## Euclid's Plan and Proposition 6

```It's interesting that although Euclid delayed any explicit use of the
5th postulate until Proposition 29, some of the earlier Propositions
tacitly rely on it.  For example, Proposition 16 says

In any triangle, if one of the sides be [extended], the
exterior angle is greater than either of the interior and
opposite angles.

A
*

*        *      *D
B        C

So he's saying the angle ACD is greater than ABC and BAC.  This is
not necessarily true in non-Euclidean geometry (as with triangles
drawn on the surface of a sphere).  Heath's comment on this proposition
seems slightly middled.  He says

As is well known, this proposition is not universally true,
under the Riemann hypothesis of a space endless in extent
but not infinite in size.

This is presumably a roundabout way of referring to a finite but
unbounded space, such as Riemann proposed as a space of constant
positive curvature (e.g., the surface of a sphere), but it seems to
me the proposition doesn't depend on the boundedness of the space,
per se, but on the possibility of intrinsic curvature, at least if
we allow the curvature to be variable.  In other words, the proposition
fails locally (not globally) under the assumption of sufficient
curvature.  For example, suppose we split up the earth's Northern
Hemisphere into four quadrants, each of which is a triangle with
three right angles.  If one of the edges is extended, the exterior
angle is also a right angle, so it is not greater than the opposite
interior angles.  This isn't because of the boundedness or finiteness
of the earth's surface (as embedded in 3-d space), it's because of
the intrinsic curvature.  That's what allows triangles to have a sum
of angles different than pi if the space is not flat.  It's the same
effect that allows the exterior angle to be not necessarily greater
than the opposite interior angles.

Still, Proposition 16 occurs long before Euclid explicitly invokes
the parallel postulate (in Proposition 29), so some people assume it
must be part of "absolute geometry", i.e., propositions that do not
depend on the parallel postulate.  I think it's more accurate to say
that Euclid actually tacitly assumed the parallel postulate prior to
invoking it explicitly in Proposition 29.  This is just one of several
example of logical problems in the Elements.  Nevertheless, in spite
of its imperfections, it remains a remarkable document.  Considering
how many bright men over the centuries were convinced that the 5th
postulate was actually redundant, it was a great vindication for
Euclid when non-Euclidean geometry was discovered (ironically).
It's easy to see why Newton concluded that "the ancients" knew more
than they were telling.

Just for fun, here's a summary of the functional dependence between
the propositions of Euclid's Book I, based on the Dover 2nd edition
of T.E. Heath's translation:

23 Definitions: d1 through d23
5 Postulates: p1 through p5
5 Common Notions: cn1 through cn5
48 Propositions: PR1 through PR48

PR1 = f (p3, p1, d15, d15, cn1)
PR2 = f (p1, PR1, p2, p3, cn3, cn1)
PR3 = f (PR2, p3, d15, cn1)
PR4 = f (cn4)
PR5 = f (p2, PR3, p1, PR4)
PR6 = f ( )                  <------[see below]
PR7 = f (PR5)
PR8 = f (PR7)
PR9 = f (PR3, PR8)
PR10 = f (PR1, PR9, PR4)
PR11 = f (PR3, PR1, PR8, d10)
PR12 = f (p3, PR10, p1, PR8, d10)
PR13 = f (d10, PR11, cn2, cn1)
PR14 = f (PR13, p4, cn1, cn3)
PR15 = f (PR13, p4, cn1, cn3)
PR16 = f (PR10, PR3, p1, p2, PR15, PR4, cn5)
PR17 = f (p2, PR13)
PR18 = f (PR3, PR16)
PR19 = f (PR5, PR18)
PR20 = f (PR5, cn5, PR19)
PR21 = f (PR20, PR16)
PR22 = f (PR20, PR3)
PR23 = f (PR22, PR8)
PR24 = f (PR23, PR4, PR5, PR19)
PR25 = f (PR4, PR24)
PR26 = f (PR4, PR16)
PR27 = f (PR16, d23)
PR28 = f (PR15, PR27, PR13, PR27)
PR29 = f (PR13, *p5*, PR15, cn1, cn2)   <--- 1st appearance of
PR30 = f (PR29, cn1)                           the 5th Postulate
PR31 = f (PR23, PR27)
PR32 = f (PR31, PR29, PR13)
PR33 = f (PR29, PR4, PR27)
PR34 = f (PR29, PR26, cn2, PR4)
PR35 = f (PR34, cn1, cn2, PR29, PR4, cn3)
PR36 = f (cn1, PR33, PR34, PR35)
PR37 = f (PR31, PR35, PR34)
PR38 = f (PR31, PR36, PR34)
PR39 = f (PR31, PR37, cn1)
PR40 = f (PR31, PR38, cn1)
PR41 = f (PR37, PR34)
PR42 = f (PR23, PR31, PR41)
PR43 = f (PR34, cn2, cn3)
PR44 = f (PR42, PR31, PR29, p5, PR31, PR43, cn1, PR15)
PR45 = f (PR42, PR44, cn1, PR29, PR14, cn2, PR34, PR30, PR33)
PR46 = f (PR11, PR31, PR34, PR29)
PR47 = f (PR46, PR14, cn2, PR4, PR41)
PR48 = f (PR47, PR8)

Interestingly, Proposition 6 does not explicitly invoke any axioms,
definitions, common notions, or prior postulates in the Dover edition
of Heath's translation, nor is it cited by any of the subsequent
propositions in Book I (although it is cited later, e.g., in Book II,
Proposition 4).  Here is the immaculately conceived Proposition 6:

If in a triangle two angle be equal to one another, the
sides which subtend the equal angles will also be equal
to one another.

Euclid's proof, as translated by Heath, is as follows:

"Let ABC be a triangle having the angle ABC equal to
the angle ACB; I say the side AB is also equal to the
side AC.  For, if AB is unequal to AC, one of them
is greater.  Let AB be greater; and from AB the greater,
let DB be cut off equal to AC the less; let DC be joined
[as shown below].
A
/\
/  \
/    \
D/      \
/ '.     \
/    '.    \
/       '.   \
/          '.  \
/             '. \
/                '.\
/____________________\
B                      C

Then, since DB is equal to AC, and BC is common, the two
sides DB, BC are equal to the two sides AC, CB respectively,
and the angle DBC is equal to the angle ACB.  Therefore,
the base DC is equal to the base AB, and the triangle DBC
will be equal to the triangle ACB, the less to the greater,
which is absurd.  Therefore AB is not unequal to AC; it is
therefore equal to it."

This is the first appearance of "reductio ad absurdum" in The Elements.
Given a triangle ABC with equal angles ABC and ACB, we make the
supposition that AB and AC have unequal lengths, from which it follows
that one is longer than the other [presumably by the definition of
equality].  Without loss of generality we assume that AB is longer
than AC, which implies that there is a point D on the line such that
the length of BD equals the length of AC.  Euclid also tacitly assumes
that this point D lies BETWEEN the points A and B, and that this
"betweenness" implies that the triangle DBC is smaller than the
triangle ABC.  It's been pointed out by Hilbert and others that this
really needs to be a postulate, but there are no postulates of
"betweenness" in The Elements.

In any case, the argument continues by noting that we have two
triangles, DBC and ACB, such that DB equals AC [by construction]
and BC equals CB [by an interesting tacit assumption of what might
be called "commutativity" of distances between points], and of
course we have our basic fact that the angles DBC and ACB are equal
[assuming that since D is on the interior segment AB the angle DBC
equals the angle ABC].  Now we invoke Proposition 4 (although the
Dover edition of Heath fails to include this reference), which says

If two triangles have the two sides equal to two sides
respectively, and have the angles contained by the equal
straight lines equal, they will also have the base equal
to the base, the triangle will be equal to the triangle,
and the remaining angles will be equal to the remaining
angles respectively, namely those which the equal sides
subtend.

Thus, Euclid claims that the triangles DBC and ACB must be equal,
and yet we know DBC is smaller than ACB, so we have a contradiction.
QED

This particular proposition has been the subject of considerable
discussion over the years, since the rigor of the argument can be
challenged in various ways.  See, for example, the related articles

Are All Triangles Isosceles?
Do Equal Bisectors Imply Isosceles?

Focusing just on the relations between the Propositions, we could
summarize Book I as follows:

1   5   10   15   20   25   30   35   40   45
|   |    |    |    |    |    |    |    |    |
************************************************
1
2  *
3  *
4
5    **
6    **
7      *
8      *
9  * *    *
10  *  *    *
11  * *    *
12         * *
13            *
14            *
15            *
16    **     *    *
17              *  *
18    * *          *
19      *            *
20    * *             *
21                 *   *
22    *                *
23         *             *
24    ***             *   *
25     *                   *
26    **           *
27    **     *    *
28              * *           *
29              * *           *
30                              *
31                        *   *
32              *               * *
33     *                      * *
34     *                     *  *
35     *                        *    *
36                                  ***
37                                *  **
38                                *  * *
39                                *     *
40                                *      *
41                                   *  *
42           *            *       *      *  *
43     *                     *  *
44                *             * *          **
45               *              **  **       * *
46    *       *                 * *  *
47     *         *                *         *    *
48    *    *  *                                   *

Of course Book I is just a small part of The Elements.  Here's a

I. Basic Propositions on Lines, Triangles, and Squares
II. Gnomon and Geometric Algebra
III. Geometry of the Circle
IV. Rectilinear Figures Inscribed or Circumscribed in Circles
V. Eudoxus' Theory of Proportions
VI. Application of the Theory of Proportions to Plane Geometry
VII. Number Theory (Greatest Common Divisor, Euclidean Algorithm)
VIII. Geometric Progressions
IX. Number Theory (unique factorization, infinitude of primes,
perfect numbers)
X. The Theory of Incommensurable (Irrational) Magnitudes
XI. Introduction to Three-Dimensional Geometry
XII. Method of Exhaustion For Areas and Volumes
XIII. Construction of the Platonic Solids

Isaac Newton's assistant at Cambridge claimed that during five years
he saw Newton laugh only once.  Newton had loaned a copy of Euclid
to an acquaintance, and the gentleman asked what use it was to study
Euclid, "upon which Sir Isaac was very merry".
```