Joyce is mistaken.
> > It MUST MEET the circumference because the distance from the centre to the circumference is ALWAYS of length RADIUS!!! There is no problem here! :-) > > Look, it's easy to prove if you must have a proof:
> > If your compass is set to ZERO length, it will never meet the other circle. Any LENGTH greater than zero will cross the path of the circle.
> > NONSENSE! Euclid's thinking is far clearer than any of the baboons who teach real analysis today. It is VERY rigorous and does not require extra proofs. Furthermore, there are no missing premises.
@Ken > I think there are. Euclid depends on the continuity of the plane and of lines > for all his constructions.
There are *no* missing premises. There are however some circular definitions (not good) and others that could be stated in a less confusing way. Even in the original Greek, those definitions are vague. I have rewritten the Elements as part of my unpublished work which includes the full New Calculus. I studied the original Greek, not David Joyce's aleph0 site. :-) His site is good, but it has errors.
There is strictly speaking, no such thing as "continuity of the plane and of lines". Both are formulated from intangible objects - the "point". They are NOT composed of points. They are defined by "paths" with the help of "points". A straight line is the shortest path between two points. A finite plane consists of parallel straight lines. Euclid defines "surface" also. It's a shame that Euclid did not define these geometric objects in terms of paths. I teach advanced mathematics in Asia and I normally give my students a handout with the correct concepts from the very start. It is imperative for definitions to be well formed, something mainstream "mathematicians" know little or nothing about. For one to subscribe to Georg Cantor's rot, explains everything.
Mainstream conceptions of continuity are absolute rot, because these are based on limits which are ill formed. A straight line is discontinuous if it consists of disjoint paths. Same with a plane or solid.
Although Euclid did not mention averages, he showed a very good understanding: the plane and solid numbers. A plane number defines area and a solid number volume. These are the reasons calculus works and the New Calculus uses these concepts masterfully.
Examples in calculus are integrals. The standard integral is a product of two averages and describes distance or area. In multi-variable calculus, the Jacobian matrix is used to transform integrals into a form where it is clear that the area or volume is a product of averages. For example, it is used where an average of "normal line lengths" are required to determine the given integral.
Alright, I have rambled on a bit. Coming back to the topic at hand:
"To construct an equilateral triangle on a given finite straight line."
A lot of the propositions (in fact almost all of them) in book I are constructions using a compass. Bk. 1, Prop.1 describes how to reify an equilateral triangle. What is vitally important for the astute scholar is to realise early, that these constructions are NOT real. They are visualisations. For example, _______________ is NOT a straight line. It is only the visualisation of a straight line. A triangle as drawn on paper is NOT a triangle, it is a visualisation of the concept where the shortest path between 3 points is described. A dot on paper is a visualisation of point or location. You cannot understand the Elements in any amount of depth, unless you grasp the main concepts right at the start. I can't tell you how many professors would fail an exam that I set on the Elements.
Now, the construction does not require continuity. It requires only the knowledge that paths intersect at given points. Let's quickly see how this is done:
1. Draw any first circle using compass 2. Draw line (using straight edge) through centre of circle, that is, a diameter. 3. Draw another congruent circle whose centre is intersection of diameter and first circle. 4. Draw radii to either point of circle intersections.
An equilateral triangle is formed because the circles are congruent and their centres lie exactly the length of one radius apart.
As previously stated, that the two circles intersect depends on showing that a line between two points, one outside the circle and one inside the circle, meets the circle. Not a straight line, any line. In particular, a portion of the circumference of the OTHER circle.
> Doesn't it? It seems obvious to the intuition, but that intuition is based on prior concepts of the continuity of the plane and of lines on the plane.
No connection at all. Continuity is irrelevant. A straight line does NOT consist of points contrary to popular academic thought. The real number line is itself a contradiction in terms. The claim that real numbers correspond to points on a number line is absurd! It's easy to reify certain endpoints on a number line, but the line segment lies between two points. It does not include the endpoints.
> Doesn't it?
Of course not.
> Maybe I'm just as you imply confusing myself.
You are confusing yourself. It has zero to do with continuity.
In the New Calculus, theorems such as Green's and Stokes' divergence are very easy to understand. While most professors can evaluate line integrals, their understanding of the same is pitifully low. Actually their comprehension of many concepts leaves much to be desired.
In my unpublished book, I include a brand new mathematics based on tangent objects. Vector calculus is greatly simplified. As you may or may not be able to tell, I *love* simple concepts. :-) If one forms good definitions, even the most complex mathematics is simple.
As long as morons continue calling me a crank, there is a good chance that you shall never get to know any of this incredible knowledge.