Much of the algebra, geometry, and number theory we use today was discovered and developed by the ancient Greeks. The study of mathematics and science flourished during Greece's Classical, Hellenistic, and Roman periods. Though there were many great minds working to advance human understanding of fundamental and esoteric math problems, seven Greek mathematicians stand out as the foremost thinkers.

### Pythagoras (570–495 BC)

Most people know him best as the namesake of the Pythagorean Theorem, which states that if *a* and *b* are the legs of a right triangle and *c* is the hypotenuse, then *a*^{2}+ *b*^{2} = *c*^{2}. This relation was actually known outside of Greece since long before his time, but Pythagoras and his followers (the Pythagoreans) were the first to devise a formal proof of the expression. They were also the first to prove that the square root of 2 is an irrational number (one that cannot be expressed as a fraction).

Direct evidence of Pythagoras's contributions to mathematics are scarce, and many scholars believe some results credited to him were really the work of his followers. Nevertheless, the Pythagorean school of thought is responsible for a very deep contribution--popularizing the style of rigorous proof-based mathematics.

### Euclid (c. 300 BC)

Euclid is often refered to as the "Father of Geometry" for his advances in classical geometry and number theory. His most famous written work, "The Elements," is a 13-volume set of tracts covering methods of logical proof, prime numbers, number theory, plane geometry, and three-dimensional geometry. The works were used as standard mathematical texts up until the 1800s.

Among his discoveries is a proof that there are infinitely many prime numbers, and that every number has a unique prime factorization.

To prove that the set of prime numbers is infinite, Euclid proceeded by contradiction, assuming that the set was finite. He called these *n* prime numbers *P*_{1}, *P*_{2}, ... *P _{n}*. Euclid noticed that if you multiply all of these primes and add 1 to the product, you end up with a number that is not divisible by any of the

*n*primes in the original set. Thus, this number is either itself a new prime, or a product of other prime factors not in the original set. Applying this algorithm over and over means you can generate infinitely many primes.

### Archimedes (287-212 BC)

Like other ancient Greek scientists, Archimedes was not only a mathematician but also an astronomer, physicist, and engineer who put his talents and discoveries to practical use to improve the lives of his fellow citizens. While he is more famous for his engineering feats, he was also a brilliant mathematician in his time.

Archimedes was the first to prove that the area of a circle is pi times the square of the radius, as well as provide a finer estimation of pi. (The area formula was known empirically but never proven before.)

He also figured out the area of a parabolic section using a method of infinitesimals, similar to concepts used in integral calculus. The geometric relation which he was most proud of discovering was the relation between a sphere and a cylinder with the same diameter and height. It turns out that the surface area of the cylinder (including both circular bases) is 1.5 times the surface area of the sphere. And likewise, the volume of the cylinder is 1.5 times the volume of the sphere.

### Eratosthenes (276-195 BC)

Erathosthenes gained fame for discovering a purely geometric method of calculating the Earth's circumference, and being the first to do so with an amazing level of accuracy, considering the limited geographical knowledge available in his day.

Though Erathosthenes employed most of his mathematical advancements in the service of cartography and geography, he also contributed ideas to pure mathematics.

For example, he devised a method to find all the prime number up to a given number *n*, the so-called Sieve of Eratosthenes algorithm.

### Heron of Alexandria (10-70 AD)

Heron was an inventor, mathematician, and theater-lover who lived in the Greek colony of Alexandria in Egypt. Historians dispute his date of birth and death, as well as the authorship of many works traditionally credited to him. Much of this confusion can be attributed to the fact that "Heron" was a very common name in his day.

Nevertheless, Heron's name has stuck to the formula for the area of a triangle in terms of its side lengths. If a triangle has sides measuring *a*, *b*, and *c*, then its area is

sqrt[*s*(*s-a*)(*s-b*)(*s-c*)]

where *s* is half the perimeter of the triangle, (*a+b+c*)/2. Heron is also the namesake of Heronian Triangles and the Heronian Mean, a type of average between two numbers that arises in the study of conical fustrums. The Heronian Mean of *x* and *y* is

[*x* + sqrt(*xy*) + *y*]/3

### Diophantus (200-284 AD)

Diophantus was another Greek mathematician from Alexandria, which was a center for scientific discovery in the early centuries AD. Diophantus's most famous work is *Arithmetica*, a treatise that contains hundreds of problems and solutions in the field of indeterminate integer equations. Such equations now bear his name, as they are called Diophantine Equations.

A typical example of a simple Diophantine problem is to find all integers solutions (*x*, *y, z*) to the equation

3*x* + 2*y* + 5*z* = 241

Diophantus also posed more challenging equations involving powers--squares, cubes, etc. Many of these problems are quadratic, for instance, finding all integer solutions to 3*x*^{2} - 2 = *y*^{2}

### Pappus of Alexandria (290-350 AD)

Often considered the last of the great ancient Greek mathematicians, Pappus worked during a time when mathematical advancements were stagnating in Greek civilization. Like Euclid, Pappus was prolific and his works have been compiled into several thick volumes.

His favorite area of study was geometry, where he made many remarkable discoveries:

(1) Hexagonal Theorem: Given three collinear points A, B, and C, and another set of collinear points D, E, and F, you can generate a third set of collinear points from the intersections of line segments between the first two sets.

(2) Chain of Pappus: Start with two circles tangent to each other with one inside the other. Next consider the infinite chain of smaller tangent circles inside the larger circle. It turns out that the centers of these circles all lie on an ellipse. (See dotted curve in figure.)

## Comments

Very interesting, we owe so many of the ideas we take for granted about the world to the Greeks.

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