The Elements, Book I (Prop I - V)

The Thirteen Books of the Elements

Euclid


Book I

Reference edition: Sir Thomas Heath, Johan Ludvig Heiberg
 The University Press, 1908

Free Google Ebook

Proposition I:  Given finite straight line AB, to construct an equilateral triangle.

Proposition II:  To place at a given point a straight line equal to a given straight line.

Given point A and straight line BC,

Construct equilateral triangle ΔABD [Prop. I]

Construct the circle with center B, and radius BC.

Extend the straight line DB to intersect the circle.  Call the point of intersection F
BC ≅ BF

Construct the circle with center D and radius DF.

Extend straight line DA to intersect this new circle.  Call the point of intersection G:

Now, DG ≅ DF

and AD ≅ BD
And therefore DG − AD ≅ DF − BD
But DG − AD = AG
and DF − BD = BF
Hence AG ≅ BF.

Proposition III:  Given two unequal straight lines, to cut off from the greater a straight line equal to the lesser.

Given straight line AB (the greater) and straight line C (the lesser),

Use an endpoint of C to draw a circle of radius C.

Connect A to the endpoint and build an equilateral triangle. (Prop I)
Extend the side of the triangle to the circle. (Prop II)

Draw a second circle with center at D and extend DA to intersect the circle. (Prop II)

Draw a third circle with center A which also intersects this point:

The intersection of this last circle with AB is labeled γ.
Aγ ≅ C and the construction is complete.

Proposition IV: Side-Angle-Side Congruence

If you can find more than one way to draw the segment BC, let me know.

Proposition V: In an isosceles triangle, the base angles are equal.  If the equal sides are extended in straight lines, the angles under the base are equal.

Given  AB

Construct AC = AB and isosceles triangle ΔABC.

Construct AC = AB and isosceles triangle ΔABC.
Extend AB  to arbitrary point F.
Place  CG = BF  [Prop II, or by definition with the compass]

Connect   FC, GB
     ΔFCA ≅ ΔGBA  [Prop IV, common angle ∠A]
The corresponding angles and sides are then equal, namely
     FC = GB,    ∠CFA ≅ ∠BGA

Then ΔFCB ≅ ΔGBC and
     ∠FBC = ∠GCB,   (the angles under the base)
     ∠BCF = ∠CBG
And  ∠ABC  = ∠ABG −∠CBG  = ∠ACF − ∠BCF  = ∠ACB (the base angles of the isosceles triangle)
Which were the things to be proved.

{Next −−>}

____________________________________
There are hundreds of propositions remaining of which I will post a selection, including only a restatement of the Proposition and a completed diagram.

Continuity: Regions

IN THE BLUE CORNER, Advanced Calculus:

"

Multiple Integrals  

§ 1. Introduction 

In this chapter, we shall discuss double and triple integrals.  We shall follow as closely as possible the analogy with the theory of simple integrals developed in the previous chapter. 

1.1 REGIONS
     We have already discussed in Chapter 1 regions of the plane.  Let us collect here the notations which will be needed in the present chapter.
     A domain D is an open connected set of points.  That is, every point of D is the center of some δ-neighborhood, all of whose points are points of D; and any two points of D can be joined by a broken line having a finite number of segments, all of whose points are points of D.  A domain is bounded if all its points lie inside some square.
     A region R is a closed point set consisting of a bounded domain plus its boundary points.  We shall assume further that the boundary of R consists of a finite number of closed curves that do not cross themselves nor each other.  Note that the regions here defined and designated by the letter R are special cases of the more general ones of §3.1, Chapter 1.  In practical problems, R will usually be given in terms of its boundary curves.  For example, R might be the set of points between two concentric circumferences plus the points on the circumferences.  More frequently, we shall meet regions that can be most simply described by use of functions. Accordingly, we shall have a special notation for these.
     Let φ(x) and ψ(x) ∈ C in a ≤ x₁ ≤ b and φ(x) <ψ(x) in a < x < b.  Then the region R_x or R[a, b, φ(x), ψ(x)], is the region bounded by the curves   

x = a,     x = b,     y = φ(x),     y = ψ(x) 

If (x₁, y₁) is a point of R_x, then a ≤ x₁ ≤ b and φ(x₁)≤ y₁ ≤ ψ(x₁).  A line x = x₁, a  < x₁ < b cuts the boundary of R_x in just two points.  For example, the region R[-1, 1, − √(1 − x²), √(1 − x²)] is the circle x² + y² ≤ 1.  We could define in an obvious way a region R_y.  The region R described above as lying between two concentric circles is neither an R_x nor an R_y. It could be divided into four regions R_x, for example, by two vertical lines tangent to the inner circle.  These vertical lines would be counted twice, as the boundary of adjoining regions.
     A region R is simply connected if its boundary consists of a single closed curve.  The concept of the area of a region R will be assumed known.  Of course, the area of R_x is known from elementary calculus, and the area of R could be defined by use of a limiting process.  The diameter of a region R is the length of the longest line segment that joins two points of R.  In the case of a circle this coincides with the elementary notion of the diameter.  Observe that, if a region R varies so that its diameter approaches zero, then its area also approaches zero.  The converse is not true. 

1.2 DEFINITIONS
     We begin  by dividing a given region of R of area A into subregions.
As in the case of simple integrals, we introduce certain simplifying notations.
"
-Daivd Widder, Advanced Calculus

IN THE RED CORNER,  Euclid of Alexandria.

"The extremities of a surface are lines."
-Euclid, The Elements, Definition 6

_______________________________________ 

Notes

There are important differences between the two quotes.  We need the symbols necessary to operate algebraically, and to perform the Calculus.  To define arbitrary surfaces.  Our pencil is the function.  Still.  The surfaces we will be operating on in Chapter 6 of Widder's Advanced Calculus are precisely those which are defined in Euclid.  The definition is rigorous.  And so,

I have some questions for you, Modern Abstract Algebra.  And I don't mean Gauss.

1. What is the difference between a Dedekind section and Euclid's Definitions?
2. What is the relationship between infinite division and continuous measure?
   (Hint: No matter what anyone says, I can prove false by counterexample any mathematical argument which concludes by proof that I, Ryan, can measure.  All I need to do is disagree.  Likewise, a proof that I cannot measure.  All I need to do is say, "one apple."
   And likewise, I can prove any assertion that integral, unbroken measure does not exist, by waiting for a new instant in time, unbroken from the previous.  Or taking a step, some distance in some direction.  Left foot, right foot.  I defeat you, fake divisor-paradox!  Time for a nap.)
3. What is the logical status of a postulate?
4. I say, mathematics is not a logical circle.  Why?

I think, Mr. Modern, you do not answer these questions accurately.  Prove me wrong.  I'm asking for it.  See?

{Thus goad I a concept.
 Poke. 
 Thus}

Euclidean Subdivision

Playing around with Proposition I two mornings ago, I constructed a curious diagram.  On nice paper it looks like this:

Euclidean SubD - Quad

The only thing given is finite segment AB.    There are two closed shapes: the square ABCD, and the shaded figure in the middle (mouse over).  Let's call the square the base geometry, and the curvy figure in the middle the subdivided geometry.  That's more or less what 3D programs call them.  The construction here is simple, but I'll list a few key details that might not be obvious:

 In Euclid's Proposition I , we draw two circles, and from these the square can be built.  For example,  AD is the radius of once circle, and tangent to the other.

   The sides of the subdivided figure are the points of equilateral triangles.

   To get the corners, I drew new circles around the four new points.

   The  point at the center and the segments connecting it are part of the new surface.  It thus has 9 points: 8 around the perimeter, and one at the center.

I have taken some liberties with Proposition  I.  For example, I have used the tangents of circles at points to construct the square.  I was doodling. 

The significance?

I think every 3D modeler recognizes this shape.   It is a smoothly subdivided surface.  Drop a Sub-D modifier onto your stack, and this is what happens to your model.  I have had subdivision on my mind for awhile, and I did not know where to start, so I started here.  I even made a plan for going through the Elements, and turning Propositions into algebra, vector equations, and algorithms.  But this is exactly what I wanted, and it only took the first Proposition!

I still need to make it an appealing and functional subdivision routine.  Several modifications should correct most of the problems:

1) As shown, none of the new points lie on the base geometry, which is too aggressive.  Next, I will split each edge at its midpoint, and leave these points on the base geometry.  Only the 'corners' will pull inside the original perimeter (or push out).  If the geometry is subdivided again, all points may lie off the original edges.

Splitting the edges at midpoints is also faster to compute.

2) The math does not need to be exact; it just has to look nice.  I used 30-60-90 triangles, so the distance to a new point is a term in √3.  I can choose friendlier numbers.  An even better solution is to build a template which gives the new topology like a stencil, tessellated across the whole mesh.

3) Triangles!  Arbitrary polygons can be divided into triangles. I need a subdivided triangle.  

Fig. 2: Euclidean Sub-D: Tri

4) Topologically, all triangles can be seen as the same, all four-sided shapes the same....and so on.  I am not yet considering polygons with more than four sides, so Figures 1 and 2 cover all cases.

5)  Order of construction:

Edges first, then vertices.

The midpoint of any edge can be found, independent of the surrounding topology.  So I will begin there.

   i) Add midpoints to each edge.  For example, in the new version of Fig 2,  m₁ will fall directly on the edge AB,  m₂ on  BC,  and  m₃  on AC.

  ii) When all edges of any polygon are split, add the center point.  This is easiest to see with Fig. 2.   If we complete the segments Am₂, Bm₃, Cm₁,  they intersect at a common point in the center.

  A segment joining a vertex to the midpoint of the opposite side is called a median.  In a triangle, the three medians always intersect at a common point inside the triangle.  This point of intersection is the geometric center (or centroid).  So I will use it.

   iii) This leaves A',  B' and C'.   They may be connected to other polygons, or they may be free edges.  The same general rule can be applied to both cases.  Say C joins four polygons.  When all four of these polygons have a geometric center,  balance C' between these centers.

      The rule is then, when all adjacent polygons to a vertex have centers, balance the new vertex between them.

   Balance:    The geometric center of a polygon is its balance point.  Then the obvious approach is to balance A', B'.... the same way.  This is (I believe) a tensor.  Say all the lines connected to C are flexible.  Where does C fall if the tension on these lines is balanced?  Or relaxed?  This question results in a system of linear equations which are most easily handled as vectors.

6) The third dimension: I plan to ignore it.  A surface is is two dimensional.  For example, you can attach an image to any surface.  However much it curves in space, you can always paste (map) a flat image to it.  Every polygon either lies in a plane, or there is a plane that is closest to it.  Viewed in this plane, a polygon will appear to subdivide as if it were perfectly flat.  The amount any vertex moves out of this plane will be completely determined by the geometry of the model.  The more rapidly the base geometry is changing perpendicular (normal) to the plane, the less the subdivided surface will be able to follow it.  Sharp angles smooth to a more gradual change.

7) The distance function is two-dimensional.  The math is fine, but how will it look?  3D models are surfaces, but they represent 3-D shapes objects.  A 2D distance function might be too tight. Consider some distances:

    From the origin (zero)  to 1 on the number line:

           1 - 0     =  1  

    From the origin (0, 0) to the point (1, 1), in the x-y plane:

           √(1² + 1²) =  √2.     (Pythagorean theorem)
     From the origin (0, 0, 0) to the point (1, 1, 1) in x-y-z space:

           √(1² + 1² + 1²) = √3.         (same)

 If AB in Fig. 1 has length of 2, then my unit subdivision length is √3.  I will keep this metric while developing the algorithm, and make the number under the root a variable.  I hope scaling the dimension of the subdivision length independently will make the algorithm both flexible and intuitive.  Scaling between "k=1.414" and "k= 1.732" on a linear slider may cover the same range of values as k = √2 , k = √3, but the relationship is unclear in the first case, and the scaling wrong.

I seek an adaptable subdivision method.  Once I have a working algorithm, I will revisit questions which I have glossed over.    #s 6 and 7  are such questions.  In general, "adaptable" implies greater sensitivity to rapid changes of direction in the base geometry (like corners).  Derivatives of higher order could be useful for a project involving assets designed from the ground up to be aggressively tessellated, and categorized by material type.

For example, how can a sub-D routine be tailored to hard surfaces?  And terrain? (...and types?)  Any spatial characteristics of the base mesh could be important factors.  One could also surely model using different subdivision templates throughout the mesh.  This would be labor-intensive for the artist, but it could also prove very efficient and intuitive, especially for hard surfaces, modeling for games.  Given by templates means a pool of scaled normal map files could potentially be used to speed normal map generation for complex surfaces, and to allow the low- and hi-poly meshes to be the same.  But that's just pretend for now;   this part of my project is not pressing.  It may be some time before I revisit subdivision and construct a basic algorithm.

Regardless, I think this provides sufficient generalization to tailor the algorithm to taste, and any problems which arise can be adjusted.  It's all made up of course, so maybe it'll be a disaster.  That's fine.  Now I have something to fix.

Make Your Own

I am happy to give the full construction of each diagram, and of my subdivision method when it is complete. It was more important to me to show how a subdivision routine can be constructed by anyone with a compass and straightedge.  The specific steps I have taken are personal preference.  I was playing a game with a compass.  Any way to break up a shape is a way to subdivide, and whatever people might want you to think, a bunch of special vocabulary is unnecessary.  It gets in the way.  If you follow that link, the authors will spend half a page describing a square with a diagonal drawn in it.  They will give it a special name.  Subdivision will appear mysterious, loaded with unforeseen complexities.*  It is not. 

Tell Me, or Ask A Question

If I have written about a problem, I have the time to go through every step.    Just ask.  I don't know what's missing from the explanation until someone tells me.  On the other hand, additional details make the broader picture harder to see.  There is a choice when writing about math: where to draw this line?

Ideally, the text should be both concise and complete.   This will take me quite some time to achieve, but that is my goal.  I tend to overcomplicate.  All suggestions about how ts can be more clearly presented are welcome.

I will add new diagrams in my next pass through subdivision.  This will provide additional clarity.

{* I had enough after a few pages.  I don't have the patience for this.}

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.