Note on Euclid's elements

Inspired by Einstein's quote, "If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker," I recently purchased the book "Euclid's Elements" for my son. With this esteemed work on our bookshelf, I hope it brings an opportunity for my children to one day open its pages and delve into a new world of wonder. This post is to share two interesting discoveries after I read the first three pages of the book.

Euclid's fourth postulate

At the beginning of the book, Euclid proposes five postulates for plane geometry, which I quote directly from this web:

1. A straight line segment may be drawn from any given point to any other. 
2. A straight line may be extended to any finite length. 
3. A circle may be described with any given point as its center and any distance as its radius. 
4. All right angles are congruent. 
5. If a straight line intersects two other straight lines, and so makes the two interior angles on one side of it together less than two right angles, then the other straight lines will meet at a point if extended far enough on the side on which the angles are less than two right angles.

Among these five postulates, the first three look trivial: they tell us how to draw a straight line segment, straight line and circle. The fifth postulate, also known as the Parallel Postulate, holds the most significance in the history of mathematics on non-Euclidean geometry. What I don't understand is the fourth postulate. What does the fourth postulate really mean?

To answer this question, let's first recall Euclid's definition of right angle: When a line segment AB intersects with another line segment CD at the point B, as shown in Fig. 1, both adjacent angles $\angle ABC$ and $\angle ABD$ are right angles when $\angle ABC=\angle ABD$.

Fig. 1  A line segment (AB) drawn so that it forms right angles with a line (CD). Take from Wiki.

Literally, the fourth postulate says, at any point on the plane, its right angles (by drawing two intersecting line segments through this point) are equal to the right angles at other points. What is a geometrical plane that the fourth postulate does not hold? i. e. there exists a point at which the right angle is different from others. The answer is a cone's surface plane!

To form a cone surface, as shown in Fig. 2, one can first cut a pie piece OAA' out of a disk and then identify the two cutting boundaries OA and OA'. From the right figure in Fig. 2,  OA and OB forms a "straight line" through the cone vertex O on the cone surface plane, and the line segment OC bisects the angle $\angle AOB$. Following Euclid's definition, $\angle AOC$ and $\angle BOC$ are right angles. 

Fig. 2  The right angles $\angle AOC$ and $\angle BOC$ at the vertex of a cone. 

Now focus on the left disk in Fig. 2, which shows the corresponding locations of lines OA, OB and OC, suppose the value of the cutting angle $\angle AOA'$ is $\theta$, then the right angles $\angle AOC$ and $\angle BOC$ are equal to ${90}^{\circ}-\frac{\theta}{4}$, which is less than ${90}^{\circ}$! 

Remarks:

1. The Mandarin translation "直角" conceals the true meaning of right angle. As shown in the above example, the right angle may not necessarily be 90 degree. The literal translation of right angle, "正正好的角", reveals the meaning of Euclid's original definition.
2. The Euclid's fourth postulate excludes the existence of such conical singularity in his geometry. Similarly in physics, the existence of Hawking temperature of a black hole also relates to the absence of conical singularity in the near-horizon metrics. I won't go to details on the Hawking temperature since it is off the topic too much. Interested readers can read the notes 2 to 4 of Prof. Hong Liu's lecture notes instead.

Euclid's first proposition

Immediately after five postulates and five common notions, the first proposition in the book is to "construct an equilateral triangle on a given finite straight line". The construction looks quite simple: given the finite straight line AB, we draw two circles with the same radius $|AB|$ but different centers: one center at the point A and the other center at the point B. The two circles intersect at a point C. Then $\Delta ABC$ is the required equilateral triangle.

What makes me upset is the lack of proof that two such circles must intersect. Obviously, not every two circles necessarily intersect. Moreover, from analytic geometry, the intersection is related to the existence of solutions of algebraic equations. Therefore, a proof of intersection is indeed required for rigorous reasoning . 

However, I am not able to figure out a proof within Euclid's framework. I searched online and it turns out it is impossible to prove the existence of intersection of two such circles from Euclid's five postulates and common notions. There needs more postulates! After all,  there is only line-line intersection in Euclid's postulate (the fifth postulate). It is David Hilbert who added additional line-circle and circle-circle axioms to the Euclid's original postulates. See section 11 of Prof. Hartshorne's "Geometry: Euclid and Beyond" for details. Honest speaking, I don't have any interest to learn the details of Hilbert's axioms. Knowing that it is indeed lack of axioms in Euclid's five postulates for the proof of the first proposition suffices for me.

Finally, note that we can prove the intersection easily from analytic geometry. For this first proposition, we have algebraic equations: \begin{equation} x^2 + y^2 = R^2\,,\quad (x-R)^2 + y^2 = R^2\,,\tag{1}\end{equation} from which we solve the coordinates of the intersecting points $(R/2, \pm \sqrt{3}R/2)$. Although it does not require the additional axiom, the equation (1), however, implies the usage of Pythagoras' theorem $a^2+b^2=c^2$. In other words, the difference between Euclidean geometry and analytic geometry is that

• Euclidean geometry starts with Euclid's postulates or Hilbert's axioms, and then prove the first proposition, and eventually prove the Pythagoras' theorem. Pythagoras' theorem is the proposition 47 in Euclid's elements.
• Analytic geometry, takes the Pythagoras' theorem as an axiom (in order to compute the distance between two points in terms of their coordinates). Euclid's postulates are not needed anymore in Analytic geometry.

Modern geometry is the generalization of the analytic geometry approach. In Riemannian geometry, the very starting point is the definition of metrics that is used to measure the distance between two points: $ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}$. And nowadays, physics students studying general relativity keep talking "Gmunu".