Euclid was one of the most prominent and influential mathematicians in history. While little is known about his life outside of mathematics, the contributions he made through the writing of his Elements have laid the foundations for both geometry and mathematics in general. The work done by Euclid helped to systematize and began to formalize the field of mathematics with his reliance on proofs, the use of definitions, and the establishment of axioms. This last element, the axiom, has become pivotal in modern mathematics, and deserves a small discussion of its own.
Axioms are statements that are used as the starting point of mathematical reasoning. Their truth is incontrovertible in that they can be accepted without proof; for instance, an axiomatic statement from geometry is that two points in a space can be connected by exactly one line. Such statements are self-evident and, without them, the formulation of more complicated statements like definitions and theorems would not be possible. Presumably aware of this, Euclid took care to implement axioms in his work, leading to his recognition as one of history's most notable figures in the field of mathematics.
As a result of Euclid's rigorous standards in his writing, The Elements long stood as a leading text in mathematics. The work, consisting of 13 books, was unmatched in terms of its ability to show and prove many geometric properties. For this reason, The Elements was used through the nineteenth century to teach concepts of geometry. This is an impressive fact, given that Euclid wrote the work in about 300 BC. Perhaps no written work, next to religious texts, has remained so prevalent throughout history.
It is clear, then, that the importance of Euclid lies in the fact that he was one of the first mathematicians to begin formalizing the subject, particularly in the area of geometry. He recognized the need to state axioms in order to write formal definitions of terms. From this point, Euclid deduced logical conclusions from his assumptions to prove statements using a logical method that could be followed by the reader. Let's look at one of his statements and its proof.
Book IV, Proposition 31 of The Elements states that "any composite number is measured by some prime number." His proof, cited here, is as follows:
"Let A be a composite number. I say that A is measured by some prime number. Since A is composite, therefore some number B measures it. Now, if B is prime, then that which was proposed is done. But if it is composite, some number measures it. Let a number C measure it. Then, since C measures B, and B measures A, therefore C also measures A. And, if C is prime, then that which was proposed is done. But if it is composite, some number measures it. Thus, if the investigation is continued in this way, then some prime number will be found which measures the number before it, which also measures A. If it is not found, then an infinite sequence of numbers measures the number A, each of which is less than the other, which is impossible in numbers. Therefore some prime number will be found which measures the one before it, which also measures A. Therefore any composite number is measured by some prime number."
The language Euclid uses is rather unconventional by our standards, so let's "translate" his proof into something we would more easily understandable with terminology we are used to:
Let A be a composite number. Since A is composite, there must exist some integer B that divides it. If B is prime, then the proof is done. However, if B is also composite, some other integer C must divide B. By the transitivity property of "divides", we have that C also divides A. Then, if C is prime, the proof is done. But if C is composite, then some other integer divides it. We can continue in this way and find that some prime number will be found that divides the number before it, which must in turn divide A. If such an integer isn't found, an infinite sequence of numbers will be found that divides A, each getting progressively smaller than the last, which is impossible in the set of integers. Therefore, some prime number will be found that divides both the one before it and A. Therefore, any composite number is divisible by some prime number.
As an example, consider the composite number 24. While 12 divides 24, 12 is not prime. Likewise, 6 divides 12 and 24 but is not prime. However, 3 divides 6, 12, and 24. Since 3 divides 24 and is prime, we have found a prime divisor of our chosen composite number. In addition to number theory, propositions in The Elements covered much in the area of geometry. Next, we look at one such proposition and an interesting application.
In class, we learned Thale's Theorem states that a triangle inscribed in a circle is a right triangle if and only if one of its sides is a diameter of the circle. We can use this theorem to draw lines that are tangent to a given circle. First, we will need to discuss a result from Book 3 of Euclid's Elements, which states that "if a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent" (cited here). That is, if we choose a point on a circle and draw the line tangent to the circle at that point, the tangent line will be perpendicular to the radius formed by the the chosen point. This is shown below in Figure 1, where AB is a radius, and the shown line is tangent to the circle at point B.
How are we certain that the radius is perpendicular to the given tangent line? For a formal proof involving some other geometric properties, see this wiki proof. However, we can see intuitively from Figure 1 that this proposition makes sense. If we rotated segment AB about the origin and moved the tangent line with it to "draw" the circle, we would see that no part of the tangent line would ever be inside the circle. (Indeed, every point in the plane other than those contained in the circle would be a point on the tangent line at some point.) To get a visual idea of this concept, we can draw several radii and their corresponding tangent lines. This is shown in Figure 2.
We see in Figure 2 that no point of any tangent line lies inside the circle. Including more tangent lines and removing their respective radii, we begin to see in Figure 3 how the tangents, when perpendicular to the radii, collectively assume the shape of the circle.
With this consideration, then, it's clear that if the angle between a radius and the corresponding tangent line were not 90 degrees, part of the tangent line would necessarily lie inside the circle (thus creating a secant line). In essence, we would be pivoting the tangent line about the point B one way or the other, and would then be unable to draw the circle by pivoting the radius and tangent line around the origin in the same way we did in Figure 3.
Imagine performing the same process to Figure 4 as we did in Figure 3. Doing this, we would find that our original circle is no longer discernible among all the secant lines. (In fact, a smaller circle is made with the collection of secant lines, but my only support for this conjecture is a mental visualization and is off-topic.) So, in conclusion, it is intuitively true that the tangent line is perpendicular to its corresponding radius. We can now move on to our main topic of using Thale's Theorem to create a tangent line for a given circle and point lying outside the circle.
Suppose we have a circle with center A, and we want to draw a line tangent to this circle through some point X.
We know that AX is a diameter of circle C, so by Thale's theorem we know triangle AXD is a right triangle with a right angle at point D. Further, since A is the center of our original circle, we see that the segment AD is a radius of the circle with center A. We can then conclude that the segment DX is perpendicular to the radius AD. We then use Proposition 18 of Book 3 of Euclid's Elements, discussed above, to conclude that the segment DX is on the line tangent to circle A at point D. Thus, we have created a line tangent to circle A that passes through point X, as desired.
Note that two such tangents can be created passing through X. The other tangent can be constructed in a similar manner by choosing the other point of intersection of the two circles in quadrant four.