Euclid's Elements

I began Euclid's Elements a few days ago.  Here is

Proposition I.  On a given finite straight line to construct an equilateral triangle.

Elements, Book I,  Proposition I

The construction is:
  (Given finite line AB)
  Draw the circle with center A and radius AB.
  Draw the circle with center B and radius AB.
  The two circles intersect at exactly two points.  Choose one of these points and call it C.
  Join the straight lines AC and BC.
  Then AC, AB are both radii of circle A :   AC = AB.
  And AB, BC are radii of circle B, and thus equal:  AB = BC.
  Hence AC = AB = BC.   The three sides are equal to one another, being the thing to be shown.


It's simple, but Euclid's method has garnered a lot of attention.  There is a great deal of nonsense said about Proposition I.  For example, consider the following passages from T.L. Heath:

"It is a commonplace that Euclid has no right to assume, without permissing some postulate, that the circles will meet in a point C.¹  To supply what is wanted we must invoke the Principle of Continuity (see note thereon above, p. 235).²"

"Zeno's remark that the problem is not solved unless it is taken for granted that two straight lines cannot have a common segment has already been mentioned....  Thus, if AC, BC meet at F before reaching C,³ and have the part FC in common, the triangle obtained, namely FAB, will not be equilateral, but FA, FB will each be less than AB⁴.  But Post. 2 has already laid it down that two straight lines cannot have a common segment.⁵
"Proclus devotes considerable space to this part of Zeno's criticism....." 

What does the note on page 235 say?  It is astonishing!  Here is a bit:

"The use of actual construction as a method of proving the existence of figures having certain properties is one of the characteristics of the Elements.  Now constructions are effected by means of straight lines and circles drawn in accordance with Postulates 1−3; the essence of them is that such straight lines and circles determine by their intersection other points in addition to those given, and these points again are used to determine new lines, and so on.  This being so, the existence of such points of intersection must be postulated or proved in the same way as that of the lines which determine them.⁶  Yet there is no postulate of this character expressed in Euclid⁷ except Post. 5.  This postulate asserts that two straight lines meet if they satisfy a certain condition.  ....if the existence of the intersection were not granted, the solutions of problems in which the points of intersection of straight lines are used would not in general furnish the required proofs of the existence of the figures to be constructed. ⁸ 

"But, equally with the intersections of straight lines, the intersections of circle with straight line, and of circle with circle, are used in constructions.  Hence, in addition to Postulate 5, we require postulates asserting the actual existence of points of intersection of circle with straight line and of circle with circle⁹.  In the very first proposition the vertex of the required equilateral triangle is determined as one of the intersections of two circles, and we need therefore to be assured that the circles will intersect.  Euclid seems to assume it as obvious,¹⁰ although it is not so; and he makes a similar assumption in I. 22."  

[The thirteen books of Euclid's Elements: Books I and II;  T.L. Heath and  J.L. Heiberg, Cambridge University Press, 1908; pp. 242, 234-5.  Available as a free Google ebook.  Footnotes mine; see below.]

That's not the end of it!  But, more to the point,

THE GRUMPY PART

{click to skip to the next post about Euclid }

this is all nonsense.  Pointwise, because these are arguments:
1.  Euclid does not assume it.  It is not in the postulates, definitions or the common notions.  Heath is upset because Euclid does not postulate the point in advance of the construction.  But no such postulate would be sufficient.   The question is, does the point, in fact exist?  It can only be given by demonstration, i.e. construction.

2. To supply what is wanted we must construct the point.

3.  !! They do not.
Are we playing pretendsies?
Suppose that instead of meeting at a point C, the lines intertwine, wrap one another in lengthening spirals.  Instead of meeting at a point C, the lines brood and tigthen, kink and curve against the paper.  The point C is not met.  There well be no more business of point C.  Suppose they push and crouch, the paper begins to flap about.  Suppose there is a pop of air as they free themselves, spring out of the page in snaking cylinders, work around the table and chair legs, hungry for rectilinear motion, dimension, line and curve.

4. There are two issues here.
First, this way of reasoning is called the fallacy of the irrelevant thesis: seeking to prove, perhaps successfully, an issue not in question.  If the point F existed, then the following problem would arise.   Indeed, let us grant this.  Then the question remains, does point F exist?  It does not.  It is an objection for a case which does not exist.  And what about the other cases which don't exist? What if AC, BC meet in a pony before arriving at C?  What if they follow the little piggy to market?

Second, a construction is an existence proof. All arguments which claim to truth rely upon facts independent of the argument itself; they admit revision.  Where something is given by construction, a valid construction is necessarily a valid counterexample.   Zeno clearly sees that revision is admissible, but does not seem to understand the structure.  All he has to do is construct the objection, and it will be valid. His constructions fail; they are pretend.
Euclid is not difficult to follow.  In the present case, all we need to do is build the point F .  Or a pony.  Or any of the infinite possible nonexistent cases which Euclid does not mention.

5. Postulate 2 is not necessary.  Zeno is ignoring Postulate 1: "To draw a straight line from any point to any point."  For example, a construction of Zeno's argument is presented here:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html
Point C is right there.  He refuses to connect it, makes up a new point and connects that one instead. If Zeno does not agree to draw a straight line from one point to another, then he should object to Postulate I.  He is free to make any geometry he wants.  He seems to want Euclid's.  It is up to him to decide whether he accepts the postulates and definitions or not.

INTERMISSION

I'm arguing with a dead guy!

[pinocchio] I'm a real boy! [/pinocchio]

Have you thought about  skipping to the next post, about subdivision?

6. Hahahahahahahaha!  This is a good one!  "Sally's existence must be postulated or proved in the same way as that of the lines which determine her location."  !!   I want to try this sentence again, without going crazy in the middle.  "The existence of such points cannot be proved by argument; the points must in fact exist.  Hence, Euclid proceeds by construction of the points."

7.  Of course not.  What is strange to me about this sentence and what follows is that Heath reinterprets Postulate 5 every time he discusses it.  His interpretation and Euclid's definition imply one another; but Heath's version is less amenable to what he wants than the postulate itself.

8.  "Furnish...the proofs of the figures to be constructed"  ?       *!!*
If we insist on the argument,  the following cases are sufficient to demonstrate the error:
    -Suppose such proofs to exist, and the diagram also can be constructed.
           Then the diagram can be constructed.
     -Suppose such proofs not to exist, but diagram can be constructed.
           Then the diagram can be constructed.
       -Suppose proofs to exist which state the diagram cannot be constructed, and the diagram can nevertheless be constructed.
           Then the diagram can be constructed and the proofs are meaningless.
The proof is the construction itself.

9. No we do not.  Strike "postulates asserting".

10.  This is the heart of the matter.  Euclid does not assume this.  He seems clear that such assumptions are neither necessary nor sufficient.  If Heath refers to those properties which can be deduced directly from the Definitions, then of course Euclid assumes them.  But I do not think this is what Heath means.


I have overstated some of the points.  Take me to task!  There are additional complexities.  Where?  I am not dismissing continuity. I find  Euclid's Definitions sufficient to meet Heath's arguments, and more elegant than most modern theory on continuity.  But this is still not good enough.  In the end, the point must be drawn.  Else, the rest is sophistry.