###### August 20, 2023

### Ancient Greece – Since The Beginning

**How ancient Greece birthed an entire foundation of thought for modern day technology & intellect **

Origins of Mathematical Knowledge

A multitude of classical and archaic communities paved the way for modern life and thought. Civilizations such as Rome, China, India, Egypt, Mesopotamia, Persia, and many more were able to start an array of practices- from SCRATCH. They were not privileged with many of the resources we have today, so they relied on futuristic thought and hard labor to create a life they were happy with (and proud of).

One of the fundamental empires is Greece, a civilization rich with philosophical thought, groundbreaking strategy, and a jubilant social nature. They provided a historical miscellany of concepts and contraptions, which lays ground for many of the practical modalities we modernized in order to build our own civilization. Humankind embodies many Greek contributions on an intellectual basis, and we don’t even know it!

There’s so much to dive into when exploring ancient Greek culture. Despite the ecosystem of knowledge they’ve nourished, there is a core contribution that initially planted the seed of their impact + influence on modern day thought.

**Major Contributions from Ancient Greece**

One of ancient Greece’s most influential contributions primarily involves the school of thought. Logic, philosophy, and academia was one of their strong suits, for many schools of thought were born and flourished. They all lay a foundation for other areas of thought to incubate. However, one of their greatest subjects of impact is mathematics.

Math is the backbone for many other areas of knowledge we’ve used such as Science, Astronomy, Architecture & Engineering, Warfare, even Agriculture. The Greeks needed to concoct a logical methodology to formulate tools, tricks, and processes that would build their empire as efficiently as possible.

LONG before modern technology, math was a slightly more laborious process. For instance, everything was written or through word of mouth; for instance, ancient civilizations kept charts and tables on clay tablets or papyrus scripts. And because the value of thought was so potent, the exchange was that much more impactful!

**Between 685-525 BCE, (before the common era), Egypt’s ports along the Nile river opened up to Greek trade**, breaking the barrier of interaction between them. With the migration of people & goods, both verbal and written communication acted as the vehicle to carry Egyptian ideas about math. That explains how and why much of Greek mathematics was adopted from the nearby civilization.

Egypt and their neighbors, like Mesopotamia and Ionia, had some of the finest math in the world. They utilized calculations for engineering purposes, to build structures for living and business such as the Great Pyramids or boats for trade & transportation. Unlike the others, the Greeks took these mathematical calculations to create practical applications for effective life skills.

Rigor was a major characteristic of Greek math. It was meticulous, exact, and at times super specific. They spent much of their effort contemplating deeper connotations behind the math they were working on. Even the word theorem evolved from the Greek word theoreo, which translates to “I contemplate”. Therefore, Greek math was intricately rooted in the association between mathematical review and analytical scrutiny.

**Here are some of the most common contributions ancient Greek math gave to modern and even Western thought:**

**Ratios of a Triangle**: Many of us have dabbled with the Pythagorean theorem, a tool proposed by one of ancient Greece’s most impactful mathematicians, Pythagoras. The 3:4:5 triangle was easily understood as a right triangle, but the Greeks were incredibly interested in the specificities of this abstract thought unlike their Egyptian benefactors.

Eventually, they expanded on it further by trying to calculate the longest side of the triangle (the hypotenuse) by calculating the similarity between the two smaller sides. This cracked open an intellectual revolution!

**Numerical System**: One of the most popular and widely-used tools the Greek created was their base system. By picking one core number, they formulated a number system for real-life usage that was easily divisible; this helped especially with fractions and proportions.

The ancient Greeks used the base number 60, which wasn’t as difficult to apply as we think. It’s a moderately divisible number with lots of other divisible factors, which made it a pretty flexible system to work with.

**Square Root**: This was an idea concocted a tad after the Pythagorean theorem swept civilization. With the new theorem begged a new question: if two sides of a right triangle are 1 unit, and the diagonal side equates to the square root of the two sides, what is it’s exact calculation?

After trying to find the square root of 2 and realizing it was irrational, this opened a world of questions regarding the square root of all numbers and what made them rational versus irrational.

**Geometry**: This is an entire discipline with math that works with the properties and relationships between lines, points, shapes, surfaces, and higher dimensional figures. With Greek architecture and engineering came the need for deep understanding of shapes and their dimensional properties.

Because it was a time of practicality, the Greeks were really using geometry as a logistical science to calculate land measurements. This was also a practice that originated with the Egyptian mathematical perspective; how do you think they built the Pyramids so beautifully?

**Proofs**: Known as one of the most tedious and difficult techniques to master, proofs are arguments based in inference and math logic to assure the answer to a problem is correct! Other theorems and math techniques can be applied to verify the validity of the proof, encouraging the practice of deductive reasoning with logic.

The first mathematical proof was credited to another Greek math icon, Thales of Miletus. He also proposed proofs that concerned ALL mathematical shapes and figures, not just the abstract ones! His contributions kickstarted the discussion of what the Universe was made of.

###### July 23, 2023

### Introducing… Euclid!

**How the Contributions of One Alexandrian Mathematician Influenced the Course of Human Thought**

**What Started As A Thought…**

Soon blossomed into an entire movement of thought and education.

We’ve all been taught about ancient Greece as well as their intellectually rich culture. They birthed and molded so many foundational leaders of thought in realms such as math, philosophy, religion, even art and architecture.

One of the most contributors to the evolution of math is Euclid. Euclid was known and referred to as the “father” or “founder” of Geometry. Geometry is a very distinct field within mathematics responsible for explaining the relationships of planes and objects. We all start to integrate geometrical knowledge when we learn the properties of shapes, points, lines, and the connection between them all!

Euclid is the namesake of Euclidean geometry, which is the basis of plane geometry like his published works allude to. But we’ll get into that later!

**About Euclid**

What we know of Euclid derives from a summary of famous mathematicians contrived by Greek philosopher Proclus (410-485 CE). Very little detail is actually known about Euclid’s life, however, has been hypothesized and framed by key events: his birth, death, and prominent mathematical contributions.

Euclid was born approximately 325 BCE and hails from the great Alexandria, a prominent civilization in Egypt. It is believed he passed there as well about 265 BC.

He taught in Alexandria at the time of Ptolemy I Soter (367/366-283/282 BC), the Macedonian ruler of Egypt.

In the summary procured by Proclus, it describes Euclid stating:

Not much younger than these [pupils of Plato] is Euclid, who put together the “Elements”… for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry…

Euclid of Alexandria is typically mistook for Euclid of Megara, who lived 100 years before he was even born! And even though much of his existence remains in question, there is absolutely no doubt that he provided one of the most revolutionary pieces of work for math to evolve in the following 2000 years.

**Euclid’s Contributions + Euclidean Geometry**

What did Euclid ultimately contribute to math?

He wrote a collection of work known as the Elements. Euclid’s Elements is a compilation of postulates, proposals, and rules of geometry. There are five postulates introduced at the beginning of this collection, assuming the existence of points and lines and how they relate to one another.

**The five postulates state:**

– A straight line segment can be drawn to connect any two points.

– Any straight line segment can extend indefinitely in a straight line.

– Given any straight line segment, a circle can be drawn having the segment as a radius and one endpoint (of the segment) as the circle’s center.

– All right angles are congruent (equal).

– If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

**
The fifth postulate defines what’s known as a parallel postulate**, and it has not been proven yet despite many attempts.

You’ll probably find many of these postulates as basic rules introduced to you when you learned what geometry was! Elements essentially highlights the fundamentals of Euclidean geometry, which is taught in secondary education.

There are thirteen books in total, highlighting definitions and propositions revolving around the theory of geometry, proportions, circles, number theory, geometric algebra, and solid figures. Euclid dives into great detail for each subject he accentuates with his writing to ensure the comprehension of how they all correlate to one another.

It’s one of the oldest surviving mathematical publications known to humankind, which is why it’s been carried into our prevalent education today.

**Why Is It Important?**

Subcategories of math such as geometry and number theory help us develop a wide array of applications. For instance, geometry helps us develop and understand spatial awareness as well as relationships. This helps us create modern structures that hold up, and it also helps us find shapes for functional inventions. Without the fundamental knowledge of circles, polygons, and solid shapes, we wouldn’t have half the stuff we use on a daily basis!